INTRODUCTION TO SENTENTIAL LOGIC
TRUTH FUNCTIONAL SENTENCE CONNECTIVES
Mathematical functions input numbers and output numbers. Anything that gives one certain output for any given input can be called a function. [pic]
Sentence connectives are like truth functions because the truth value of a compound sentence (such as A & B) is determined solely by the truth values of its simple claims. That is, if you know whether sentence A and sentence B are true, then you know whether sentence A & B is true.
AND: The truth function for the connective “AND” gives a truth value of true (T) whenever both conjuncts are true and false whenever one of the conjuncts are false (F). For example, if sentence A is true and sentence B is true, then sentence A & B is also true.
OR: The truth function for the connective “OR” gives a truth value of true (T) whenever one of the disjuncts are true and false whenever both disjuncts are false (F). For example, if sentence A is true and sentence B is false, the sentence A V B is true (since only one needs to be true).
IF/THEN: The IF/THEN or IMPLIES connective produces a CONDITIONAL sentence, which connects an antecedent (the condition, after the ‘if’) and a consequent (after the ‘then’) in a particular way. Namely, a conditional sentence asserts that whenever the antecedent is true, so is the consequent. For example, the sentence A ( B is true if A is true and B is true. The only way for a conditional claim to be false is for the antecedent (before the ‘if’) to be TRUE and the consequent to be FALSE. Otherwise the conditional sentence is always true (T).
Many other words can be used to show a relationship of implication between two simple claims. For example,
P implies Q P ( Q
Q, provided that P P ( Q
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A FUNCTION
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