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philosophy note

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Buffalo State College
James Hildebrand

1. a. If the tree for a finite set, Γ, of sentences of SL is closed then Γ is t-f inconsistent. b. There is no sentence, P, such that both P and ~P can be proven to be theorems using the tree method (i.e., such that both {~P} and {P} have closed trees). c. If P is truth functionally equivalent to Q, and R contains one or more occurrences of P, then the result, R(Q//P), of replacing one or more of the occurrences of P in R with occurrences of Q will be truth functionally equivalent to R. d. If a set, Γ, of sentences of SL truth functionally entails a sentence, P, of SL, then there is some way of deriving P from Γ using the derivation system, SD. 2. b ((a) is appealed to in proving soundness of the tree method; (c) defines decidability of the tree method; (d) defines compactness of sentences of SL; (e) defines consistency of the tree method) 3. a. 3. b. 6. Length figures in the proof that the tree method is decidable. In that proof, attention is drawn to the fact that all the decomposition rules are such that longer sentences are checked off (so not decomposed again) and decomposed into at most 4 shorter sentences. Combined with a proof that the tree for a finite set of sentences cannot have infinitely many branches, this helps to establish that decomposition cannot go on forever. Even if the tree does not close, eventually only sentences of lengths 1 and 2 (literals) will be left, which means the branches must be completed open. Length also figures in the proof that the tree method is complete and the related proof that SL is compact.
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