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MIT24_241F09_lec10

MIT24_241F09_lec10 - Log ic I Session 10 Thursday 1 Plan Re...

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Logic I - Session 10 1 Thursday, October 15, 2009

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Plan Re: course feedback Review of course structure Recap of truth-functional completeness? Soundness of SD 2 Thursday, October 15, 2009
The course structure Basics of arguments and logical notions (deductive validity and soundness, logical truth, falsity, consistency, indeterminacy, equivalence SL: syntax and semantics Derivation system SD (and SD+) Meta-logic: proofs about SL and SD / SD+ PL: syntax and semantics Derivation system PD (and PD+, PDE) Meta-logic: proofs about PL and PD / PD+ / PDE Thursday, October 15, 2009 3

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Last time Mathematical induction Strategy: (1) Insert relevant definitions in the claim you want to prove. (2) Arrange a sequence for the induction. (3) Formulate basis clause and inductive hypothesis. (4) Prove basis clause. (5) Prove inductive hypothesis by assuming its antecedent (n case) and deducing its consequent (n+1 case). Truth-functional completeness Thursday, October 15, 2009 4
Truth-functional completeness Truth-function: a mapping, for some positive integer n, from each combination of TVs n sentences can have to a TV. E.g. for two sentences: {T,F}X{T,F} {T,F}. More generally: {T,F} n {T,F} SL is truth-functionally complete iff for every truth-function f, there is an SL sentence P that expresses f. P expresses f iff P ’s truth-table is the characteristic truth- table for for f Thursday, October 15, 2009 5

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Truth-functional completeness We can state this more formally than in TLB: An truth-function f is a set of ordered pairs like this: { < <T,T>, T > , < <T,F>, F > , < <F,T>, F > , < <F,F>, F > } P expresses f iff for any i that is a member of f, when the atomic components of P are assigned the TVs in the 1st member of i, P receives the TV that’s the 2nd member of i. Thursday, October 15, 2009 6
Truth-functional completeness Why care? We want to use SL and truth-tables to test for TF- truth, validity, consistency, etc. Suppose we couldn’t express some TF in SL, e.g. neither/nor. Then we would have no sentence of SL that expressed the same truth-function as ‘Neither Alice nor Bill can swim.’ But then SL wouldn’t let us use a TT to show that the sentence is TF-entailed by {`Alice can swim if and only if Bill can swim’, `If Alice can swim, then Carol can’t swim’, `Carol can swim’}. Similar points apply to other truth-functions and tests for truth-functional properties and relations Thursday, October 15, 2009 7

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Truth-functional completeness So we want to know that we can express every truth-function We know this because we can set out an algorithm that, for any truth-function f, generates a sentence that expresses f.
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MIT24_241F09_lec10 - Log ic I Session 10 Thursday 1 Plan Re...

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