Chapter 5: More Efficient Semantic Methods

 Truth tables provide a completely effective method for determining the semantic status of any wff, for determining whether a set of wffs is consistent or inconsistent, for determining whether a pair of wffs are logically equivalent, and for assessing the validity/invalidity of arguments.  By characterizing the method as effective, we mean that if one follows the rules for constructing truth tables, a the semantic property in question can be determined by a finite application of those rules, for any wff, regardless of its length.  The fact that there is an effective method or procedure, or what logicians often call a decision procedure is an important theoretical result.  We will see later in this text that there are logical systems for which there is no such decision procedure.

Though completely effective, truth tables are unwieldy. Recall that the size of the truth table is determined by the number of distinct sentence letters in the wffs we evaluate. The truth table will have 2ⁿ rows, where n is the number of distinct sentence letters. Suppose we were to examine a set of sentences for consistency which had 8 distinct sentence letters. That's 2⁸ or 256 rows! Clearly such a large truth table is much to unwieldy. Hence the search for more efficient methods. 

We will canvass three methods in this chapter. They are the Gappy Truth Table Method, the Counterexample Method, and the Truth Tree Method.