# Logic1019 - Tableaux for first-order]= logic 1 2 3 Validity...

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Midterm today, 9pm Room 403 (bring your student id for entrance to the building)
13.9 Find a formula that is valid in all non- empty domains (hence simply valid) but not valid in an empty domain. ∀xPx => ∃xPx. In any non-empty domain, if all x is P, then there is x which is P. The formula is valid. In an empty domain, ∀xPx is vacuously true, but ∃xPx is false because there is no such x. The conditional is false in an empty domain.

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Tautologies We can obtain a first-order formula X by substituting first-order formulas for the propositional variables in Y. Then X is an instance of a formula Y. Ex. If we substitute ∀xPx for p in pv~p, we obtain ∀xPx v ~∀xPx. A first-order formula is a tautology if it is an instance of a tautology of proposi-
However, not all valid formulas are tau- tologies. Exs. ∀x(Px v ~Px), ∀xPx => ∃xPx. T FX doesn’t close with eight tableaux rules in propositional logic. In order to prove valid formulas that are not tautologies, we need additional tableau rules for ∀ and ∃. tautolo- gies Valid formu- las

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Ex. proof of the formula ∀xPx => ∃xPx. (1) F∀xPx => ∃xPx
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Logic1019 - Tableaux for first-order]= logic 1 2 3 Validity...

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