Lec1.5-7 - L1.5-7 Lecture Notes: Methods of Proof All...

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L1.5-7 Lecture Notes: Methods of Proof All proofs are alike! This sounds like a broad and capricious generalization but it is nonetheless true. Any proof begins with “axioms” or “premises,” (or whatever you wish to call them), which are (assumed to be) true propositions. These premises are accepted unconditionally as true, or as “given.” That’s how the game is played. The truth of premises is a matter for science or mathematics, not logic. We then apply “logic” or “common sense” or “rules of inference” which guarantee that further truths follow indubitably from previous truths, starting with the premises. More formally: Rule of inference: Two Parts: (1) Premises = One or more propositions given as True. (2) Conclusion that follows logically from the premises. Follows logically : We say that the conclusion C follows logically from the premises P 1 . . P n whenever P 1 . . P n C is a tautology. Sometimes we say the premises P 1 . . P n logically imply the conclusion and write: P 1 . . P n C. (NB: The double shafted arrow is not a logical operation. It is a statement about a logical operation — a metastatement. A B means that A B is a tautology. It is the same type of situation as with logical equivalence, where A B means A B is a tautology.) If you understand how the truth table for works, you see that an implication can only be a tautology when for all possible values of atomic propositions, the right side of the arrow (the consequent) can never be false as long as the left side of the arrow (the antecedent) is true. That is exactly the effect we want: The conclusion cannot be false as long as all of the premises – i.e., their conjunction – are true. Well, this is a bit abstract. Here is a simple example of what’s going on. Let P 1 be a premise = p q Let P 2 be another premise = p 1
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Let C be a conclusion = q In words this rule of inference says whenever the statement “p implies q” is true, and at the same time “p” is true, then “q” must be true, regardless of what the propositions represented by p and q happen to be. This is a rule of inference because C follows logically from P 1 and P 2 because the conjunction of the premises implies the conclusion is a tautology, as we can see from the truth table: p q p q p/\(p q) [p (p q)] q T T T T T T F F F T F T T F T F F T F T This rule of inference corresponds to one form of simple reasoning that people do everyday in their use of language. Consider: If it rains the dog gets wet. We saw earlier that this can be symbolized logically using two propositions: p: it rains q: the dog gets wet p q: If it rains, the dog gets wet. If this last statement is true, and if
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This note was uploaded on 03/26/2008 for the course CSC 226 taught by Professor Watkins during the Spring '08 term at N.C. State.

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Lec1.5-7 - L1.5-7 Lecture Notes: Methods of Proof All...

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