Introduction to Logic, Unit 2 Lectures Kevin C. Klement UMass–Amherst A. Introduction to Derivations Problems with Truth Tables Truth tables are great. They can tell us almost anything we’d wish to know about a statement or argument in propositional logic. But they have two problems. 1. They grow in size exponentially. (Six atomics means 64 rows.) 2. They’re alien to the way we ordinarily think. Our new unit focuses on a new way of establishing the validity of an argument without these Faws. Step by Step Reasoning Consider the following form: p ! q p q This is an obviously valid form close to how we in fact reason. Now consider: A ! B B ! C C ! D A D This argument is not, strictly speaking, of the same form as the one above, and it is not quite as obviously valid. It is valid, nevertheless, and we can prove that by showing how the conclusion follows from a chain of smaller arguments similar to our ±rst example. To put it roughly:
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