08 Slides--Boolean Connectives

08 Slides--Boolean Connectives - CS103A HO# 8 Boolean...

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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 worldthey 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 worldthey 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 Henkin-Hintikka 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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08 Slides--Boolean Connectives - CS103A HO# 8 Boolean...

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