A Logic of Arbitrary and Indefinite Objects Stuart C. Shapiro Department of Computer Science and Engineering, and Center for Cognitive Science University at Buffalo, The State University of New York 201 Bell Hall, Buffalo, NY

January, 2005S. C. Shapiro2 Based On Stuart C. Shapiro, A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004,

January, 2005S. C. Shapiro3 Collaborators Jean-Pierre Koenig David R. Pierce William J. Rapaport The SNePS Research Group

January, 2005S. C. Shapiro4 What Is It? A logic For KRR systems Supporting NL understanding & generation And commonsense reasoning LALA Sound & complete (via translation to Standard FOL) Based on Arbitrary Objects, Fine (’83, ’85a, ’85b) And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93)

January, 2005S. C. Shapiro5 Outline of Talk Introduction and Motivations Informal Introduction to L A with Examples Examples of Proof Theory Implementation as Logic of SNePS 3

January, 2005S. C. Shapiro6 Basic Idea Arbitrary Terms (any x R (x)) Indefinite Terms (some x (y 1 … y n ) R (x))

January, 2005S. C. Shapiro7 Motivation 1 Uniform Syntax Standard FOL ( L s ): Dolly is white. White(Dolly) Every sheep is white.  x(Sheep(x)  White(x)) Some sheep is white.  x(Sheep(x)  White(x))

January, 2005S. C. Shapiro8 Motivation 1 Uniform Syntax FOL with Restricted Quantifiers ( L R ): Dolly is white. White(Dolly) Every sheep is white.  x Sheep White(x) Some sheep is white.  x Sheep White(x)

January, 2005S. C. Shapiro9 Motivation 1 Uniform Syntax L A : Dolly is white. White(Dolly) Every sheep is white. White(any x Sheep(x)) Some sheep is white. White(some x ( ) Sheep(x))

January, 2005S. C. Shapiro10 Motivation 2 Locality of Phrases Every elephant has a trunk. Standard FOL  x(Elephant(x)   y(Trunk(y)  Has(x,y)) L R :  x Elephant  y Trunk Has(x,y))

January, 2005S. C. Shapiro11 Motivation 2 Locality of Phrases Every elephant has a trunk. Logical Form, or FOL with “complex terms” ( L C ): Has(, ) L A : Has(any x Elephant(x), some y (x) Trunk(y))

January, 2005S. C. Shapiro12 Motivation 3 Prospects for Generalized Quantifiers Most elephants have two tusks. Standard FOL ?? L A : Has(most x Elephant(x), two y Tusk(y)) (Currently, just notation.)

January, 2005S. C. Shapiro13 Motivation 4 Structure Sharing any x Elephant(x) some y ( ) Trunk(y) Has(, )Flexible( ) Every elephant has a trunk. It’s flexible. Quantified terms are “conceptually complete”. Fixed semantics (forthcoming).

January, 2005S. C. Shapiro14 Motivation 5 Term Subsumption Hairy(any x Mammal(x)) Mammal(any y Elephant(y))  Hairy(any y Elephant(y)) Pet(some w () Mammal(w))  Hairy(some z () Pet(z)) Hairy Mammal Elephant Pet

January, 2005S. C. Shapiro15 Outline of Talk Introduction and Motivations Informal Introduction to L A with Examples Examples of Proof Theory Implementation as Logic of SNePS 3

January, 2005S. C. Shapiro16 Quantified Terms Arbitrary terms: (any x [ R (x)]) Indefinite terms: (some x ([y 1 … y n ]) [ R (x)])

January, 2005S. C. Shapiro17 (Q v ([a 1 … a n ]) [ R (v)]) (Q u ([a 1 … a n ]) [ R (u)]) (Q v ([a 1 … a n ]) [ R (v)]) Compatible Quantified Terms different or same All quantified terms in an expression must be compatible.

January, 2005S. C. Shapiro18 Quantified Terms in an Expression Must be Compatible Illegal: White(any x Sheep(x))  Black(any x Raven(x)) Legal White(any x Sheep(x))  Black(any y Raven(y)) White(any x Sheep(x))  Black(any x Sheep(x))

January, 2005S. C. Shapiro19 Capture White(any x Sheep(x)) Black(x) White(any x Sheep(x))  Black(x) bound free same Quantifiers take wide scope!

January, 2005S. C. Shapiro20 Examples of Dependency Has(any x Elephant(x), some(y (x) Trunk(y)) Every elephant has (its own) trunk. (any x Number(x)) < (some y (x) Number(y)) Every number has some number bigger than it. (any x Number(x)) < (some y ( ) Number(y)) There’s a number bigger than every number.

January, 2005S. C. Shapiro21 Closure  x …  contains the scope of x Compatibility and capture rules only apply within closures.

January, 2005S. C. Shapiro22 Closure and Negation  White(any x Sheep(x)) Every sheep is not white.   x White(any x Sheep(x))  It is not the case that every sheep is white.  White(some x () Sheep(x)) Some sheep is not white.  x White(some x () Sheep(x))  No sheep is white.

January, 2005S. C. Shapiro23 Closure and Capture Odd(any x Number(x))  Even(x) Every number is odd or even.  x Odd(any x Number(x))    x Even(any x Number(x))  Every number is odd or every number is even.

January, 2005S. C. Shapiro24 Tricky Sentences: Donkey Sentences Every farmer who owns a donkey beats it. Beats(any x Farmer(x)  Owns(x, some y (x) Donkey(y)), y)

January, 2005S. C. Shapiro25 Tricky Sentences: Branching Quantifiers Some relative of each villager and some relative of each townsman hate each other. Hates(some x (any v Villager(v)) Relative(x,v), some y (any u Townsman(u)) Relative(y,u))

January, 2005S. C. Shapiro26 Closure & Nested Beliefs (Assumes Reified Propositions) There is someone whom Mike believes to be a spy. Believes(Mike, Spy(some x ( ) Person(x)) Mike believes that someone is a spy. Believes(Mike,  x Spy(some x ( ) Person(x)  ) There is someone whom Mike believes isn’t a spy. Believes(Mike,  Spy(some x ( ) Person(x)) Mike believes that no one is a spy. Believes(Mike,   x Spy(some x ( ) Person(x)  )

January, 2005S. C. Shapiro27 Outline of Talk Introduction and Motivations Informal Introduction to L A with Examples Examples of Proof Theory Implementation as Logic of SNePS 3

January, 2005S. C. Shapiro28 Proof Theory: anyE (abbreviated) From B (any x A (x)) and A (a) conclude B (a)

January, 2005S. C. Shapiro29 Proof Theory: anyI (abbreviated) From A (a) as Hyp and derive B (a) Conclude B (any x A (x))

January, 2005S. C. Shapiro30 Example Proof From Every woman is a person. Every doctor is a professional. Some child of every person all of whose sons are professionals is busy. Conclude Some child of every woman all of whose sons are doctors is busy. [Based on an example of W. A. Woods]

January, 2005S. C. Shapiro31 Example Proof 1.Person(any x Woman(x)) 2.Professional(any y Doctor(y)) 3.Busy(some u (v) childOf(u, any v Person(v)  Professional(any w sonOf(w,v)))) 4.Woman(a)Hyp 5.Doctor(any z sonOf(z,a)) Hyp 6.Person(a)anyE,1,4 7.Professional(any z sonOf(z,a))anyE,2,6 8.Busy(some u ( ) childOf(u,a))anyE3,6  7 9.Busy(some u (v) childOf(u, any v Woman(v)  Doctor(any w sonOf(w,v)))) anyI,4  5—8 QED

January, 2005S. C. Shapiro32 Syllogistic Reasoning as Subsumption (Derived Rules of Inference) Barbara: From A (any x B (x)) and B (any y C (y)) conclude A (any y C (y))

January, 2005S. C. Shapiro33 Syllogistic Reasoning as Subsumption (Derived Rules of Inference) Darii: From A (any x B (x)) and C (some y φ B (y)) conclude A (some y φ C (y))

January, 2005S. C. Shapiro34 Outline of Talk Introduction and Motivations Informal Introduction to L A with Examples Examples of Proof Theory Implementation as Logic of SNePS 3

January, 2005S. C. Shapiro35 Current Implementation Status Partially implemented as the logic of SNePS 3

January, 2005S. C. Shapiro36 SNePS 3 Example snepsul(25): #L#!(build object (any x (build member x class Mammal)) property hairy) Is((any Arb1 Isa(Arb1, Mammal)), hairy) snepsul(26): #L#!(build member (any y (build member y class Elephant)) class Mammal) Isa((any Arb2 Isa(Arb2, Elephant)), Mammal) snepsul(27): #L#?(build object (any y (build member y class Elephant)) property hairy) Is((any Arb2 Isa(Arb2, Elephant)), hairy) snepsul(28): #L#!(build member Clyde class Elephant) Isa(Clyde, Elephant) snepsul(29): #L#?(build object Clyde property hairy) Is(Clyde, hairy)

January, 2005S. C. Shapiro37 Summary L A is A logic For KRR systems Supporting NL understanding & generation And commonsense reasoning Uses arbitrary and indefinite terms Instead of universally and existentially quantified variables.

January, 2005S. C. Shapiro38 Arbitrary & Indefinite Terms Provide for uniform syntax Promote locality of phrases Provide prospects for generalized quantifiers Are conceptually complete Allow structure sharing Support subsumption reasoning.

January, 2005S. C. Shapiro39 Closure Contains wide-scoping of quantified terms

January, 2005S. C. Shapiro40 Implementation Status Partially implemented as the logic of SNePS 3

January, 2005S. C. Shapiro41 For More Information The SNePS Research Group web site: The SNePS 3 Project page: