The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in logic refers generally to a property of particular statements and deductive arguments. Although validity and logical truth are synonymous concepts, the terms are used variously in different contexts. Whether or not logical truth is analytic truth is a matter of clarification.
Validity of Arguments
When an argument is set forth to prove 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, whenever all premises are true, the conclusion is also necessarily true.
An argument that is not valid is said to be “invalid’’.
An example of a valid argument is given by the following well-known syllogism:
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 necessity of the conclusion, given the two premises. 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, the conclusion does not follow inescapably from the premise: a universe is easily imagined in which ‘Socrates’ is not a man but a woman, so that in fact the above premises would be true but the conclusion false. This possibility makes the argument invalid. (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 such that for each interpretation under which the premises are all true also the conclusion is true.
As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.