Contradictory Premises and the Notion of Validity

When evaluating arguments in philosophy, it can be tempting to call an argument 'invalid' if you determine that it has contradictory premises. For example, in an introductory philosophy course at the University of Sydney, students are taught that a particular argument for the existence of God - called the Argument from Causation - is invalid because two of its premises contradict each other. It is tempting to call such an argument invalid because we can determine a priori that it is not sound, i.e. that it isn't both valid and such that its premises are true. But on a classical conception of validity, any argument with contradictory premises counts as valid, since it is impossible for all the premises of an argument with contradictory premises to be true, and so a fortiori impossible for the argument to have true premises and false conclusion.

I have heard this anomaly explained away by appeal to the fact that, while an argument with contradictory premises may count as formally valid, we are looking at informal validity, and in an informal sense perhaps any argument with contradictory premises should count as invalid. But I don't think that's right. If 'formal' is meant to signal that we are not interested in the meanings of non-logical terms and are only interested in what can be shown on the basis of the form of the argument, then that is clearly a different issue: premises could be determined to be contradictory on the basis of form alone, or in part on the basis of the meanings of the non-logical terms. The issue of contradictory premises is similarly orthogonal to the issue of 'formality' if 'formal' is instead meant to signify something like 'in an artificial language' or 'in a precise mathematical sense'. 

In fact, it's arguable that the standard treatment of validity of arguments in classical formal logic should be supplemented, so that an argument counts as valid iff it has no countermodel and its premises are jointly satisfiable.

If we defined 'valid' that way in classical logic, then to test an argument for validity using the tree method, you might have to do two trees. First, one to see if the premises can all be true together. If the tree says No, the argument is invalid and we can stop, but if the tree says Yes, then we do another tree to see if the premises together with the negation of the conclusion can all be true together, and if the tree says No, the argument is valid.

Whether or not it's worth adopting in practise, it is worth noting that this augmented definition of 'valid' in classical logic seems to correspond more closely to the ordinary, informal notion of deductive validity than the usual definition. This even delivers at least one of the desiderata which motivate relevance logic.

However, note that while we seem pretty disposed to call an argument invalid if it has contradictory premises, there is no equally strong tendency to say corresponding things using 'follows from', 'is a consequence of', or 'implies'. This is interesting in itself. It looks like, when we're talking about implication, our focus is on the putative implier or impliers and what can be got out of them, whether or not they're true. By contrast, when we talk about arguments, we're often more focused on the conclusion and whether it is shown to be true by the argument in question, so that validity is treated as one of the things we need to verify along the way. If validity is playing that role, it makes sense to declare an argument invalid if we work out that its premises can't all be true.