Quantifiers & Cognition · Schedule 1.Monday, November 22, 13-15, room F347 2.Tuesday, November 23, 13-15, room F363 3.Wednesday, November 24,10-12 & 13-15, room F355 4.Thursday,

Quantifiers & Cognition

Jakub Szymanik

Quantifiers

Logic Psycholinguisticscomputability

Quantifiers

Logic

Psycholinguisticscomputability

Quantifiers

Logic Psycholinguistics

computability

Quantifiers

Logic Psycholinguisticscomputability

Plan

Lecture 1 Quantifiers and cognitive strategiesLecture 2 Quantifiers and approximationLecture 3 Quantifiers and countingLecture 4 Quantifiers and monotonicityLecture 5 Quantifiers and computational complexity

Schedule

1. Monday, November 22, 13-15, room F3472. Tuesday, November 23, 13-15, room F3633. Wednesday, November 24, 10-12 & 13-15, room F3554. Thursday, November 25, 16-18, room D700.5. Friday, November 26, 10-12, room F263.

A joint meeting with Logic, Language, and Mind Seminar.

Practicalities

I Website:http://www.jakubszymanik.com/quacog.htm

I E-mail: [email protected] Office hours: Mon., Tue. and Thur. 10-12.I Assignment for Wednesday: suggest an experiment.I Final assignment: (experimental) research proposal.

http://www.jakubszymanik.com/quacog.htm

mailto:[email protected]

Quantifiers are usefulEveryone knows everyone here.

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Lecture 1:Quantifiers and Cognitive Strategies

Outline

Understanding quantifiersQuantifier sentencesQuantifier meaningForm and meaningProcedural perspective

Generalized Quantifier Theory

Definability of GQs

Outline



Definability of GQs

Outline



Definability of GQs

1. All poets have low self-esteem.2. Some dean danced nude on the table.3. At least 3 grad students prepared presentations.4. An even number of the students saw a ghost.5. Most of the students think they are smart.6. Less than half of the students received good marks.7. Many of the soldiers have not eaten for several days.8. A few of the conservatives complained about taxes.

Determiners

DefinitionExpressions that appear to be descriptions of quantity.

ExampleAll, not quite all, nearly all, an awful lot, a lot, a comfortablemajority, most, many, more than n, less than n, quite a few,quite a lot, several, not a lot, not many, only a few, few, a few,hardly any.

Outline



Definability of GQs

Main question

QuestionHow people understand quantifiers?

Aspects of meaning

MEANINGcomprehension

reasoning

use

VERIFICATION

. . .

How are people doing it?

I They apply some strategies/procedures/algorithms.I Those depend on:

I visual clues;I level of precision subjects want to achieve;I quantifiers;I . . .

Let’s see a few examples

How are people doing it?

I They apply some strategies/procedures/algorithms.I Those depend on:

I visual clues;I level of precision subjects want to achieve;I quantifiers;I . . .

Let’s see a few examples

Reasoning

1. At the party every girl was paired with a boy.2. Peter came alone.3. There were more boys than girls at the party.

1. Most villagers are communists.2. Most townsmen are capitalists.3. All communists and all capitalists hate each other.4. Most villagers and most townsmen hate each other.

Reasoning

1. At the party every girl was paired with a boy.2. Peter came alone.3. There were more boys than girls at the party.

1. Most villagers are communists.2. Most townsmen are capitalists.3. All communists and all capitalists hate each other.4. Most villagers and most townsmen hate each other.

Counting

More than half of the cars are yellow.

Approximating 1

Approximating 2

Outline



Definability of GQs

‘Most’ vs. ‘More than half’

I Different distribution in corpus:I ‘Most’ is associated with higher proportions.I ‘Most’ is generic.I ‘More than half’ has a ‘survey results’ interpretation.

I They trigger different verification strategies.I They differ pragmatically.

Hackl, On the grammar and processing of proportional quantifiers, NaturalLanguage Semantics, 2009

Solt, On orderings and quantification: the case of most and more than half,manuscript, 2010

‘Most’ vs. ‘More than half’

I Different distribution in corpus:I ‘Most’ is associated with higher proportions.I ‘Most’ is generic.I ‘More than half’ has a ‘survey results’ interpretation.

I They trigger different verification strategies.I They differ pragmatically.

Hackl, On the grammar and processing of proportional quantifiers, NaturalLanguage Semantics, 2009

Solt, On orderings and quantification: the case of most and more than half,manuscript, 2010

Hackl’s self-paced counting

Results reported: equivalence

Let’s look into the verification process

Results: difference on 4 screens

Discussion

I Speakers treat expressions as equivalent;I but they use different verification strategies.

Outline



Definability of GQs

Computability and cognition

I Cognitive science is concerned with cognitive performance.

I A cognitive task is a computational task.

Computability and cognition

I Cognitive science is concerned with cognitive performance.I A cognitive task is a computational task.

Marr’s 3 levels of explanation

1. computational level:I problems that a cognitive ability has to overcome

2. algorithmic level:I the algorithms that may be used to achieve a solution

3. implementation level:I how this is actually done in neural activity

Marr, Vision: A Computational Investigation into the Human Representation andProcessing Visual Information, 1983











Algorithmic level

An algorithm is likely to be understood more readily byunderstanding the nature of the problem being solved than byexamining the mechanism (and the hardware) in which it isembodied. (Marr, 1981)

The aim of a computational theory is to single out a function thatmodels the cognitive phenomenon to be studied. Within theframework of a computational approach, such a function must beeffectively computable. However, at the level of the computationaltheory, no assumption is made about the nature of the algorithmsand their implementation. (Frixone, 2001)

Algorithmic level

An algorithm is likely to be understood more readily byunderstanding the nature of the problem being solved than byexamining the mechanism (and the hardware) in which it isembodied. (Marr, 1981)

The aim of a computational theory is to single out a function thatmodels the cognitive phenomenon to be studied. Within theframework of a computational approach, such a function must beeffectively computable. However, at the level of the computationaltheory, no assumption is made about the nature of the algorithmsand their implementation. (Frixone, 2001)

Meaning and computation

I Ability of understanding sentences.I Capacity of recognizing their truth-values.

I Long-standing tradition.I Meaning is a procedure for finding extension in a model.I Adopted often with psychological motivations.


I Ability of understanding sentences.I Capacity of recognizing their truth-values.I Long-standing tradition.I Meaning is a procedure for finding extension in a model.

I Adopted often with psychological motivations.


I Ability of understanding sentences.I Capacity of recognizing their truth-values.I Long-standing tradition.I Meaning is a procedure for finding extension in a model.I Adopted often with psychological motivations.

Explicit formulation

Pavel Tichý “Intension in terms of Turing machines”, 1969:

[. . . ] the fundamental relationship between sentence andprocedure is obviously of a semantic nature; namely, the purposeof sentences is to record the outcome of various procedures.Thus e.g. the sentence “The liquid X is an acid” serves to recordthat the outcome of a definite chemical testing procedure appliedto X is positive.

Explicit formulation

Pavel Tichý “Intension in terms of Turing machines”, 1969:[. . . ] the fundamental relationship between sentence andprocedure is obviously of a semantic nature; namely, the purposeof sentences is to record the outcome of various procedures.Thus e.g. the sentence “The liquid X is an acid” serves to recordthat the outcome of a definite chemical testing procedure appliedto X is positive.

For what does it mean to understand, i.e. to know the sense of anexpression? It does not mean actually to know its denotation butto know how the denotation can be found, how to pinpoint thedenotation of the expression among all the objects of the sametype. E.g. to know the sense of “taller” does not mean actually toknow who is taller than who, but rather to know what to dowhenever you want to decide whether a given individual is tallerthan another one. In other words, it does not mean to know whichof the binary relations on the universe is the one conceived by thesense of “taller”, but to know a method or procedure by means ofwhich the relation can be identified. (Tichy, 1969)

Psychological motivation

The basic and fundamental psychological point is that, with rareexceptions, in applying a predicate to an object or judging that arelation holds between two or more objects, we do not considerproperties or relations as sets. We do not even consider them assomehow simply intensional properties, but we have proceduresthat compute their values for the object in question. Thus, ifsomeone tells me that an object in the distance is a cow, I have aperceptual and conceptual procedure for making computations onthe input data that reach my peripheral sensory system [. . . ]Fregean and other accounts scarcely touch this psychologicalaspect of actually determining application of a specific algorithmicprocedure. (Suppes 1982)

Meaning as a collection of procedures

I have defended the thesis that the meaning of a sentence is aprocedure or a collection of procedures and that this meaning inits most concrete representation is wholly private and idiosyncraticto each individual. (Suppes 1982)

Automata perspective

An attractive, but never very central idea in modern semantics hasbeen to regard linguistic expressions as denoting certain“procedures” performed within models for the language. (VanBenthem, 1986)

QuestionWhat are we computing in the case of quantifiers?

Outline



Definability of GQs

What is the semantic assigned to quantifiers?

1. Every poet has low self-esteem.2. Some dean danced nude on the table.3. At least 7 grad students prepared presentations.4. An even number of the students saw a ghost.5. Most of the students think they are smart.6. Less than half of the students received good marks.

Generalized Quantifiers

DefinitionA quantifier Q is a way of associating with each set M afunction from pairs of subsets of M into {0,1} (False, True).

Example

everyM [A,B] = 1 iff A ⊆ B

evenM [A,B] = 1 iff card(A ∩ B) is even

mostM [A,B] = 1 iff card(A ∩ B) > card(A− B)



Example






Example




Illustration

U A B

S0

S1 S2S3c1

c2 c3

c4

c5

Full definition

DefinitionA generalized quantifier Q of type t = (n1, . . . ,nk ) is a functorassigning to every set M a k -ary relation QM between relationson M such that if (R1, . . . ,Rk ) ∈ QM then Ri is an ni -ary relationon M, for i = 1, . . . , k . Additionally, Q is preserved by bijections,i. e., if f : M −→ M ′ is a bijection then (R1, . . . ,Rk ) ∈ QM if andonly if (fR1, . . . , fRk ) ∈ QM′ , for every relation R1, . . . ,Rk on M,where fR = {(f (x1), . . . , f (xi)) | (x1, . . . , xi) ∈ R}, for R ⊆ M i .

DefinitionIf in the above definition for all i : ni = 1, then we say that aquantifier is monadic, otherwise we call it polyadic.

Full definition

DefinitionA generalized quantifier Q of type t = (n1, . . . ,nk ) is a functorassigning to every set M a k -ary relation QM between relationson M such that if (R1, . . . ,Rk ) ∈ QM then Ri is an ni -ary relationon M, for i = 1, . . . , k . Additionally, Q is preserved by bijections,i. e., if f : M −→ M ′ is a bijection then (R1, . . . ,Rk ) ∈ QM if andonly if (fR1, . . . , fRk ) ∈ QM′ , for every relation R1, . . . ,Rk on M,where fR = {(f (x1), . . . , f (xi)) | (x1, . . . , xi) ∈ R}, for R ⊆ M i .

DefinitionIf in the above definition for all i : ni = 1, then we say that aquantifier is monadic, otherwise we call it polyadic.

Quantifiers are second-order relations

ObservationIf we fix a model M = (M,AM ,BM), then we can treat ageneralized quantifier as a relation between relations over theuniverse.

Example

every[A,B] = 1 iff AM ⊆ BM

even[A,B] = 1 iff card(AM ∩ BM) is even

most[A,B] = 1 iff card(AM ∩ BM) > card(AM − BM)



Example






Example




Outline



Definability of GQs

Definability

DefinitionLet Q be a generalized quantifier of type t and L a logic. Wesay that the quantifier Q is definable in L if there is a sentenceϕ ∈ L of vocabulary τt such that for any τt -structure M:

M |= ϕ iff M ∈ Q.

Elementary GQs

Some GQs, like ∃≤3, ∃=3, and ∃≥3, are expressible in FO.

Example

some x [A(x),B(x)] ⇐⇒ ∃x [A(x) ∧ B(x)].

Non-elementary GQs

TheoremThe quantifiers ‘there exists (in)finitely many’, most and evenare not first-order definable.

We can use higher-order logics:

ExampleIn M = (M,AM ,BM) the sentence

most x [A(x),B(x)]

is true if and only if the following condition holds:

∃f : (AM−BM) −→ (AM∩BM) such that f is injective but not surjective.

Non-elementary GQs

TheoremThe quantifiers ‘there exists (in)finitely many’, most and evenare not first-order definable.We can use higher-order logics:


most x [A(x),B(x)]



Non-elementary GQs

TheoremThe quantifiers ‘there exists (in)finitely many’, most and evenare not first-order definable.We can use higher-order logics:


most x [A(x),B(x)]



We know what GQs denote. Now, it’s time to see how wecompute those denotations.

Documents

Quantifiers & Cognition · Schedule 1.Monday, November 22, 13-15, room F347 2.Tuesday, November 23, 13-15, room F363 3.Wednesday, November 24,10-12 & 13-15, room F355 4.Thursday,