DEFINING VALIDITY

What constitutes a valid argument? Obviously, if we are to be persuaded by an argu­ment—if we are to accept its conclusion—we must believe the premises to be true. But this will not be enough by itself.

No NDP leader has been elected as Canadian Prime Minister.

Jack Layton was not elected as Canadian Prime Minister.

Therefore Jack Layton was an NDP leader.

At the time of writing, both these premises are true, but they give us no reason to accept the conclusion, even though that too is true. And even if the premises were not true, we could certainly imagine them to be so without feeling at all obliged to accept the conclu­sion. Thus whether or not the conclusion is true, we have not been given good grounds for accepting it. In such a case we say that the argument is invalid, or that the inference from the premises to the conclusion is invalid.

The root notion of validity is that accepting the premises ought to lead one to accept the conclusion. That is, it cannot be good reasoning if we can accept all the premises (even if we accept them only for the sake of argument) and still deny the truth of the conclusion. This motivates the following definitions:

An argument or inference is valid if and only if denying its conclusion is incompatible with accepting all its premises.

An invalid argument or inference is one that is not valid.

Here is an example to help clarify the definition.

A body’s mass increases without bound as it approaches the speed of light, c. But mass is directly proportional to energy, according to Einstein’s formula.

Therefore it would take an infinite amount of energy to accelerate a body to the speed of light.

Here if we understand and accept both the premises, this is incompatible with denying the conclusion, since the energy we use to accelerate the body must also increase with­out bound as the body’s velocity approaches the speed of light.

This definition of validity harks all the way back to the ancient Greek Stoic logician, Chrysippus. It has very wide applicability, because we will judge differently in different contexts whether statements are incompatible. In the above example, for instance, the context is that of modem physics. Given an understanding of mass, energy, acceleration, etc. the denial of the conclusion is incompatible with an acceptance of the premises, even if it is not logically inconsistent with them. Two statements are logically inconsis­tent if there is a logical contradiction between them, that is, if they are a statement and its negation, or if together they entail both a statement and its negation. For example, the statement “This figure is a circle” is logically inconsistent with the statement “This figure is not a circle,” since these two statements contradict one another. “This figure is a circle” is also incompatible with the statement “This figure is a square,” since if you know the meanings of ‘circle’ and ‘square,’ its being a circle precludes its being a square. Butjudging them incompatible depends on a knowledge of circles and squares, whereas the inconsistency of the first two statements does not depend on meaning or context. In purely formal contexts such as formal logic and mathematics, logical inconsistency is the appropriate standard of incompatibility. When we interpret incompatibility as logical inconsistency, we obtain a narrower kind of validity, formal validity, as we shall discuss in the next section. But if we adopt this criterion of validity universally, a very large class of arguments whose reasoning appears perfectly valid will come out as formally invalid. This is because they implicitly appeal to other standards of evidence appropriate to their contexts, and not to purely formal criteria.

We will also apply the over-arching Chiysippean definition to establish the validity of rules of inference in formal contexts, where we will interpret incompatibility as logical inconsistency. As a consequence, validity in such formal contexts will simply be identi­cal with formal validity. This has an important consequence for arguments with logically inconsistent premises. For since one cannot accept a contradiction, it is not possible for one to accept the premises of such an argument and also deny the conclusion, thus making the argument valid—whatever the conclusion might be. As we shall see, this agrees with the rules of formal validity, according to which anything at all follows from a contradiction.

Our definition of validity differs very subtly from one often found elsewhere, where a valid argument is defined as one “such that it is impossible that its premises are all true and its conclusion false.” One objection to this rival definition is that it is not easy to see how to apply it if the premises are not actually all true. A particular argument is only given once, with its premises variously having the values of true or false that they do; we cannot change their values to make them all true and then see whether the conclusion is false. We can, however, imagine an argument of the same form to have all true premises and a false conclusion, in which case the definition in terms of truth and falsity could then be applied to arguments of this form. We will take this approach in the next section, where we will see that arguments that are instances of a valid form are formally valid.^[7]

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