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If p is false, the second sentence is true and the whole sentence is true. RULE OF INFERENCE: Disjunction The rules of disjunctive syllogism and addition emerge directly from the fact that when two sentences are connected by a DISJUNCTION, what’s being asserted is that at least one of the disjuncts are true. As a result, if we know that one of the disjuncts is false, we also know that the other disjunct must be true. This is the rule of DISJUNCTIVE SYLLOGISM. In addition, if we know that one sentence is true, then we know that the sentence formed using that sentence, a disjunction, and another sentence whose truth value we do not know, is also true. We know this is true because only one disjunct has to be true for the disjunctive compound to be true. This is the rule of ADDITION. DISJUNCTIVE SYLLOGISM ADDITION Given p V q and Given p , we can deduce ~p we can deduce p V q q Because one of the disjuncts because p is true and pV q is must be true, and p is not true regardless of the truth value of q, just because p is true RULES OF REPLACEMENT: DISJUNCTION Disjunctions are commutative. In other words, the order of conjuncts or disjuncts in any given formula doesn’t make a difference to the truth of the formula. As a result, the commutation rule of replacement allows us to substitute one formula for another when the only difference between the two is the order of the conjuncts/disjuncts. Furthermore, when a formula contains just conjunction or just disjunctions, it doesn’t matter whether you put one part of the whole formula within parentheses, or another part. As a result, when given a formula of at least three sentences, two of which are in parentheses, you COMMUTATION ASSOCIATION Given p V q, we can deduce Given p V (q V r), we can deduce q V p (p V q) V r RULES OF REPLACEMENT: V, &, ~ DISTRIBUTION The formula p & (q V r) can (1) replace or can (2) be replaced at any time during a deduction by the formula (p & q) V (p & r). The formula p V (q & r) can (1) replace or can (2) be replaced at any time during a deduction by the formula (p V q) & (p V r). 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