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University of Michigan - Ann Arbor

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Problems deal with Philosophy of Symbolic Logic, more specifically, the Soundness Theorem. Also deals with compactness theorem. Problems stem from work done in The Logic Book 6th Edition

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Philosophy 303 Problem set 7a Winter, 2018 Due Monday March 26 at 4:30 pm Note: This is one of two parts. Next weekend a second short set of questions will be posted, as foci for the discussion sessions of the subsequent week. 1. Say that you have the language of all the strings in the infinite alphabet {a1 , a2 , a3 , . . . an . . .} (with no empty string). State a correlation that associates a distinct natural number with each string in this language. Remark: Here is an obvious first try, modelled on the strategy for the finite alphabet of PL, as used on p. 254 - 255 of the textbook. It doesn’t work, but it is instructive to see why not. The simplest try would be to just let each letter be associated with its subscript, and then string together the associated numbers. That is, associate a1 ↔ 1, a2 ↔ 2, a3 ↔ 3, a4 , ↔ 4 . . . , ai ↔ i . . . then string together the associated numbers to correspond to an associated string: ai1 ai2 ai3 . . . ain ↔ i1 i2 i3 . . . in . The reason this strategy doesn’t work is that it doesn’t associate a unique number to each string of symbols. For example, both a1 a1 and a11 get assigned 11 by this scheme. Both a3 a2 a6 a83 and a32 a683 get assigned 32683. There are lots of other examples of diﬀerent strings getting the same number. Your answer has to avoid this problem. There are a bunch of diﬀerent ways to do this. Potentially useful fact: Remember that n is a prime number if it is a natural number greater than 1, and the only numbers that evenly divide it are 1 and n itself. Every natural number has a unique prime power decomposition. That is, n can be written in exactly one way as αm a product q1α1 · q2α2 · q3α3 . . . qm where each qi is a prime number and each αi a natural number greater than 0. So for example: 12 = 22 · 3, 1960 = 23 · 5 · 72 and these are the only ways to decompose 12 and 1960 into products of powers of primes. 2. For this question, you may assume the truth of the completeness theorem and the soundness theorem: (Putting them together gives: Γ ⊢ S ⇔ Γ ! S with Γ a set of sentences of SL and S a sentence of SL.) Say that S is a sentence of SL and Γ is a set of sentences of SL, and Γ is infinite. Problem: Show that if Γ ! S then there is some finite set ∆ ⊆ Γ such that ∆ ! S. This is a consequence of the compactness theorem 6.4.12, but I want you to show how the statement follows directly from the completeness and soundness theorems. The argument takes just four or five lines. (Hint: Derivations are only finitely many lines long.) 3. Show, using the soundness theorem that you cannot derive a contradiction in SD from just the sentence letter A. (To put it in other words: Prove that the set {A} is consistent in SD.) 2
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Tags: logic philosophy symbolic logic