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Lecture_14_10

# Lecture_14_10 - b = 3 n P ={1,4 n R =(1,2(2,3(3,4(4,1 The...

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Predicate Logic Without Quantifiers n Atomic sentences are made of smaller units:  individual constants  and  relation symbols . n The language is interpreted in a structure  consisting of the following: a non-empty set, U. a distinguished element of U for each constant  symbol. a distinguished n-ary relation on U for each n-ary
Example Language n Signature:  Constant symbols: a,b. Relation symbols: P(-), R(-,-). n Language: If t and t’ and constant symbols, then the following  are formulas: t   t’, P(t), R(t,t’). If   and   are formulas, then the following are  φ ψ formulas: ~ φ φ∧ . ψ

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Example Interpretation Let  be the following interpretation of the  language given above.: n U = {1,2,3,4} n a  = 2,  b  = 3. n P  = {1,4} n R  = {(1,2),(2,3),(3,4),(4,1)}

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Unformatted text preview: b = 3. n P = {1,4} n R = {(1,2),(2,3),(3,4),(4,1)} The interpretation determines a function from Example Interpretation n Atoms • [t t’]I = 1 iff ≈ t = t’. • [P(t)]I = 1 iff t is an element of P . • [R(t,t’)]I = 1 iff ( t , t’ ) is an element of R . n [~ ]I = 1 iff [ ]I = 0. φ φ n [ φ ]I = 1 iff ∧ψ [ φ ]I = [ ]I = 1. ψ Some types of questions concerning n Basic conceptual questions n Given an interpretation I and a sentence , determine if is true in I. n Counterexamples, i.e. show that a given implication does not hold. n More generally, given a consistent set of sentences, find an interpretation that satisfies all of the sentences in that set....
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Lecture_14_10 - b = 3 n P ={1,4 n R =(1,2(2,3(3,4(4,1 The...

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