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Today’s Topics

Today’s Topics. Thinking about proofs Strategies and hints Notes on symbolization and proof construction. Thinking About Proofs. Proofs in logic work just like proofs in geometry

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Today’s Topics

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  1. Today’s Topics • Thinking about proofs • Strategies and hints • Notes on symbolization and proof construction

  2. Thinking About Proofs • Proofs in logic work just like proofs in geometry • The 18 rules we have allow us to manipulate a basic set of assumptions (the premises) so as to show that the conclusion is a logical consequence of them. • A proof is a set of instructions on how to get from the premises to the conclusion.

  3. Constructing a proof is like giving instructions. The question is “How do I get there (the conclusion) from here (the premises)?” • The rules are the allowable moves or turns you can take. • Proceed stepwise. Suppose you want to get to D from A, B, and C. Well, if from A and B you can get to E, and from E and C you can get to D, you have your instructions. • That is all there is to constructing proofs

  4. Basic Strategic Hints • Argument forms are patterns. Learn the patterns and look for them. • Inference rules can be grouped according to the types of statements on which they operate. • Short statements are your friends! • Work backwards from the conclusion. • BE FLEXIBLE. When stuck, experiment. Try steps and then search for familiar patterns.

  5. Develop Goal Lines and work toward them. • Ask yourself, “What line, if I had it, would allow me to get to the conclusion?” • Make that line a goal and work towards it. • Think in terms of equivalences—ask yourself “To what is the conclusion (or the line you want) equivalent?” Can you get to that Version?

  6. if you need one disjunct of a disjunction, scan the remaining lines for the negation of the other disjunct and use DS • If you need the consequent of a conditional, look for the antecedent and use MP • If you need the negation of the antecedent of a conditional, think MT • If there is a statement letter in the conclusion that occurs nowhere in the premises, use addition

  7. A Few More Strategic Hints • Simplify conjucntions • Use DeMorgan to turn negations of disjunctions into conjunctions that can be simplified • Use commutation and association to isolate components that fit other patterns (DS or Simp) • To derive a conditional, think HS or IMPL • To derive a disjunction, think ADD or CD

  8. Justifying Steps in a Proof • Each line in a proof must be justified. • Premises justify themselves, we assume them to be true. • Derived Lines (those lines after the premises) must be justified according to valid rules of inference or equivalence as following from previous lines.

  9. Supplying Justifications

  10. Homework, due Friday, October 5, 2001 • Justification of Proofs, 1-30 • Constructing Proogs, 1-15

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