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James Hildebrand
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Lecture

# philosophy note

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Buffalo State College

Philosophy

1002

James Hildebrand

Spring

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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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