Download Predicate Calculus

Rules of Inference
Goal for predicate logic
1. Introduce rules of inference
2. Distinguish between correct & incorrect arguments.
Universal Specification
• If x P( x ) is true, then P( c ) is true for
arbitrary c in the universe of discourse.
• This can be written:
x P( x )
________
 P( c ) (for arbitrary c in the domain).
• Example: M( x ): x is mortal.
From x M( x ), infer M( Socrates ): Socrates
is mortal. (assumes Socrates is in domain)
Copyright © Peter Cappello
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Universal Generalization
• If P( c ) is true for each element c in the
domain, then x P( x ).
• This can be written:
P( c ) for arbitrary c in the domain
________
 x P( x ).
Copyright © Peter Cappello
3
Existential Specification
• If x P( x ) is true then there is an
element c such that P( c ) is true.
• This can be written:
x P( x )
________
 P( c ), for some c.
• Element c is not arbitrary.
– We know only that some c satisfies P.
– We do not necessarily know which one
(e.g., from a non-constructive proof).
Copyright © Peter Cappello
4
Existential Generalization
• If P( c ) is true for some c, then x P( x ).
• This can be written:
P( c ), for some c
________
 x P( x ).
Copyright © Peter Cappello
5
Example Argument
• In English:
– All CS courses are easy.
– CS 2 is a CS course.
– Therefore, CS 2 is easy.
• A more compact representation:
x ( C( x )  E( x ) ).
C( CS 2 ).
Therefore, E( CS 2 ).
Copyright © Peter Cappello
6
Proof
WHAT
WHY
1. x ( C( x )  E( x ) )
[premise 1]
2. C( CS 2 )  E( CS 2 )
[step 1, U.S.]
3. C( CS 2 )
[premise 2]
4. E( CS 2 )
[steps 2, 3, &
modus ponens]
Copyright © Peter Cappello
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