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# Logic1118 - Completeness for prob positional logic 1 2 3...

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% O q° °°° °° Completeness for pro- positional logic 1. Completeness 2. Truth set 3. Compactness

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Read Ch.18 for the next week HW #8 (due Monday, 11/23) 1. Let A be a formula in the propositional language (no quantifiers). Let cA be the number of places at which a binary connective (one of , v, =>) occurs. Let sA be the number of places at which propositional variables occur. Show that sA = cA+1. 2. Exercise 17.1 3. Exercise 17.2 (a)
Completeness The tableau method proves all tautolo- gies. C If X is a tautology, then the tableau proves X. C If X is a tautology, then there is a closed tableau for FX. C (contrapositive) If there is no closed tableau for FX, then FX is satisfiable. More generally, for any completed tableau, if it is not closed, then the origin

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A branch Θ is completed iff For every α in Θ, there are α1 and α2 in Θ. For every β in Θ, there is β1 or β2 in Θ. A tableau is completed iff every open branch is completed. Strategy of the proof A completed open branch forms what lo- gicians call a Hintikka set .

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