Validity

The term validity as it occurs in logic refers generally to a property of deductive arguments, although many logic texts apply the term to statements as well (a statement is a sentence that “has a truth value,” i.e., that is either true or false). For the purposes of this article, an argument is a set of statements, one of which is the conclusion and the rest of which are premises. The premises are reasons intended to show that the conclusion is, or is probably, true.

When an argument is set forth to show that its conclusion is true (as opposed to probably true), then the argument is intended to be deductive. An argument set forth to show that its conclusion is probably true may be regarded as inductive. To say that an argument is valid is to say that the conclusion really does follow from the premises. That is, an argument is valid precisely when it cannot possibly lead from true premises to a false conclusion. The following definition is fairly typical:


 * An argument is deductively valid if it cannot possibly have all true premises and a false conclusion.

An argument that is not valid is said to be ‘’invalid’’.

The following is a famous example of a deductively valid argument:


 * All men are mortal


 * Socrates is a man


 * Therefore, Socrates is mortal.

What makes this a valid argument is not the mere fact that it has true premises and a true conclusion, but the fact of the logical impossibility of things being otherwise. No matter how the universe might be constructed, it could never be the case that this argument should turn out to have simultaneously true premises but a false conclusion. The above argument may be contrasted with the following invalid one:


 * All men are mortal


 * Socrates is mortal


 * Therefore, Socrates is a man

In this case, there is no impossibility of true premises but false conclusion: it is easily imagined that there is a woman named ‘Socrates’, so that in fact the above premises would be true but the conclusion false—hence it is possible that the argument has true premises and a false conclusion. This possibility is what constitutes invalidity. (Although whether or not an argument is valid does not depend on what anyone could actually imagine to be the case, this approach helps us evaluate some arguments.)

A standard view is that whether an argument is valid is a matter of the argument’s logical form. Many techniques are employed by logicians to represent an argument’s logical form. A simple example, applied to the above two illustrations, is the following: Let the letters ‘P’, ‘Q’, and ‘s’ stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:


 * All P are Q


 * s is a P


 * Therefore, s is a Q

Similarly, the second argument becomes:


 * All P are Q


 * s is a Q


 * Therefore, s is a P.

These abbreviations make plain the logical form of each respective argument. At this level, notice that we can talk about any arguments that may take on one or the other of the above two configurations, by replacing the letters P, Q and s by appropriate expressions. Of particular interest is the fact that we may exploit an argument's form to help discover whether or not the argument from which it has been obtained is or is not valid. To do this, we define an “interpretation” of the argument as an assignment of sets of objects to the upper-case letters in the argument form, and the assignment of a single individual member of a set to the lower-case letters of the argument form. Thus, letting P stand for the set of men, Q stand for the set of mortals, and s stand for Socrates is an interpretation of each of the above arguments. Using this terminology, we may give a formal analogue of the definition of deductive validity:


 * An argument is formally valid if its form is one for which no interpretation exists under which the premises are all true but the conclusion false.

As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.