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/
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, 565-575.
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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 Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Basic Idea
Arbitrary Terms
(any x R(x))
Indefinite Terms
(some x (y1 … yn) R(x))
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Motivation 1
Uniform Syntax
Standard FOL (Ls ):
Dolly is white.
White(Dolly)
Every sheep is white.
x(Sheep(x)  White(x))
Some sheep is white.
x(Sheep(x)  White(x))
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Motivation 1
Uniform Syntax
FOL with Restricted Quantifiers (LR ):
Dolly is white.
White(Dolly)
Every sheep is white.
xSheep White(x)
Some sheep is white.
xSheep White(x)
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Motivation 1
Uniform Syntax
LA :
Dolly is white.
White(Dolly)
Every sheep is white.
White(any x Sheep(x))
Some sheep is white.
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))
LR :
xElephant yTrunk Has(x,y))
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Motivation 2
Locality of Phrases
Every elephant has a trunk.
• Logical Form,
or FOL with “complex terms” (LC):
Has(<x Elephant(x)>, <yTrunk(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
Examples of Proof Theory
Implementation as Logic of SNePS 3
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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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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Proof Theory:
anyE (abbreviated)
From B(any x A(x))
and A(a)
conclude B(a)
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Proof Theory:
anyI (abbreviated)
From A(a) as Hyp
and derive B(a)
Conclude B(any x A(x))
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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]
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Example Proof
1.
2.
3.
4.
5.
6.
7.
8.
9.
Person(any x Woman(x))
Professional(any y Doctor(y))
Busy(some u (v)
childOf(u, any v Person(v)
 Professional(any w sonOf(w,v))))
Woman(a)
Hyp
Doctor(any z sonOf(z,a))
Hyp
Person(a)
anyE,1,4
Professional(any z sonOf(z,a))
anyE,2,6
Busy(some u ( ) childOf(u,a))
anyE3,67
Busy(some u (v)
childOf(u, any v Woman(v)
 Doctor(any w sonOf(w,v))))
anyI,45—8
QED
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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))
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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))
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Outline of Talk
Introduction and Motivations
Informal Introduction to LA
with Examples
Examples of Proof Theory
Implementation as Logic of SNePS 3
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Current Implementation Status
Partially implemented as the logic of SNePS 3
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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)
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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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Implementation Status
Partially implemented as the logic of SNePS 3
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For More Information
The SNePS Research Group web site:
http://www.cse.buffalo.edu/sneps/
The SNePS 3 Project page:
http://www.cse.buffalo.edu/sneps/Projects/sneps3.html
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