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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 . – Take-Home Mid-Term will be posted on Friday. – Please submit any extra-credit solutions with (any one of) your homework assignment(s). There is no deadline for extra-credit. • 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 (truth-tables) 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