Lexical Resource Semantics - User contributions [en]

SoSE15: Term paper project: Determiners

2015-08-30T14:19:23Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://www.youtube.com/watch?v=bFDFcYmQqek</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans.|Every woman that is red-haired dances with Hans. }

{  
|type="{}"}
''like2((ιx : iceman1(x)), Sven)''
{ The iceman likes Sven. }

</quiz>

=References=

<references />

:Kearns, Kate. 2000. ''Semantics.'' 42-47, 68-119. New York: Palgrave Macmillan.

:Levine, Robert D., Frank Richter & Manfred Sailer. 2015. ''Formal Semantics. An Empirically Grounded Approach.'' 203-248. Stanford: CSLI Publications.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-30T14:18:28Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://www.youtube.com/watch?v=bFDFcYmQqek</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''like2((ιx : iceman1(x)), Sven)''
{ The iceman likes Sven. }

</quiz>

=References=

<references />

:Kearns, Kate. 2000. ''Semantics.'' 42-47, 68-119. New York: Palgrave Macmillan.

:Levine, Robert D., Frank Richter & Manfred Sailer. 2015. ''Formal Semantics. An Empirically Grounded Approach.'' 203-248. Stanford: CSLI Publications.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-30T14:06:04Z

Monique Lanz: /* The difference between the existential quantifier and the definite article */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://www.youtube.com/watch?v=bFDFcYmQqek</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate. 2000. ''Semantics.'' 42-47, 68-119. New York: Palgrave Macmillan.

:Levine, Robert D., Frank Richter & Manfred Sailer. 2015. ''Formal Semantics. An Empirically Grounded Approach.'' 203-248. Stanford: CSLI Publications.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-30T13:57:10Z

Monique Lanz: /* The difference between the existential quantifier and the definite article */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://youtu.be/OH3fc4lwSQA</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate. 2000. ''Semantics.'' 42-47, 68-119. New York: Palgrave Macmillan.

:Levine, Robert D., Frank Richter & Manfred Sailer. 2015. ''Formal Semantics. An Empirically Grounded Approach.'' 203-248. Stanford: CSLI Publications.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-30T13:37:08Z

Monique Lanz: /* References */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://www.youtube.com/watch?v=I04rakQe4R8&feature=youtu.be</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate. 2000. ''Semantics.'' 42-47, 68-119. New York: Palgrave Macmillan.

:Levine, Robert D., Frank Richter & Manfred Sailer. 2015. ''Formal Semantics. An Empirically Grounded Approach.'' 203-248. Stanford: CSLI Publications.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-27T15:03:21Z

Monique Lanz:

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

<mediaplayer>https://www.youtube.com/watch?v=I04rakQe4R8&feature=youtu.be</mediaplayer>

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

:Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-23T17:59:55Z

Monique Lanz: /* Model from the scenario "Frozen" */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2013.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

:Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T21:57:39Z

Monique Lanz: /* References */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2014.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

:Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T21:57:25Z

Monique Lanz: /* References */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2014.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

::Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

::Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T21:57:14Z

Monique Lanz: /* References */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2014.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

:Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

:Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T18:23:36Z

Monique Lanz:

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2014.<ref>''Frozen Movie Official Disney Site.'' The Walt Disney Company. Web. 21 Aug. 2015. <[http://frozen.disney.com Frozen Disney]></ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

<references />

Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T18:11:34Z

Monique Lanz: /* Model from the scenario "Frozen" */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

Inspired by the Disney movie "Frozen" released in 2014.<ref>...</ref>

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:55:13Z

Monique Lanz: /* References */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate: Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:54:18Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false since there is only one reindeer in our model, namely Sven, that is male but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our model since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our model. Elsa is the only queen in our model and she likes Anna, thus every queen in our model likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:51:41Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for these paraphrases.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:50:45Z

Monique Lanz: /* The Difference between the existential quantifier and the definite article */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:49:32Z

Monique Lanz: /* Example for the universal quantifier */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example of the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:49:01Z

Monique Lanz: /* The difference between logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end of the page there will be a few exercises and further references to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:47:09Z

Monique Lanz: /* The difference between logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article should be treated like a noun phrase or name and not like a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:45:46Z

Monique Lanz: /* The difference between logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article has to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:44:18Z

Monique Lanz: /* The difference between logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope: the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article has rather to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:43:37Z

Monique Lanz: /* The difference between logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope – the number of possible individuals for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article has rather to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:41:15Z

Monique Lanz: /* The difference between the logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope – the number of possible individual for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article has rather to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T17:38:07Z

Monique Lanz: /* The difference between the logical quantifiers and definite descriptions */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope. If there were individuals with the features of the restrictor and not of the scope or the other way around than the formula would be false.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope – the number of possible individual for this formula are endless. But if there was no such individual at all, the formula would be false.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features. This individual can ideally be named and is a fixed part of the chosen universe. There is no other individual that fits the description, otherwise this formula would be false.

Thus the definite article has rather to be treated like a noun phrase or name than a quantifier. In the following, the differences will be shown giving examples and further explanations. At the end, there will be a few exercises to deepen the understanding of those differences.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:47:21Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:45:19Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans.|All red-haired women dance with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:44:15Z

Monique Lanz: /* Example for the universal quantifier */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:35:46Z

Monique Lanz:

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

=References=

Kearns, Kate. Semantics. New York: Palgrave Macmillan, 2000. 42-47, 68-119. Print.

Levine, Robert D., Frank Richter & Manfred Sailer: Formal Semantics. An Empirically Grounded Approach. Stanford: CSLI Publications, 2015. 203-248. Print.

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:18:43Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:18:26Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:18:00Z

Monique Lanz: /* Example for the universal quantifier */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col", style="width:80px; text-align:left;"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row", style="text-align:left;"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row", style="text-align:left;"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row", style="text-align:left;"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:10:29Z

Monique Lanz: /* Example for the universal quantifier */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | ✗
| style="width:100px; text-align:center;" | ✗
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:06:33Z

Monique Lanz: /* Example for the universal quantifier */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' Every royal is human.

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:05:20Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2((ιx : iceman1(x)), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:04:50Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''friend-of2(ιx : iceman1(x), Sven)''
{ The iceman is a friend of Sven. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T16:01:24Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

{  
|type="{}"}
''∀x ((woman1(x) &and; red-haired1(x)) : dance-with2(x, Hans))''
{ Every red-haired woman dances with Hans. }

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:56:16Z

Monique Lanz: /* Model from the scenario "Frozen" */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:54:33Z

Monique Lanz:

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''
* ''male1 = {<x> | x is male} = {<Kristoff>, <Sven>, <Olaf>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))''
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

</quiz>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:53:57Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''
* ''male1 = {<x> | x is male} = {<Kristoff>, <Sven>, <Olaf>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

'''Task 3:''' Write down the correct sentences for the logical formulae in the boxes.

<quiz display=simple>

{  
|type="{}"}
''∃x (prince1(x) : love2(Elsa, x))
{ Elsa loves a prince.|Elsa loves some prince.|Some prince is love by Elsa.|A prince is loved by Elsa. }

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:46:35Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''
* ''male1 = {<x> | x is male} = {<Kristoff>, <Sven>, <Olaf>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

# Some reindeer is male and royal.
# The snowman sings.
# Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
# ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.
# ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.
# ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:43:16Z

Monique Lanz: /* Exercises */

{{MaterialUnderConstruction}}

= Short description of the project =

* Difference between "every", "some" and the definite article;
* Video about how to differentiate "some" and the definite article;
* Three exercises for each operator

= The difference between the logical quantifiers and definite descriptions =

The universal and existential quantifiers have to be interpreted differently than the definite article.

The universal quantifier (every, all → ∀) indicates that '''every single individual''' in a model that has the features of the restrictor, also has the features of the scope.

The existential quantifier (some, a → ∃) states that there is '''at least one individual or more''' in a model that has both the features of the restrictor and the scope.

The definite article (the → ⍳) states that there is '''absolutely one individual and no more or less''' in a model that fits exactly the described features.

==Model from the scenario "Frozen"==

'''Individuals:'''
* Elsa, the Snow Queen of Arendelle
* Anna, the Princess of Arendelle
* Kristoff, an iceman
* Sven, a reindeer
* Olaf, a snowman
* Hans, the Prince of the Southern Isles

'''Properties:'''
* ''royal1 = {<x> | x is royal} = {<Elsa>, <Anna>, <Hans>}''
* ''prince1 = {<x> | x is a prince} = {<Hans>}''
* ''human1 = {<x> | x is human} = {<Elsa>, <Anna>, <Kristoff>, <Hans>}''
* ''male1 = {<x> | x is male} = {<Kristoff>, <Sven>, <Olaf>, <Hans>}''

'''Relations:'''
* ''sibling2 = {<x, y> | x and y are siblings} = {<Elsa, Anna>, <Anna, Elsa>}''
* ''get-engaged2 = {<x, y> | x and y get engaged} = {<Anna, Hans>, <Hans, Anna>}''

==Example for the universal quantifier==

'''Example:''' ''Every royal is human.''

Here, ''every'' is our determiner, ''royal'' is our restrictor and ''human'' is the scope. We will choose ''x'' as our variable. Therefore, the paraphrase would look like that:

For every ''x'' such that ''x'' is royal ''x'' is human.

The overall formula for this expression would look like that:

''∀ x (royal1(x) : human1(x))''

Now we will interpret our formula, so we have to check the truth values for all our individuals in the model:

{| class="wikitable"
|-
! scope="col"| g(x)
! scope="col"| royal1(x)
! scope="col"| human1(x)
|-
! scope="row"| Elsa
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Anna
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Kristoff
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | ✓
|-
! scope="row"| Sven
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Olaf
| style="width:100px; text-align:center;" | '''x'''
| style="width:100px; text-align:center;" | '''x'''
|-
! scope="row"| Hans
| style="width:100px; text-align:center;" | ✓
| style="width:100px; text-align:center;" | ✓
|}

Since every character that is royal – Elsa, Anna and Hans – is also human the overall formula is true. If not every royal was human, the formula would be false.

==The Difference between the existential quantifier and the definite article==

Here is a video about how to differentiate the existential quantifier and the definite article:

=Exercises=

After having read this page and watched the video, work on the following tasks:

'''Task 1:''' Identify the correct formula for this paraphrase.

<quiz display=simple>

{Every reindeer which eats carrots sings.
|type="()"}
- ''∀x (reindeer1(x) : (eat-carrots1(x) &and; sing1(x)))''
- ''∀x (reindeer1(x) &and; eat-carrots1(x) &and; sing1(x))''
+ ''∀x ((reindeer1(x) &and; eat-carrots1(x)) : sing1(x))''

{A snowman loves summer.
|type="()"}
- ''love2((∃x snowman1(x)), summer)''
+ ''∃x (snowman1(x) : love2(x, summer))''
- ''∃x (snowman1(x) : love2(summer))''

{The iceman loves a princess.
|type="()"}
- ''∃x (princess1(x) : love2(iceman, x)''
+ ''∃x (princess1(x) : love2((ιy : iceman1(y)), x))''
- ''ιx : (iceman1(x) &and; love2(x, ∃y (princess1(y))))''

</quiz>

'''Task 2:''' Write down the logical formulae for the following sentences. Explain further, if and why the formulae are true or false.

(a) Some reindeer is male and royal.

(b) The snowman sings.

(c) Every queen likes Anna.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
(a) ''∃x (reindeer1(x) : (male1(x) &and; royal1(x)))'' → This formula is false, since there is only one reindeer in our scenario, namely Sven, that is male, but not royal.

(b) ''sing1(ιx : snowman1(x))'' → This formula is true in our scenario, since there is exactly one snowman, namely Olaf, that sings.

(c) ''∀x (queen1(x) : like2(x, Anna))'' → This formula is also true in our scenario. Elsa is the only queen in our scenario and she likes Anna, thus every queen in our scenario likes Anna.</div>
</div>

= Participants =

* [[User:Monique_Lanz|Monique Lanz]]

<hr>
Back to the [[Semantics_2,_SoSe_2015|Semantics 2]] page.

SoSE15: Term paper project: Determiners

2015-08-21T15:38:49Z

Monique Lanz: /* Exercises */

SoSE15: Term paper project: Determiners

2015-08-21T15:23:04Z

Monique Lanz: /* Exercises */

SoSE15: Term paper project: Determiners

2015-08-21T15:22:22Z

Monique Lanz: /* Exercises */

SoSE15: Term paper project: Determiners

2015-08-21T14:48:56Z

Monique Lanz: /* Exercises */

SoSE15: Term paper project: Determiners

2015-08-21T14:46:49Z

Monique Lanz: /* Exercises */

SoSE15: Term paper project: Determiners

2015-08-21T14:45:40Z

Monique Lanz: /* Exercises */

Exercise Quantifiers

2015-08-21T14:39:06Z

Monique Lanz: Undo revision 6176 by Monique Lanz (talk)

= Introduction to the topic =

== Input ==

Watch the following video on logical determiners:

<mediaplayer> http://youtu.be/5PRL23XcaFY</mediaplayer>


== Exercises ==

After having watched the video, work on the following tasks.

'''Task 1''' Identify the determiners in the following sentence.

(a) Juliet talked to some stranger at the party.

(b) Every Capulet is an enemy to some Montague.

(c) Many people in Verona are not happy about the Capulet-Montague feud.

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
(a) ''some''

(b) ''every'', ''some''

(c) ''many''</div>
</div>

'''Task 2''' Identify the formula that corresponds to the translation of the sentence.

<quiz display=simple>

{''Some Montague who was at the party fell in love with Juliet.''
|type="()"}
- ∃''x'' ('''montague1'''(''x'') : ('''at-party1'''(''x'') &and; '''fall-in-love-with2'''(''x'','''juliet''')))
|| In restricted quantifier notation, the "complete" semantic representation of the noun phrase (NP) appears in the restrictor (-> square brackets).
+ ∃''x'' (('''montague1'''(''x'') &and; '''at-party1'''(''x'')) : '''fall-in-love-with2'''(''x'','''juliet'''))
- ∃''x'' ('''montague1'''(''x'') : ('''at-party1'''(''x'') &and; '''fall-in-love-with2'''(''x'','''juliet'''))
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor.
- ∃''x'' (('''montague1'''(''x'') &and; '''fall-in-love-with2'''(''x'','''juliet''')) : '''at-party1'''(''x''))
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor, that of the VP in the scope.

</quiz>

'''Task 3''' The sentence: ''Some Tybalt loved some Montague.'' is translated into the formula ∃ y ('''montague1'''(''y'') : '''love2'''('''tybalt''',''y'').

<quiz display=simple>
{Mark all the cells in the table that stand for a true statement.
|type="[]"}
| '''montague1'''(''y'') zwisch| '''love2'''('''tybalt''',''y'')zwisch
+- ''Romeo''
+- ''Mercutio''
-- ''Juliet''
-- ''Tybalt''
-- ''Laurence''
-- ''Paris''
</quiz>

Given this table, is the overall formula true or false? (Give a reason for your answer.)
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
The formula is false, because there is no individual in our model for which both the restrictor and the scope are true.
</div>
</div>

'''Task 4''' Variable assignment function 
Start with the following variable assigment function ''g'':
''g(u) = Romeo, g(v) = Juliet, g(w) = Romeo, g(x) = Laurence, g(y) = Mercutio, g(z) = Juliet''

Provide the changed variable assignment function ''g''[''v/Paris''].

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">
''g''[''v/Paris'']''(u)'' = ''g(u)'' = ''Romeo'' ''g''[''v/Paris'']''(v)'' = ''Paris'' ''g''[''v/Paris'']''(w)'' = ''g(w)'' = ''Romeo'' ''g''[''v/Paris'']''(x)'' = ''g(x)'' = ''Laurence'' ''g''[''v/Paris'']''(y)'' = ''g(y)'' = ''Mercutio'' ''g''[''v/Paris'']''(z)'' = ''g(z)'' = ''Juliet''
</div>
</div>

= More exercises on quantifiers =

{{CreatedByStudents1213}} ''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]

==Restricted Quantifiers==

Find the right formula for the sentence below.

<quiz display=simple>

{Some students who heard the concert were interviewed by Holmes.
|type="()"}
- ∃''x'' ('''student'''(''x'') : ('''hear'''(''x'','''concert''') &and; '''interview'''('''holmes''',''x'')))
|| In restricted quantifier notation, the "complete" semantic representation of the noun phrase (NP) appears in the restrictor (-> square brackets).
+ ∃''x'' (('''student'''(''x'') &and; '''hear'''(''x'','''concert''')) : '''interview'''('''holmes''',''x''))
- ∃''x'' ('''student'''(''x'') &and; '''hear'''(''x'','''concert''') &and; '''interview'''('''holmes''',''x''))
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor.
- ∃''x'' (('''student'''(''x'') &and; '''interview'''('''holmes''',''x'')) : '''hear'''(''x'','''concert'''))
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor, that of the VP in the scope.

</quiz>

==Different types of Quantifiers==

Which type(s) of quantifiers does the sentence below have?

<quiz display=simple>
{Ramon signs every sculpture he makes.
|type="[]"}
- existential
|| Existential quantifiers are used for sentences that represent something that exists.
|| Of course, you could argue that there is a person x such that x is called Ramon and x makes (and then signs) sculptures - but this is not what we were going for.
+ universal

{Some playwright also wrote famous sonnets.
|type="[]"}
+ existential
- universal

{Shakespeare wrote for King James.
|type="[]"}
- existential
- universal
||There is no explicit quantifier in the sentence. Both ''Shakespeare'' and ''King James'' are proper names.

{All pupils read some plays by Shakespeare in school.
|type="[]"}
+ existential
||The sentence contains the existential quantifier ''some plays by Shakespeare''.
+ universal
||The sentence contains the universal quantifier ''all pupiles''.

</quiz>

'''2.''' Write down the logical formula(e) that correspond to the sentence ''Ramon signs every sculpture he makes.''

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Check your solutions here
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Sentence: ''Ramon signs every sculpture he makes.''

'''Universal Quantifier'''

∀''x'' (('''sculpture'''(''x'') &and; '''make'''('''ramon''', ''x'')) ⊃ '''sign'''('''ramon''', ''x''))

Paraphrse: ''"For every thing ''x'', if ''x'' is a sculpture and ''x'' is made by Ramon then ''x'' is signed by Ramon."''

We use the name constant '''ramon''' for both the name (''Ramon'') and the personal pronoun ''he'' that referes to Ramon.

'''In restricted quantifier notation'''

∀''x'' (('''sculpture'''(''x'') &and; '''make'''('''ramon''', ''x'')) : '''sign'''('''ramon''', ''x''))

Here, the N' is "sculpture he makes" and therefore its translation appears in the part before the colon.

</div>
</div>

==Scopal Ambiguity==

'''1.''' In which way is the following sentence ambiguous? 
 
'''Everyone loves someone.''' 
 
The following pictures may help you:

<gallery>
File:Everyone_loves_someone_1.jpeg|One Reading
File:Everyone_loves_someone_2.jpeg|Another Reading
</gallery>

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">In this sentence, the scopal ambiguity is created by the two quantifiers ''everyone'' and ''someone''.

When looking at the two pictures that try to help you, you can see two possible readings:

1. For every person there is, there is at least one other person who loves him / her.

2. There is one person that is loved by everyone else.</div>
</div>

 

'''2.''' Write down the two possible logical forms.
 

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here:
<div class="mw-collapsible-content">1. For every person there is at least one person who loves him / her:


∀''x'' ('''person'''(''x'') ⊃ ∃''y'' ('''person'''(''y'') &and; '''love'''(''x'',''y'')

Or, in restricted-quantifier notation: ∀''x'' ('''person'''(''x'') : ∃''y'' ('''person'''(''y'') : '''love'''(''x'',''y'')

2. There is one person that is loved by everyone:


∃''y'' ('''person'''(''y'') ⊃ ∀''x'' ('''person'''(''x'') &and; '''love'''(''x'',''y'')

Or, in restricted-quantifier notation: ∀''x'' ('''person'''(''x'') : ∃''y'' ('''person'''(''y'') : '''love'''(''x'',''y'')

</div>
</div>

==== Navigation ====

* [[Exercise-ch1|To the exercise page for chapter 1]]
* [[Textbook-chapters|To the material by chapters overview]]
* [[FSEGA|To the main page of the textbook]]

SoSE15: Term paper project: Determiners

2015-08-21T14:30:24Z

Monique Lanz: /* Exercises */

Exercise Quantifiers

2015-08-21T14:26:45Z

Monique Lanz: /* Exercises */