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Unformatted text preview: CS103A HO# 8 Boolean Connectives 1/16/06 1 Truth Tables for Boolean Connectives Negation P ¬ P T F F T ¬P is true if and only if P is false. Disjunction P Q P ∨ Q T T T T F T F T T F F F P ∨ Q is true if and only if P is true, Q is true, or both are true. Conjunction P Q P ∧ Q T T T T F F F T F F F F P ∧ Q is true if and only if both P and Q are true. In the truth table, we are not concerned with what P and Q mean in some world—they stand for two independent propositions that can be true or false. Conjunction P Q P ∧ Q T T T T F F F T F F F F P ∧ Q is true if and only if both P and Q are true. In the truth table, we are not concerned with what P and Q mean in some world—they stand for two independent propositions that can be true or false. To say that the connectives are truth functional means that the value of a sentence using the connective can be determined by the truth value of its constituent parts. Truth Tables for Boolean Connectives The HenkinHintikka Game Playing the game may help you understand a difficult sentence. The real significance is that the game adds to you understanding of how the connectives work. CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false P ∧ Q false One of P, Q false Both of P, Q true P ∨ Q true One of P, Q true Both of P, Q false P ∨ Q false Both of P, Q false One of P, Q true ¬P true P false P true ¬P false P true P false CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false P ∧ Q false One of P, Q false Both of P, Q true To show that P ∧ Q is false, we show that one of P, Q is false. ¬(P ∧ Q) ¬P ∨ ¬Q CS103A HO# 8 Boolean Connectives 1/16/06 2 CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false P ∧ Q false One of P, Q false Both of P, Q true To show that P ∧ Q is false, we show that one of P, Q is false. ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false P ∧ Q false One of P, Q false Both of P, Q true To show that P ∧ Q is false, we show that one of P, Q is false. ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q Our shorthand representation of this fact is ¬(P ∧ Q) ⇔ ¬P ∨ ¬Q CLAIM DEMONSTRATION REFUTATION P ∧ Q true Both of P, Q true One of P, Q false P ∧ Q false One of P, Q false Both of P, Q true To show that P ∧ Q is false, we show that one of P, Q is false....
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 Winter '07
 Plummer,R
 Computer Science

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