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lecture 5 2009

# lecture 5 2009 - truth value under τ If B is a...

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1. Any atomic sentence of FOL is a sentence. 2. If A and B are sentences of FOL, then so are . A, (A . B), and (A . B). A truth value assignment is the assignment of exactly one of the truth values True (T), False (F) to the atomic sentences of a language. A sentence is TT-possible if that sentence is true under at least one truth value assignment. If A is a sentence of FOL, then A is a tautology if and only if A is true under every truth value assignment.

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A sentence B is a tautological consequence of sentences A 1 , …,A n if and only if every truth value assignment that results in all of A 1 ,…,A n being true also results in B being true. Two sentences A, B are tautologically equivalent if and only if B is a tautological consequence of A and A is a tautological consequence of B. i.e. For each truth value assignment τ, A and B have the same

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Unformatted text preview: truth value under τ. If B is a tautological consequence of A 1 ,…,A n , then B is a logical consequence of A 1 ,…,A n . (Recall: B is a logical consequence of A 1 ,…,A n if and only if it is impossible for all of A 1 ,…,A n to be true but B to be false.) Examples: Tet(a) Tet(a) . Cube(b) . Dodec(d) , Dodec(d) . Cube(d) Cube(d) Two sentences A,B are logically equivalent if and only if there is no logically possible circumstance in which A, B have different truth values. Examples: . (Tet(a) . Small(a)) . Tet(a) . . Small(a) . Cube(b) . (Dodec(c) . Tet(d)) (Cube(b) . Dodec(c)) . (Cube(b) . Dodec(c)) (a = b) . ( b = c) (a =b) . ( b = c) . ( a = c) These two sentences are logically equivalent but they are not tautologically equivalent....
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