Symbolic Logic

Equivalent propositions or statements can be symbolized with the two-headed arrow " ." In the preceeding section we showed the first of De Morgan's rules:

Rules such as these are useful for simplifying and clarifying complicated expressions. Other useful rules are

Each of these rules can be verified by writing out its truth table.

A truth table traces each of the various possibilities. To check rule 4 with its three different statements, p, q, and r, would require a truth table with eight lines. On occasion one may want to know the truth value of an expression such as ((T V F) L Λ (F V T)) V ~ F where the truth values of particular statements have been entered in place of p1 q, etc. The steps in evaluating such an expression are as follows:

Such a compound expression might come from the run-on sentence, "Roses are red or daisies are blue, and February has 30 days or March has 31 days; or it is not the case that May is in the fall." Admittedly, one is not likely to encounter such a sentence in ordinary conversation, but it illustrates how the rules of symbolic logic can be used to determine the ultimate truth of a complex statement. It also illustrates the process of replacing statements with known truth values instead of filling out a truth table for all the possible truth values. Since this example incorporates five different statements, a truth table of 32 lines would have been needed to run down every possibility.