Exercise Quantifiers: Difference between revisions

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= Introduction to the topic =
== Input ==
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 logica 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'') &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>
= More exercises on quantifiers =
{{CreatedByStudents1213}}<br />''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]
{{CreatedByStudents1213}}<br />''Involved participants: [[User:AnKa| AnKa]], [[User:Katharina| Katharina]], [[User:Lara| Lara]]


==Restricted Quantifiers==
==Restricted Quantifiers==
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{Some students who heard the concert were interviewed by Holmes.
{Some students who heard the concert were interviewed by Holmes.
|type="()"}
|type="()"}
- '''Some''' ''x'' ('''student'''(''x'') : '''hear'''(''x'','''concert''') &and; '''interview'''('''holmes''',''x'')
- &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).
|| In restricted quantifier notation, the "complete" semantic representation of the noun phrase (NP) appears in the restrictor (-> square brackets).
+ [Some x: STUDENT (x) & HEAR (x, c)] INTERVIEW (h, x)
+ &exist;''x'' (('''student'''(''x'') &and; '''hear'''(''x'','''concert''')) : '''interview'''('''holmes''',''x''))
- Some x: STUDENT (x) & HEAR (x, c) & INTERVIEW (h, 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.
- [Some x: STUDENT (x) & INTERVIEW (h, x)] HEAR (x,c)
|| In restricted quantifier notation, the semantic representation of the noun phrase (NP) appears in the restrictor.
|| 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>
</quiz>
Line 22: Line 106:
==Different types of Quantifiers==
==Different types of Quantifiers==


Which type(s) of Quantifiers does the sentence below have?
Which type(s) of quantifiers does the sentence below have?


<quiz display=simple>
<quiz display=simple>
Find the right formula for the sentence below.
{Ramon signs every sculpture he makes.
{Ramon signs every sculpture he makes.
|type="[]"}
|type="[]"}
- existential
- existential
|| Existential quantifiers are used for sentences that represent something that exists. <br />
|| 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. <br />
|| 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.
|| Maybe you want to check the possible answers once more.<br />
+ universal
+ universal
+ restricted
 
{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>
</quiz>


'''2.''' Write down the corresponding logical formula(s). [[Solutions_Quantifiers2_2|Check your solutions here]].
'''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==
==Scopal Ambiguity==
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</gallery>
</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:


[[Solutions_Quantifiers3_1|Check your solutions here]]
<!-- ∃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'')


'''2.''' Write down the two possible logical forms. [[Solutions_Quantifiers3_2|Check your solutions here]]


</div>
</div>


==== Navigation ====
==== Navigation ====

Latest revision as of 19:23, 30 May 2021

Introduction to the topic

Input

Watch the following video on logical determiners:

Exercises

After having watched the video, work on the following tasks.

Task 1 Identify the logica 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.

Check your solutions here:

(a) some

(b) every, some

(c) many


Task 2 Identify the formula that corresponds to the translation of the sentence.

Some Montague who was at the party fell in love with Juliet.

x (montague1(x) : (at-party1(x) ∧ fall-in-love-with2(x,juliet)))
x ((montague1(x) ∧ at-party1(x)) : fall-in-love-with2(x,juliet))
x ((montague1(x) ∧ fall-in-love-with2(x,juliet)) : at-party1(x))


Task 3 The sentence: Some Tybalt loved some Montague. is translated into the formula
∃ y (montague1(y) : love2(tybalt,y).

Mark all the cells in the table that stand for a true statement.

montague1(y) zwisch love2(tybalt,y)zwisch
Romeo
Mercutio
Juliet
Tybalt
Laurence
Paris


Given this table, is the overall formula true or false? (Give a reason for your answer.)

Check your solutions here:

The formula is false, because there is no individual in our model for which both the restrictor and the scope are true.


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].

Check your solutions here:

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

More exercises on quantifiers

The following material is an adapted form of material created by student participants of the project e-Learning Resources for Semantics (e-LRS).
Involved participants: AnKa, Katharina, Lara

Restricted Quantifiers

Find the right formula for the sentence below.

Some students who heard the concert were interviewed by Holmes.

x (student(x) : (hear(x,concert) ∧ interview(holmes,x)))
x ((student(x) ∧ hear(x,concert)) : interview(holmes,x))
x (student(x) ∧ hear(x,concert) ∧ interview(holmes,x))
x ((student(x) ∧ interview(holmes,x)) : hear(x,concert))


Different types of Quantifiers

Which type(s) of quantifiers does the sentence below have?

1 Ramon signs every sculpture he makes.

existential
universal

2 Some playwright also wrote famous sonnets.

existential
universal

3 Shakespeare wrote for King James.

existential
universal

4 All pupils read some plays by Shakespeare in school.

existential
universal


2. Write down the logical formula(e) that correspond to the sentence Ramon signs every sculpture he makes.

Check your solutions here

Sentence: Ramon signs every sculpture he makes.

Universal Quantifier

x ((sculpture(x) ∧ 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) ∧ make(ramon, x)) : sign(ramon, x))

Here, the N' is "sculpture he makes" and therefore its translation appears in the part before the colon.

Scopal Ambiguity

1. In which way is the following sentence ambiguous?

Everyone loves someone.

The following pictures may help you:

Check your solutions here:

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.



2. Write down the two possible logical forms.

Check your solutions here:

1. For every person there is at least one person who loves him / her:

x (person(x) ⊃ ∃y (person(y) ∧ 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) ∧ love(x,y)

Or, in restricted-quantifier notation: ∀x (person(x) : ∃y (person(y) : love(x,y)


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