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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
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
EVERYDAY LANGUAGE IS THE BASIC CORE OFNL.
Math
Philosophy
Logic
FIGURE: Natural language ;-)
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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 .
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
ELEMENTARY SENTENCE
(1.) There are at most two villagers.
(2.) ∃x∃y [V (x) ∧ V (y) ∧ ∀z(V (z) ⇒ (z = x ∨ z = y))].
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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)).
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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”.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
SOME CONSEQUENCES
Σ11 is not closed on negation.
Sentence belongs to everyday language – its negation not.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
BARWISE’ S OBSERVATION
OBSERVATION
Negation of some simple quantifier sentencescan easily be formulated as a simple quantifier sentences.In some cases it is impossible.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
NEGATION NORMAL SENTENCES
(9.) Everyone owns a car.
(10.) Someone doesn’t own a car.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
NOT NEGATION NORMAL SENTENCES
(11.) Most villagers and most townsmen hate each other.
(12.) It is not the case that most villagers and most townsmenhate each other.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
DIRECT PRACTICAL COMPUTABILITY
EDMOND’ S THESIS
Practically computable problems are in PTIME.
THEOREM
FO ⊆ PTIME
Therefore, elementary sentences are directly verifiable.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
DIRECT PRACTICAL COMPUTABILITY
EDMOND’ S THESIS
Practically computable problems are in PTIME.
THEOREM
FO ⊆ PTIME
Therefore, elementary sentences are directly verifiable.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
INDIRECT VERIFICATION AGAIN
(16.) Most villagers are A.
(17.) Most townsmen are B.
(18.) All A and all B hate each other.
(19.) Most villagers and most townsmen hate each other.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
OUTLINE
1 MOTIVATIONS
2 PRELIMINARIES
3 MAIN THESIS
4 ARGUMENT FROM NEGATION NORMALITY
5 ARGUMENT FROM PRACTICAL COMPUTABILITY
6 OUTLOOK
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
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.
Jakub Szymanik Everyday Fragments of Natural Language
MotivationsPreliminaries
Main ThesisArgument From Negation Normality
Argument From Practical ComputabilityOutlook
FEW RELATED PROBLEMS
Extend the notions for arbitrary SOD-quantifiers.
Associate with evaluation games in finite models.
Workout connections with cognitive science.
Jakub Szymanik Everyday Fragments of Natural Language
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.
Jakub Szymanik Everyday Fragments of Natural Language