Verifiable Fragments of Language - Everyday Language ... · Verifiable Fragments of Language - Everyday Language Quantifiers Author: Jakub Szymanik Created Date: 2/22/2007 12:21:33

MotivationsPreliminaries

Main ThesisArgument From Negation Normality

Argument From Practical ComputabilityOutlook

VERIFIABLE FRAGMENTS OFLANGUAGE

EVERYDAY LANGUAGE QUANTIFIERS

Jakub Szymanik

Institute for Logic, Language and ComputationUniversiteit van Amsterdam

22nd February 2007

Jakub Szymanik Everyday Fragments of Natural Language




ABSTRACT

Everyday language is easy to understand.

We claim that it is bounded by Σ11-properties.

Only then Barwise’s test of negation normality works.

In finite universe Σ11-sentences are (in)directly verifiable.





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS

4 ARGUMENT FROM NEGATION NORMALITY

5 ARGUMENT FROM PRACTICAL COMPUTABILITY

6 OUTLOOK





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





WHAT DO WE MEAN BY “ EVERYDAY LANGUAGE”?

“Natural language” as opposed to technical languages.

E. g. “natural language quantifiers” vs. “logical quantifiers”.

“Majority” and “many” but not “infinitely many”.

We use “everyday language” instead.





EVERYDAY LANGUAGE IS THE BASIC CORE OFNL.

Math

Philosophy

Logic

FIGURE: Natural language ;-)





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





BASIC MODEL

M = (U, V M , T M , HM), where:

the universe U of M is the set of human beings,

V M is the set of villagers,

T M is the set of townsmen,

HM is the relation of hating each other.

Predicates V , T , H are interpreted in M as: V M , T M , HM .





ELEMENTARY SENTENCE

(1.) There are at most two villagers.

(2.) ∃x∃y [V (x) ∧ V (y) ∧ ∀z(V (z) ⇒ (z = x ∨ z = y))].





EXISTENTIAL SECOND-ORDER SENTENCES

(3.) Every other person is a townsman.

(4.) ∃P one-to-one mapping from T M into U − T M .

DEFINITION

We call the formulae of the form ∃Pϕ(P)existential second–order formulae.We denote the class of such formulae as Σ1

1.





EXISTENTIAL SECOND-ORDER SENTENCES

(3.) Every other person is a townsman.

(4.) ∃P one-to-one mapping from T M into U − T M .

DEFINITION

We call the formulae of the form ∃Pϕ(P)existential second–order formulae.We denote the class of such formulae as Σ1

1.





ONE MORE EXAMPLE OFΣ11-SENTENCE

(5.) Most villagers and most townsmen hate each other.

(6.) ∃A∃B containing most villagers and townsmen s.t.∀x∀y(A(x) ∧ B(y) ⇒ H(x , y)).





THERE IS LIFE OUTSIDEΣ11

(7.) There are at most countably many entities in the universe.

(8.) There exists a well–ordering such thateach element has a predecessor except for the least one.





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





MAIN THESIS

MAIN THESIS

Everyday language is bounded by the Σ11-properties.

“Every other” and “most” belong to everyday language.The counterexample is “there exist countably many”.





SOME CONSEQUENCES

Σ11 is not closed on negation.

Sentence belongs to everyday language – its negation not.





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





BARWISE’ S OBSERVATION

OBSERVATION

Negation of some simple quantifier sentencescan easily be formulated as a simple quantifier sentences.In some cases it is impossible.





NEGATION NORMAL SENTENCES

(9.) Everyone owns a car.

(10.) Someone doesn’t own a car.





NOT NEGATION NORMAL SENTENCES


(12.) It is not the case that most villagers and most townsmenhate each other.





TEST FORFO-DEFINABILITY

Only negation normal sentences are FO-definable.

It agrees with our experience.

It works only if simple sentences are bounded by Σ11.





MAIN THEOREM

THEOREM

If ϕ and its negation are both definable in Σ11, then ϕ is FO.

It gives argument for main thesis in arbitrary universes.





OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





MEANING AS ALGORITHM

Meaning of ϕ is “the mode of presenting” its truth-value.

Meaning ≡ model-checking procedure.

Classifying meanings by intractability.

Everyday sentences are relatively easy.





MEANING AS ALGORITHM

Meaning of ϕ is “the mode of presenting” its truth-value.

Meaning ≡ model-checking procedure.

Classifying meanings by intractability.

Everyday sentences are relatively easy.





DIRECT PRACTICAL COMPUTABILITY

EDMOND’ S THESIS

Practically computable problems are in PTIME.

THEOREM

FO ⊆ PTIME

Therefore, elementary sentences are directly verifiable.





DIRECT PRACTICAL COMPUTABILITY

EDMOND’ S THESIS

Practically computable problems are in PTIME.

THEOREM

FO ⊆ PTIME

Therefore, elementary sentences are directly verifiable.





BEYOND ELEMENTARY LOGIC

THEOREM (FAGIN , 1974)

Σ11 captures NPTIME.

What are NPTIME computations?

Choose certificate (proof).

Check in PTIME its correctness.

Accept if it is correct.

Therefore, Σ11 sentences are indirectly verifiable.





BEYOND ELEMENTARY LOGIC

THEOREM (FAGIN , 1974)

Σ11 captures NPTIME.

What are NPTIME computations?

Choose certificate (proof).

Check in PTIME its correctness.

Accept if it is correct.

Therefore, Σ11 sentences are indirectly verifiable.





INDIRECT VERIFICATION

(13.) There were more boys than girls at the party.

(14.) At the party every girl was paired with a boy.

(15.) Peter came alone to the party.





INDIRECT VERIFICATION AGAIN

(16.) Most villagers are A.

(17.) Most townsmen are B.

(18.) All A and all B hate each other.






OUTLINE

1 MOTIVATIONS

2 PRELIMINARIES

3 MAIN THESIS



6 OUTLOOK





SUMMARY

MAIN THESIS

Everyday language is bounded by the Σ11-properties.

Arguments:

Everyday language is not closed on negation.

Everyday sentences are (in)directly verifiable.





FEW RELATED PROBLEMS

Extend the notions for arbitrary SOD-quantifiers.

Associate with evaluation games in finite models.

Workout connections with cognitive science.


Appendix References

REFERENCES

J. BarwiseOn Branching Quantifiers in English.Journal of Philosophical Logic, 8(1979).

M. Mostowski & J. SzymanikSemantical bounds for everyday language.Semiotica, to appear.


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Verifiable Fragments of Language - Everyday Language ... · Verifiable Fragments of Language - Everyday Language Quantifiers Author: Jakub Szymanik Created Date: 2/22/2007 12:21:33