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Handout 2: Propositional Equivalences and Proofs

January 13, 2003

Tautology and Contradiction

An important class of compound proposition consists of those that are always true no matter what the truth values of the variables p, q, etc., are. Such a compound proposition is called a tautology.

Why would we ever be interested in a proposition that is always true, and hence is pretty boring? The answer is that we are going to be dealing with some rather complicated-looking propositions which we hope to show are true, and the way that we will show their truth will be by using other propositions that are known to be always true.

Some classical tautologies are: , , .

A compound proposition that is always false is called a contradiction. Clearly a compound proposition P is a contradiction iff is a tautology.

Some classical contradictions are: .

Logical Equivalence and Implication

Two compound compositions P and Q are regarded as logically equivalent if they have the same truth values for all choices of truth values of the variables p, q, etc. In other words, the final columns of their truth tables must be the same. When this occurs we write . Since has truth values true precisely when the truth values of P and Q agree, we see that:

if and only if is a tautology.

Some logically equivalent propositions are: , , .

The expression is an assertion, namely that P and Q are logically equivalent, i.e., is a tautology. With , it can be true or false. But with , we say it is true. And so, must be a tautology.

Similarly, given two compound propositions P and Q, we say that P logically implies Q provided Q has truth value true whenever P has truth value true. We write when this occurs. Note that

if and only if is a tautology.

You must understand the difference between and and the difference between and clearly!

Theorems and Proofs

We use two useful substitution rules:

(a) If a compound proposition P is a tautology and if all occurrences of some variable of P, say, q, are replaced by the same proposition E, then the resulting compound proposition P* is also a tautology.

(b) If a compound proposition P contains a proposition Q and if Q is replaced by a logically equivalent proposition Q*, then the resulting compound P* is logically equivalent to P.

To see (a), consider a mathematical equation. If x+m = y+m, then you can replace m with n+2+k, and the equation will still hold.

To see (b), consider a Honda Accord car. It contains a Honda Accord engine. If you replace the engine with another Honda Accord engine, then the car with the new engine is still a Honda Accord.

These two substitution rules are the underlying principles with which we prove a theorem in propositional logic.

A theorem (argument) consists of some propositions , called it hypotheses (premises), and a proposition C, called its conclusion. A theorem with hypotheses and conclusion C is true provided

.

Thus the theorem is true if and only if is a tautology.

A formal proof of a theorem consists of a sequence of propositions, ending with the conclusion C, which are regarded as valid for any of several reasons. To be valid, a proposition may be one of the hypotheses, may be a known tautology, may be derived from propositions earlier in the sequence via the substitution rules or may be inferred from earlier propositions according to certain rules of inference. A proposition Q can be inferred from propositions provided .

A formal proof with a valid sequence of propositions is called a valid proof or valid argument. Regardless of what the conclusion is, if one or more of the propositions is invalid, then the argument is called a fallacy.

• Based on (Ross and Wright 1988).