Predicate_Logic_Rules

# Predicate_Logic_Rules - P k holds for some arbitrary k...

This preview shows page 1. Sign up to view the full content.

CMPSC 360 Discrete Mathematics for Computer Science Fall 2009 Penn State University Proof Rules - II x D.P ( x ) t D P ( t ) ( ∀- elim ) P ( k ) x D.P ( x ) ( ∀- intro ) x D.P ( x ) P ( b ) ( ∃- elim ) P ( t ) t D x D.P ( x ) ( ∃- intro ) The formula P ( x ) represents a formula that contains the variable x . The formula P ( t ) represents a formula after replacing all free occurrences of x with t . In the ∀- elim rule, we are allowed to use variables for the term t . The use of b in the ∃- elim should be viewed as an unknown constant. We cannot assume anything about it, other than that it is some element from the domain D . It cannot already occur in P . The ∀- intro rule is typically used as follows: We start with a universally quantiﬁed formula that we are trying to prove. We reduce this to showing that
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: P ( k ) holds, for some arbitrary k , about which we know nothing, other than that it belongs to the domain D . The term k is sometimes called an eigenvariable . An important constraint on this rule is that k cannot appear anywhere in P (or in any formula of the proof below the point where this rule is used. Example Proof: (1) ∃ x.likes ( x,fritos ) ( premise ) (2) ∀ x.likes ( x,fritos ) → likes ( x,doritos ) ( premise ) (3) likes ( b,fritos ) ( ∃-elim-1) (4) likes ( b,fritos ) → likes ( b,doritos ) ( ∀-elim-2) (5) likes ( b,doritos ) ( modusponens-3 , 4) (6) ∃ x.likes ( x,doritos ) ( ∃-intro-5)...
View Full Document

• Fall '08
• HAULLGREN
• Pennsylvania State University, Modus ponens, Penn State University, Rule of inference, universally quantiﬁed formula

{[ snackBarMessage ]}

Ask a homework question - tutors are online