MIT24_241F09_lec22

# MIT24_241F09_lec22 - Logic I Session 22 Meta-theory for...

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Unformatted text preview: Logic I - Session 22 Meta-theory for predicate logic 1 The course so far Syntax and semantics of SL English / SL translations TT tests for semantic properties of SL sentences Derivations in SD Meta-theory: SD is adequate for SL (sound, complete) Syntax and semantics of PL English / PL translations Derivations in PD Next: PD is adequate for PL (sound, complete) 2 Soundness, Completeness There are meta-theoretical results for PD as well as PDE. In particular: If Γ is a set of PL sentences and P is a PL sentence, then Γ ⊨ P iff Γ ⊢ P in PD. If Γ is a set of PLE sentences and P is a PLE sentence, then Γ ⊨ P iff Γ ⊢ P in PDE. We’ll focus on PL and PD, coming back to PLE and PDE later if we have time. 3 Soundness We’ll focus on soundness today. If Γ ⊢ P in PD, then Γ ⊨ P . To prove: If there’s a PD derivation all of whose primary assumptions are members of Γ and in which P occurs only in the scope of those assumptions, then P is quantiFcationally entailed by Γ . 4 Soundness As with soundness for SD, we prove our result by proving something stronger: Every sentence in a PD derivation is q-entailed by the set of assumptions with scope over it. Our proof of this will appeal to a mathematical induction analogous to the one we used to prove the soundness of SD. 5 Soundness Let Γ i be the set of assumptions open at line i in a derivation, and let P i be the sentence on line i. Basis clause: Γ 1 ⊨ P 1. Inductive step: If Γ i ⊨ P i for all i ≤ k, then Γ k+1 ⊨ P k+1. We’ll prove this by cases, one case for each rule that could have justiFed line k+1. Conclusion: ¡or every line k in a derivation, Γ k ⊨ P k. I.e.: Every sentence in a PD derivation is q-entailed by the set of assumptions with scope over it. 6 Soundness: Basis clause To prove: Γ 1 ⊨ P 1. = No interpretation mem Γ 1 true but makes P 1 false. The Frst line of any derivation is an assumption....
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MIT24_241F09_lec22 - Logic I Session 22 Meta-theory for...

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