Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

This Section: 2. Logical Equivalence, Tautologies and Contradictions

1. Statements and Logical Operators Section 2 Exercises 3. The Conditional and the Biconditional Main Logic Page "Real World" Page

2. Logical Equivalence, Tautologies and Contradictions

We have already hinted in the previous section that certain statements are equivalent; for example, we claimed that (pq)r and p(qr) are equivalent — a fact we called the associative law for conjunction. In this section, we use truth tables to say precisely what we mean by logical equivalence, and we also study certain statements that are either "self-evident" ("tautological"), or "evidently false" ("contradictory").

We start with some more examples of truth tables.





Now we say that two statements are logically equivalent if, for all possible truth values of the variables involved, both statements are true or both are false. If s and t are equivalent, we write st. This is not another logical statement. It is simply the claim that the two statements s and t are equivalent. Here are some examples to explain what we mean.




Here are the two equivalences known as DeMorgan's Laws:

DeMorgan's Laws

If p and q are statements, then

    ~(pq) (~p)(~q)

    ~(pq) (~p)(~q)

Mechanically speaking, this means that, when we distribute a negation sign, it reverses and , and the negation applies to both parts.

A compound statement is a tautology if its truth value is always T, regardless of the truth values of its variables. It is a contradiction if its truth value is always F, regardless of the truth values of its variables. Notice that these are properties of a single statement, while logical equivalence always relates two statements.




When a statment is a tautology, we also say that the statement is tautological. In common usage this sometimes means simply that the statement is convincing. We are using it for something stronger: the statement is always true, under all circimstances. In contrast, a contradiction, or contradictory statement, is never true, under any circumstances.
Note that most statements are neither tautologies nor contradictions, as in the first three examples in this section.

Before turning to the exercises, we give a list of some important logical equivalences, most of which we have already encountered. (The verification of some of these appear as exercises below.) We shall add to this list as we go along.

Important Logical Equivalences: First List

The following logical equivalences apply to any statements; the p's, q's and r' s can stand for atomic statements or compound statements.

~(~p)pthe Double Negative Law
pqqpthe Commutative Law for conjunction
pqqpthe Commutative Law for disjunction
(pq)rp(qr) the Associative Law for conjunction
(pq)rp(qr) the Associative Law for disjunction
~(pq)(~p)(~q) DeMorgan's Laws
~(pq)(~p)(~q)
p(qr)(pq)(pr) the Distributive Laws
p(qr)(pq)(pr)
ppp Absorption Laws
ppp

1. Statements and Logical Operators Section 2 Exercises 3. The Conditional and the Biconditional Main Logic Page "Real World" Page

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble