Lecture-10

# Lecture-10 - Lecture 10 Further clarifications common...

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1. Negation 2. Statement variables 3. Operator specificity 4. Component order 6. Examples 5. Subproof requirements Lecture 10 Further clarifications, common errors, and examples The appearance of a tilde in the inference or replacement rules only indicates that one statement is the negation of another, not that it has to have a tilde in front of it. Negation ! ! ! 㱤! ! x. (P v Q) ~ T y. (P v Q) z. ~ T x, y MP Negation x. (P v Q) ~ T y. T z. ~ (P v Q) x, y MT ! ! ~ ! 㱤! ~ ! Negation x. ~ (P v Q) y. ~ P ! ~ Q ~ ( ! v ! ) :: ~ ! ! ~ ! y. ~ (P v Q) x. ~ P ! ~ Q y. ~ ( ~ P v ~ Q) x. P ! Q Greek letters are statement variables. They stand for any statement, simple or compound. If the same variable shows up more than once, this implies that the statement must be the same. Statement variables But, different variable letters don’t require that the statements be different. Statement variables ! ! ! " 㱤! ! " y. Z (R v S) x. ~ (P ! Q) Z z. ~ (P ! Q) (R v S) y. Z (R v S) x. ~ (P ! Q) Z z. ~ (P ! Q) R y. Z (R v S) x. (R v S) Z z. (R v S) (R v S)

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The opertors specified in the inference and replacement rules are not substitutable (except negation, as noted above). Operator specificity ! ! ! 㱤! ! x. (P v Q) v ~ T y. (P v Q) z. ~ T x, y MP Operator specificity x. ~ (P Q) ~ ( ! v ! ) :: ~ ! ! ~ ! y. ~ P ! ~ Q x DeM The disjunction, conjunction, and biconditional operators are order invariant, so the order of disjuncts, conjuncs, and bicoonditional components as they appear in the rules does not matter. Component order Conditionals are not order invariant, so their order matters. Component order ! v ! ~ ! 㱤! ! y. H K x. P v ~ (H K) z. P y. ~ P x. P v ~ (H K) z. ~ (H K) Component order y. D v B x. (D v B) ! ~ (H K) y. ~ (H K) x. (D v B) ! ~ (H K) ! ! ! 㱤!
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Lecture-10 - Lecture 10 Further clarifications common...

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