Truth tables can be used not only to test arguments for validity; they can also be used to determine if two expressions are equivalent in terms of truth value. The rules of equivalence permit us to substitute one expression for another. If we have two logical expressions which are turth-functionally equivalent to one another, then we can freely substitute them as needed. (The full significance of equivalence will become apparent at the end of Lesson 12 in the derivation that we will do there.)
A truth table for testing equivalence is set up in the same way as one which tests arguments for validity. But since we are testing expressions for equivalence rather than arguments for validity, we will not be identifying lines and writing "OK" or "No". Instead, we will notice whether the columns of truth values are exactly the same or different. If two expressions have exactly the same truth-values, then they are truth-functionally equivalent--and can be substituted for one another.
As an example, let's set up a truth table with two variables and display three expressions:
1 2 3 p q ~p Ú q ~(p É q) p & ~q T T T F F T F F T T F T T F F F F T F F
The expressions in columns two and three have identical truth values. They are truth-functionally equivalent. To show this, draw a box around each of them and write "eqivalent". Or, write a sentence, such as, "~(p É q) and p & ~q are equivalent, for they have identical truth values."
Exercise 11 provides you with an opportunity to answer some questions about truth tables.
Copyright © 1999, Michael Eldridge