Validity

Validity is a property of arguments.

An argument is valid just if it would be impossible for its premises all to be true and its conclusion false simultaneously.

Or, which comes to the same thing:

An argument is valid just if the set consisting of its premises and the negation of its conclusion is inconsistent.

(When is one sentence the negation of another? Unfortunately there is no uniform way of forming a negation in ordinary English. The most general thing we can say is that the negation of a declarative sentence is true if the original sentence is false, and false if the original sentence is true. So "It is not raining" is the negation of "It is raining"; but "Someone is not here" is not the negation of "Someone is here", because they could both be true; and "You must not do that" is not the negation of "You must do that", because they could both be false. One of the virtues of the language of the propositional calculus, which we shall encounter shortly, is that it provides a uniform way of expressing negation.)

The set consisting of the premises of an argument together with the negation of its conclusion is called its counterexample set.

So the following is yet another way of defining validity:

An argument is valid just if its counterexample set is inconsistent.

Notice a number of things:

1. An argument may be valid or invalid, but not consistent or inconsistent. A set of sentences may consistent or inconsistent but not valid or invalid.

2. In a valid argument, there is no need for the premises to be true. (One may want the premises of one's argument to be true; but their not being true does not affect its validity.

            Socrates is a woman. All women are mortal. So, Socrates is mortal.

            Socrates is a man. All men are immortal. So, Socrates is immortal.

            Socrates is a woman. All women are immortal. So, Socrates is immortal.

So one can have valid arguments with one or more false premises; and one can have valid arguments with a false conclusion. The one thing one cannot have is a valid argument with true premises and a false conclusion.

3. (This may come as a surprise.) If the premises of an argument are inconsistent, the argument is valid according to our definitions. Why? Because, if the premises are inconsistent, it would be impossible for them all to be true together; but in that case it would also be impossible for them all to be true and the conclusion false simultaneously whatever the conclusion is.

So the following are both valid arguments:

            2+2=5. So it is raining.

            2+2=5. So it is not raining.

Here is another way of seeing this. If a set of sentences is inconsistent, and one adds another member to the set, the new set must be inconsistent whatever the new member is. So, if the set of premises of an argument is inconsistent, its counterexample set will be too.

4. (This should not be surprising now.) Suppose that it would be impossible for the conclusion to be false. Then the argument must be valid, whatever the premises may be. For, if it would be impossible for the conclusion to be false, it would also be impossible for the premises to be true and the conclusion false simultaneously.