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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements & Such • The Mountain Goats : The Sunset Tree • Administrative Stuff – HW #3 will be returned today. Resubs due Friday (as usual). + When you turn in resubmissions, make sure that you staple them to your original homework submission . – TakeHome MidTerm will be posted on Friday. – Please submit any extracredit solutions with (any one of) your homework assignment(s). There is no deadline for extracredit. • Today: Chapter 4 — Natural Deduction Proofs for LSL – Validity ( ) vs Proof ( ` ) — some introductory remarks. – Our natural deduction proofs system for LSL. – Soon — MacLogic — a useful computer program for proofs. + Natural deductions are the most challenging topic of the course. UCB Philosophy Chapter 4 10/08/08 Branden Fitelson Philosophy 12A Notes 2 ' & Chapter 4 Introduction: Truth vs Proof ( vs ` ) • Recall: p q iff it is impossible for p to be true while q is false. • We have methods (truthtables) for establishing and claims. These methods are especially good for claims, but they get very complex for claims. Is there another more “natural” way to prove ’s? Yes! • In Chapter 4, we will learn a natural deduction system for LSL. This is a system of rules of inference that will allow us to prove all valid LSL arguments in a purely syntactical way (no appeal to semantics). • The notation p ` q means that there exists a natural deduction proof of q from p in our natural deduction system for sentential logic. • [ p ` q is short for [ p deductively entails q . • While has to do with truth , ` does not . ` has only to do with what can be deduced , using a fixed set of formal, natural deduction rules. UCB Philosophy Chapter 4 10/08/08 Branden Fitelson Philosophy 12A Notes 3 ' & $ % • Happily, our system of natural deduction rules is sound and complete : – Soundness . If p ` q , then p q . [no proofs of in validities!] – Completeness . If p q , then p ` q . [proofs of all validities!] • We will not prove the soundness and completeness of our system of natural deduction rules. I will say a few things about soundness as we go along, but completeness is much harder to establish (140A!). • We’ll have rules that permit the elimination or introduction of each of the connectives &, → , ∨ , ∼ , ↔ within natural deductions. These rules will make sense, from the point of view of the semantics. • A proof of q from p is a sequence of LSL formulas, beginning with p and ending with q , where each formula in the sequence is deduced from previous lines, via a correct application of one of the rules ....
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This note was uploaded on 09/23/2009 for the course PHIL 12A taught by Professor Fitelson during the Spring '08 term at Berkeley.
 Spring '08
 FITELSON
 Philosophy

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