View source for Semantics 1, SoSe 2017 (Sailer)

Olat course: https://olat-ce.server.uni-frankfurt.de/olat/auth/RepositoryEntry/4778393609


= Assignment sheets =

Assignment logic (due June 13): [http://user.uni-frankfurt.de/~sailer/teaching/assignment-logic-SoSe17.pdf assignment-logic-SoSe17.pdf]

= Material for week 8 (6.6.2017) =

== Input: Truth tables ==

Truth tables are also useful to compute the truth value of complex formulae.
This is shown in the following podcast, created by [[User:Lisa|Lisa Günthner]].

<embedvideo service="youtube" dimensions="400">http://www.youtube.com/watch?v=ZWdltj5Mqdc</embedvideo>


== Input: Determiners/quantifiers ==

Watch the following video on logical determiners:

<embedvideo service="youtube" dimensions="400">http://youtu.be/5PRL23XcaFY</embedvideo>
<!-- old video with less optimal audio: http://youtu.be/b0iLejXP9C8 -->

== 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="()"}
- &exist;''x'' ('''montague<sub>1</sub>'''(''x'') : ('''at-party<sub>1</sub>'''(''x'') &and; '''fall-in-love-with<sub>2</sub>'''(''x'','''juliet''')))
|| In restricted quantifier notation, the "complete" semantic representation of the noun phrase (NP) appears in the restrictor (-> square brackets).
+ &exist;''x'' (('''montague<sub>1</sub>'''(''x'') &and; '''at-party<sub>1</sub>'''(''x'')) : '''fall-in-love-with<sub>2</sub>'''(''x'','''juliet'''))
- &exist;''x'' ('''montague<sub>1</sub>'''(''x'') : ('''at-party<sub>1</sub>'''(''x'') &and; '''fall-in-love-with<sub>2</sub>'''(''x'','''juliet'''))
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor.
- &exist;''x'' (('''montague<sub>1</sub>'''(''x'') &and; '''fall-in-love-with<sub>2</sub>'''(''x'','''juliet''')) : '''at-party<sub>1</sub>'''(''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<br>&exist; y ('''montague<sub>1</sub>'''(''y'') : '''love<sub>2</sub>'''('''tybalt''',''y'').

<quiz display=simple>
{Mark all the cells in the table that stand for a true statement.
|type="[]"}
| '''montague<sub>1</sub>'''(''y'') <span style="color:white">zwisch</span>| '''love<sub>2</sub>'''('''tybalt''',''y'')<span style="color:white">zwisch</span>
+- ''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<br>
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''<br>''g''[''v/Paris'']''(v)'' = ''Paris''<br>''g''[''v/Paris'']''(w)'' = ''g(w)'' = ''Romeo''<br>''g''[''v/Paris'']''(x)'' = ''g(x)'' = ''Laurence''<br>''g''[''v/Paris'']''(y)'' = ''g(y)'' = ''Mercutio''<br>''g''[''v/Paris'']''(z)'' = ''g(z)'' = ''Juliet''
</div>
</div>



==Restricted Quantifiers==

{{CreatedByStudents1213}}<br />''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]

Find the right formula for the sentence below.

<quiz display=simple>

{Some students who heard the concert were interviewed by Holmes.
|type="()"}
- &exist;''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).
+ &exist;''x'' (('''student'''(''x'') &and; '''hear'''(''x'','''concert''')) : '''interview'''('''holmes''',''x''))
- &exist;''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.
- &exist;''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==

{{CreatedByStudents1213}}<br />''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]

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.''

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your solutions here
<div class="mw-collapsible-content">
Sentence: ''Ramon signs every sculpture he makes.''

'''Universal Quantifier'''

&forall;''x'' (('''sculpture'''(''x'') &and; '''make'''('''ramon''', ''x'')) &sup;  '''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'''

&forall;''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==

{{CreatedByStudents1213}}<br />''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]

'''1.''' In which way is the following sentence ambiguous? <br />
<br/ >
'''Everyone loves someone.''' <br/ >
<br/ >
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>

<br/ >


'''2.''' Write down the two possible logical forms.
<br/ >

<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) &sup; ∃y (PERSON (y) & LOVE (x, y)) -->
&forall;''x'' ('''person'''(''x'') &sup; &exist;''y'' ('''person'''(''y'') &and; '''love'''(''x'',''y'')

Or, in restricted-quantifier notation: &forall;''x'' ('''person'''(''x'') : &exist;''y'' ('''person'''(''y'') : '''love'''(''x'',''y'')

2. There is one person that is loved by everyone:

<!-- ∃x (PERSON (x) → ∀y (PERSON (y) & LOVE (y, x)) -->
&exist;''y'' ('''person'''(''y'') &sup; &forall;''x'' ('''person'''(''x'') &and; '''love'''(''x'',''y'')

Or, in restricted-quantifier notation: &forall;''x'' ('''person'''(''x'') : &exist;''y'' ('''person'''(''y'') : '''love'''(''x'',''y'')


</div>
</div>

= Material for week 6 (23.5.2017) =


== Video == 


The following video presents the step-by-step computation of the truth value of two formulae with connectives. 
The example uses a model based on Shakespeare's play ''Macbeth''.
The two formulae are:
* '''&not; king(lady-macbeth)'''
* '''king(duncan) &or; king(lady-macbeth)'''

<embedvideo service="youtube" dimensions="400">http://youtu.be/ABXPMzHFYxU</embedvideo>
<!-- https://www.youtube.com/watch?v=K14D7VllA8M -->

The next video shows how the truth value of a more complex formula can be computed. The example contains two connectives:

'''kill(malcom,lady-macbeth) &or; &not;thane(macbeth)'''

The video shows two different methods: top down and bottom up.

<embedvideo service="youtube" dimensions="400">http://youtu.be/C1rjU104R54</embedvideo>

= Material for week 5 (16.5.2017) =

== Why it is too difficult to go directly from language to the world ==

The following architecture is extremely useful when talking about semantics:

# A natural language expressions: ''Daenerys loves Drogo.''
# ... is mapped to some expression from a formal language (here: predicate logic): '''love2'''('''daenerys''','''drogo''')
# This logical expression is then interpreted with respect to our scenario/world: The formula '''love2'''('''daenerys''','''drogo''') is true, because, in our scenario, Daenerys loves Drogo.


The following properties of natural language make it useful to use the intermediate step of a logical language:
# The same expression can have different meanings (ambiguity).
# Different expressions can have the same meaning (synonyms, paraphrases)

Find examples for the above-mentioned properties (ambiguity, synonymy, paraphrases).

<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">
1. one form, two meaingns: Ambiguity: (see earlier in this meeting and the slides of last week's meeting)

1.a Ambiguous words: ''date'' (fruit or point in time); ''bank'' (financial institute or bank of a river)

1.b. Ambiguous sentences: ''Sycorax and Prospero were stranded on the island with their children.''

2. two forms, one meaning:

2.a Synonymous words: ''couch'' - ''sofa''; ''instant'' - ''moment''

2.b Paraphrases: 
* active-passive pairs: ''Prospero set Ariel free.'' - ''Ariel was set free by Prospero.''
* cleft sentences: ''Prospero set Ariel free.'' - ''It was Prospero who set Ariel free.''
* different ways to express a possessor: ''Sycorax was the first inhabitant of the island.'' and ''Sycorax was the island's first inhabitant.''
</div>
</div>

== Towards a formal model ==


=== First steps ===

{{CreatedByStudents1213}} Involved participants: [[User:Lisa| Lisa]], [[User:Marthe| Marthe]], [[User:Elisabeth.krall| Elisabeth]], [[User:IsaB|Isabelle]].

You can think of building a formal model like being the producer of a film who has to collect everything that should be included in the film.

Here is a very simple story from which we can derive an example model.

<embedvideo service="youtube" dimensions="400">http://youtu.be/4a3mXelw7H4</embedvideo>

<quiz display="simple">
{Mark those elements that we need in a model.
|type="[]"}
+ relations
|| Yes. We use relations to express what is true between various individuals. For example the relation ''grandmother-of''.
+ individuals
||Yes. In the video, we have three individuals, ''Red Riding Hood'', ''Grandmother'', and ''Wolf''.
- nouns
|| nouns are a syntactic category and as such part of the language, not of the "world".
+ properties
||Yes. The video mentions some properties such as having a red hood.
- relatives
|| (this is a nonsense alternative)
</quiz>


<quiz display="simple">
{What is the status of the following entities in the video on Little Red Riding Hood?
|type="[]"}
|individual|property|relation
+-- ''Red Riding Hood''
-+- ''lives in the forest''
+-- ''Grandmother''
--+ ''is afternoon snack for''
-+- ''has a red hood''
-+- ''has a big mouth''
--+ ''is grandmother of''
</quiz>

== The universe and name symbols ==

'''Task:''' Assume '''three''' individuals from our ''Game of Thrones''-scenario.

Formally we collect the individuals of our model in a so-called ''universe'' (''U''). For the fairy-tale story, we can define the universe as follows:

''U'' = {''Redridinghood'', ''Grandmother'', ''Wolf''}

Do a similar definition for your own scenario.


We can introduce ''name symbols'' for some of our individuals. For example: '''redridinghood''', '''grandmother''', '''wolf'''.

We link the name symbols to the individuals in our modal. To do this, we introduce the '''interpretation function'''. We will written the interpretation function as as ''I''.<br />This function can be defined in the following way:

''I''('''grandmother''') = '''Grandmother'''<br />''I''('''redridinghood''') = ''Red Riding Hood''<br />''I''('''wolf''') = ''Wolf''<br />

=== Relations and predicate symbols ===

In the fairy-tale scenario we express a relation between Little Red Riding Hood and the Wolf, namely that Little Red Riding Hood is the Wolf's afternoon snack. To formalize this, we collect all '''pairs''' of individuals which are such that the first element in the pair is the afternoon snack of the second. '''Note:''' A ''pair'' is written in between pointy brackets.


Formally we can write this down as follows:<br />
{< ''x'', ''y'' > | ''x'' is ''y'' 's afternoon snack} = { < ''Redridinghood'', ''Wolf'' >, < ''Grandmother'', ''Redriding hood'' >.}

We can also assume empty relations:

{< ''x'', ''y'' > | ''x'' is ''y'' 's father } = { }


Note, if a relation works both ways, two pairs must be added:

{< ''x'', ''y'' > | ''x'' talks with ''y''} = { <''Redridinghood'', ''Wolf'' >, < ''Wolf'', ''Redridinghood'' >}


<!-- '''Task:''' Using your ''Game of Thrones''-universe from above, introduce one binary and one ternary relation. -->

Just like with names, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols '''afternoon-snack-of_2''' and '''father-of_2''', and '''talks-with_2'''. (The number 2 indicates that the interpretation consists of pairs, not just of single individual) There interpretation is defined as follows:<br>

''I''('''afternoon-snack-of_2''') = { < ''x'', ''y'' > | ''x'' is ''y'' 's afternoon snack } = {  <''Redridinghood'', ''Wolf'' >, <''Grandmother'', ''Wolf'' > }.

'''Task:''' For each of your properties, invent an appropriate ''predicate symbol''. Define its interpretation.

== Properties and predicate symbols ==

A property is a specification that either holds of an individual or not. In the little story, having a big mouth is a property of the Wolf, but of noone else in the story. Being female holds of both Little Red Riding Hood and the Grandmother.

We can think of a property as the set of individuals that have this property. Under this view, the property of being female would be the set {''Redridinghood'', ''Grandmother''}.

Alternatively it is convenient to think of properties as '''1-place relations'''. Under this view, the property of being female would be a set of lists of length 1. This is what the property of being female then looks like:  { <''Redridinghood''>, <''Grandmother''> }

'''Task:''' Using your ''Game of Thrones'' universe, define ''two'' properties in the format of 1-place relations.

Just like before, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols '''female_1''' and '''has-big-mouth_1'''. There interpretation is defined as follows:<br>

''I''('''female_1''') = { < ''x'' > | ''x'' is female } = {  <''Redridinghood''>, <''Grandmother''> }.

'''Task:''' For each of your properties, invent an appropriate ''predicate symbol''. Define its interpretation.

== Computing the truth value of atomic formulae ==


The following video presents the step-by-step computation of the truth value of two atomic formulae. 
The example uses a model based on Shakespeare's play ''Macbeth''.
The two formulae are:
* '''kill(macbeth,duncan)'''
* '''kill(lady-macbeth,macbeth)'''

<embedvideo service="youtube" dimensions="400">http://youtu.be/8HGCB9urmbg</embedvideo>

<hr>
Back to the [[Semantics_1,_WiSe_2016/17_(Sailer) | course page]].