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Logic in Proofs (Valid arguments) A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is taken as being true, and C is a statement which follows from H by logical reasoning. Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) An argument in logic is a way to reach a conclusion based on prior statements. An argument which has no counterexamples is said to be valid. Rules of inference correspond to tautologies Modus ponens: (p v (p6q)) 6 q is a tautology. For compound propositions P and Q, (P v (P6Q)) 6 Q is also a tautology. This rule may be expressed in short form as modus ponens: P and P 6 Q Q Hypothetical Syllogism [(p6q) v (q6r)] 6 (p6r) is a tautology In short form, hypothetical syllogism: P 6 Q and Q 6 R P 6 R. Modus tollens ((p6q) v ¬q) 6 ¬p is a tautology. In short form, modus tollens: P 6 Q and ¬Q ¬P Other Rules of Inference in short form: Addition: P P w Q simplification: P v Q P disjunctive syllogism: P w Q and ¬P Q conjunction: P and Q P v Q A formal proof is a sequence of propositions P1, P2, ..., P n, C in which each proposition is either a hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule of inference. In creating a formal proof we use Substitution Rules Names don’t matter in a tautology (only the form)! Equivalences do not change truth value! Consider a proof of [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) A sequence of statements that does not correspond to a formal proof is called a fallacy. Fallacies often imitate the form of valid rules of inference, but will have a counterexample. That is, there is at least one instance of truth values of the component variables which allows for true hypotheses and a false conclusion. Common fallacies: Affirming the conclusion (reasoning by converse) [(p6q)vq]6p is not a tautology, so the argument P6 Q / Q / P is a fallacy. Denying the hypothesis (reasoning by inverse) [(p6q)v¬p]6¬q is not a tautology, so the argument P6 Q / ¬P / ¬Q is a fallacy. Circular reasoning (begging the question) occurs when an implicit assumption is strong enough to get the desired conclusion. [L6(A6L), (Assume L) then (A6L)] is fallacious because L could be false. Jumping to conclusion (hasty generalization): [P6Z, Q6Z, ...,Y6Z; Z] is fallacious because all of P, Q, ..., Y could be false.