MIT24_241F09_lec09 - Log ic I - Session 9 Plan One more...

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Logic I - Session 9
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Plan One more example to practice meta-language proofs Explain mathematical induction Examples
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More practice 3.6E (2b) If Γ {P} Q, then Γ P Q 1. Γ {P} Q 2. Every TVA that makes every member of Γ {P} true also makes Q true 3. Every TVA that makes every member of Γ true and P true also makes Q true. 4. No TVA makes every member of Γ true and P true but makes Q false. 5. So no TVA that makes every member of Γ true makes P Q false. 6. So every TVA that makes every member of Γ true also makes P Q true. 7. So Γ P Q
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Remember: Truth-functional notions are all defined in terms of TVAs (not derivability or possibility) E.g.: Γ is truth-functionally inconsistent iff there’s no TVA that makes every member of Γ true. vs. (The symbol for derivability looks like a piece of a derivation.)
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Mathematical induction If we have an ordered sequence of infinitely many things, we can prove that all of them have a property F by proving two simpler claims: Basis clause : Prove that the first thing in the sequence is F Inductive step: Prove that if the nth thing is F, then so is the n+1st. Conclude that everything in the sequence is F The things in the sequence can b f SL or sets of sentences SL e anything, e.g. sentences o
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First Example of MI 6.1E (1a) Prove: No sentence of SL that contains only binary connectives, if any, is truth-functionally false. First step: insert relevant definitions in the thesis to be proven Prove: No sentence of SL that contains only binary connectives is false on every truth-value assignment Equivalently: Prove: Every sentence of SL that contains only binary connectives is true on some TVA
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Second step: Arrange our sentences into a sequence. We can do that by using the number of connectives in the sentences. (This will be a common strategy) Third step: Find our basis clause and inductive clause. We sometimes get them by directly applying our thesis to the members of our sequence. So we might start with:
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MIT24_241F09_lec09 - Log ic I - Session 9 Plan One more...

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