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 14260-2000
shapiro@cse.buffalo.edu
http://www.cse.buffalo.edu/~shapiro/
Collaborators
Jean-Pierre Koenig
David R. Pierce
William J. Rapaport
The SNePS Research Group
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What Is It?
A logic
For KRR systems
Supporting NL understanding & generation
And commonsense reasoning
LA
Sound & complete via translation to Standard FOL
Based on Arbitrary Objects, Fine (’83, ’85a, ’85b)
And ANALOG, Ali (’93, ’94), Ali & Shapiro (’93)
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Outline of Paper
Introduction and Motivations
Introduction to Arbitrary Objects
Informal Introduction to LA
Formal Syntax of LA
Translations Between and LA Standard FOL
Semantics of LA
Proof Theory of A
Soundness & Completeness Proofs
Subsumption Reasoning in LA
MRS and LA
Implementation Status
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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with examples
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Basic Idea
Arbitrary Terms
(any x R(x))
Indefinite Terms
(some x (y1 … yn) R(x))
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Motivations
See paper for other logics
that each satisfy some of these motivations
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Motivation 1
Uniform Syntax
Standard FOL:
White(Dolly)
x(Sheep(x)  White(x))
x(Sheep(x)  White(x))
LA :
White(Dolly)
White(any x Sheep(x))
White(some x ( ) Sheep(x))
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Motivation 2
Locality of Phrases
Every elephant has a trunk.
Standard FOL
x(Elephant(x)  y(Trunk(y)  Has(x,y))
LA :
Has(any x Elephant(x), some y (x) Trunk(y))
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Motivation 3
Prospects for Generalized Quantifiers
Most elephants have two tusks.
Standard FOL
??
LA :
Has(most x Elephant(x), two y Tusk(y))
(Currently, just notation.)
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Motivation 4
Structure Sharing
Every elephant has a trunk. It’s flexible.
Has( , )
Flexible( )
some y ( ) Trunk(y)
any x Elephant(x)
Quantified terms are “conceptually complete”.
Fixed semantics (forthcoming).
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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))
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Hairy
Mammal
Pet
Elephant
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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with examples
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Quantified Terms
Arbitrary terms:
(any x [R(x)])
Indefinite terms:
(some x ([y1 … yn]) [R(x)])
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Compatible Quantified Terms
(Q v ([a1 … an]) [R(v)])
(Q u ([a1 … an]) [R(u)])
different
or
same
(Q v ([a1 … an]) [R(v)])
(Q v ([a1 … an]) [R(v)])
All quantified terms in an expression must be compatible.
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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))
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Capture
free
bound
White(any x Sheep(x))
Black(x)
White(any x Sheep(x))  Black(x)
same
Quantifiers take wide scope!
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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.
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Closure
x …  contains the scope of x
Compatibility and capture rules
only apply within closures.
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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.
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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.
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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)
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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))
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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, xSpy(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,  xSpy(some x ( ) Person(x))
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Current Implementation Status
Partially implemented as the logic of SNePS 3
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Summary
LA 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.
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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.
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Closure
Contains wide-scoping of quantified terms
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