Rules of Inference Goal for predicate logic 1
- Slides: 7
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 2
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 7
