ABSTRACT
Sentences may be combined in various ways to form more complicated sentences. We shall consider only truth-functional combinations, in which the truth or falsity of the new sentence is determined by the truth or falsity of its component sentences. Negation is one of the simplest operations on sentences. Although a sen-
tence in a natural language may be negated in many ways, we shall adopt a uniform procedure: placing a sign for negation, the symbol :, in front of the entire sentence. Thus, if A is a sentence, then :A denotes the negation of A. The truth-functional character of negation is made apparent in the follow-
ing truth table:
A :A T F F T
When A is true, :A is false; when A is false, :A is true. We use T and F to denote the truth values true and false. Another common truth-functional operation is the conjunction: ‘‘and.’’ The
conjunction of sentences A and B will be designated by A ^ B and has the following truth table:
A B A ^ B T T T F T F T F F F F F
A ^ B is true when and only when both A and B are true. A and B are called the conjuncts of A ^ B. Note that there are four rows in the table, corresponding to the number of possible assignments of truth values to A and B. In natural languages, there are two distinct uses of ‘‘or’’: the inclusive and
the exclusive. According to the inclusive usage, ‘‘A or B’’ means ‘‘A or B or both,’’ whereas according to the exclusive usage, the meaning is ‘‘A or B, but
_, for Its truth table is as follows:
A B A _ B T T T F T T T F T F F F
Thus, A _ B is false when and only when both A and B are false. ‘‘A _ B’’ is called a disjunction, with the disjuncts A and B. Another important truth-functional operation is the conditional: ‘‘if A, then
B.’’ Ordinary usage is unclear here. Surely, ‘‘if A, then B’’ is false when the antecedent A is true and the consequent B is false. However, in other cases, there is no well-defined truth value. For example, the following sentences would be considered neither true nor false:
1. If 1þ 1 ¼ 2, then Paris is the capital of France. 2. If 1þ 1 6¼ 2, then Paris is the capital of France. 3. If 1þ 1 6¼ 2, then Rome is the capital of France.
