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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Overview of Today’s Lecture • Today’s Music: Cymande • The mid-term is tomorrow, 6/10 (in class). Bring blue books! – I’ve 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. – It’s 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 . . . – We’ll 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 (Cont’d) 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 (Cont’d) 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!). • 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 . • 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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This note was uploaded on 11/26/2011 for the course PHILOSOPHY 101 taught by Professor Buechner during the Fall '06 term at Rutgers.
- Fall '06