Predicate_Logic_Rules

Predicate_Logic_Rules - P ( k ) holds, for some arbitrary k...

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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
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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)...
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