This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Logic I  Session 22 Metatheory 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 Metatheory: 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 metatheoretical results for PD as well as PDE. In particular: If is a set of PL sentences and P is a PL sentence, then P iff P in PD. If is a set of PLE sentences and P is a PLE sentence, then P iff P in PDE. Well focus on PL and PD, coming back to PLE and PDE later if we have time. 3 Soundness Well focus on soundness today. If P in PD, then P . To prove: If theres a PD derivation all of whose primary assumptions are members of and in which P occurs only in the scope of those assumptions, then P is quantiFcationally entailed by . 4 Soundness As with soundness for SD, we prove our result by proving something stronger: Every sentence in a PD derivation is qentailed 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 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: 1 P 1. Inductive step: If i P i for all i k, then k+1 P k+1. Well prove this by cases, one case for each rule that could have justiFed line k+1. Conclusion: or every line k in a derivation, k P k. I.e.: Every sentence in a PD derivation is qentailed by the set of assumptions with scope over it. 6 Soundness: Basis clause To prove: 1 P 1. = No interpretation mem 1 true but makes P 1 false. The Frst line of any derivation is an assumption....
View Full
Document
 Fall '10
 DerekAllen
 Philosophy

Click to edit the document details