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Unformatted text preview: Lecture 5: Equivalence, consistency, implication, & validity 1. Relations between two statements: equivalence, consistency, implication 2. Relations between three or more statements: equivalence, consistency, joint implication 3. Argument validity Relations between two statements Just as it is often helpful to know whether an individual statement is a tautology, contradiction, or contingency, it is also often helpful to know what relations hold between two statements. Equivalence Two statements are equivalent iff they have identical truth columns. To test for equivalence, construct a joint truth table for the two statements and compare their truth columns. If the columns are identical, then the statements are equivalent. If they are not identical, then they are not equivalent. Consistency Two statements are consistent iff it is possible for them to both be true at the same time. To test for consistency, do a joint truth table for the two statements. If there is a row (one or more) on which both statements are T, then they are consistent. If Implication Statement ! implies statement ! iff: if ! is true, then ! must be true. To see whether ! implies ! , do a joint truth table for ! and ! , and look for a row on which ! is T and ! is F (a counter-example row ). If there IS a counter-example row, then then ! does NOT imply ! ; if there is NOT a counter- example row, then ! DOES imply ! . 1. ~ (P ! Q) 2. ~ P ! ~ Q P Q ~ P ~ Q P ! Q ~ (P ! Q) ~ P ! ~ Q T T F F T F F T F F T F T F F T T F F T F F F T T F T T Are (1) and (2) equivalent? No 1. ~ (P ! Q) 2. ~ P ! ~ Q P Q ~ P ~ Q P ! Q ~ (P ! Q) ~ P ! ~ Q T T F F T F F T F F T F T F F T T F F T F F F T T F T T Are (1) and (2) consistent? Are (1) and (2) consistent?...
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- Spring '08