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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Overview of Todays Lecture Todays Music: Cymande The mid-term is tomorrow, 6/10 (in class). Bring blue books! Ive posted (and discussed) a sample mid-term. It has the same structure and complexity as the actual mid-term (good study guide). I have posted HW #3, which is due on tomorrow @ 4pm in the drop box. Its all chapter 3 problems truth-table methods for validity-testing. I posted revised versions of lecture #6 and my short method handout. MacLogic a useful computer program for natural deduction. You might want to download MacLogic at this point . . . Well be using it very soon (and for the rest of the term) . . . See http://fitelson.org/maclogic.htm Today: Chapter 4 Continued UCB Philosophy Chapter 4 (Contd) 06/09/08 Branden Fitelson Philosophy 12A Notes 2 ' & 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 (Contd) 06/09/08 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!). Well 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 . Generally, we will be talking about deductions of formulas q from sets of premises p 1 ,...,p n . We call these [ p 1 ,...,p n ` q s sequents ....
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- Fall '06