Monday, February 16, 2009

Validity as a Modal Operator

Since validity is a modal notion, it makes sense for it to be described by a modal operator. Here's a gesture at one way one might do it.


The validity operator applies to arguments, of course, rather than propositions. There are two tricky things, related to each other, that you have to work out in order to have a modal logic that takes arguments rather than propositions:

(1) What is the default modality (in most modal systems 'true' or 'actual' is used as the default modality, neither of which make sense here)?
(2) What is the negation of a, ~a?

Without good answers to these, there's really no hope of an interesting modal formal description of validity. However, they can both be answered. Think through a moment what validity means. If an argument is valid, then in all the kinds of situations to which it may relevantly be applied, the conjunction of the premises and the conclusion will correctly describe the situation. By 'correct description' I don't mean, true, exactly; rather that the premises have a bearing on the conclusion so that either it is true or it would be true if the premises were true. (Or, in other words, a means that the argument 'works' in the relevant kind of situation.) So this gives us the default modality: a, read modally, tells us that the conjunction of the premises and the conclusion correctly describes a given relevant kind of situation. Given this we can easily add our validity operator, with the following interpretations:

Va: any relevant kind of situation is one in which a (a is a valid argument)

~Va: there is a relevant kind of situation in which ~a (a is an invalid argument)

Thus we have validity and invalidity. But we'd like to know what ~V~a is. To pin this down, we start with our interpretation of the default modality: the conjunction of premises and conclusion correctly describes a given relevant kind of situation. The negation of this, ~a, is then suggestive of a counterexample: a given relevant kind of situation which the conjunction of the premises and the conclusion does not correctly describe. This gives us the other operator; we can call it viability, and understand it in the following way:

*a: this is shorthand for ~V~a

*a: there is some relevant kind of situation in which a (a is a viable argument)

~*a: any relevant kind of situation is one where ~a (a is a nonviable argument)

Viability is actually as important as validity, although more often overlooked; an argument may be viable but not valid, in which case it does not guarantee its conclusion in every relevant kind of situation, but the truth of its premises is not irrelevant to the truth of the conclusion, either.


It's all well and good to have interpretations of modal operators, but you need rules in order to do things with them. It's clear enough that the standard sorts of rules of inference won't work here, since they are designed to handle propositions rather than arguments. Exactly what the best group of operators will be, I'm not sure, although I suspect a final system would have a way of conjoining and disjoining arguments that is analogous, but not identical, to propositional conjunction and disjunction. What I want to do here is simply identify two rules with which you can do a great deal. First, let |- indicate that one can infer the right from the left. Then we can have a rule:

Va |- a |- *a

This makes the modal system analogous to M, and can easily be seen to be right under our interpretation: If an argument is valid, the conjunction of its premises and conclusion correctly describes the kind of situation, in the sense above; and if this defaul modality is true, there is some relevant kind of situation that is correctly described.

In addition, we want to be able to make comparisons between arguments. Let (Va : Vb) indicate that the validity of a and b are linked, so that if a is valid, so is b, and vice versa. In other words, they have the same relevant structure. Then we want an inference rule such that

Va, (Va : Vb) |- Vb

This is a typical way we argue from scratch about validity: we identify a particular argument as valid and generalize this conclusion to all arguments with the same relevant structure; combined with the fact that ~a indicates a counterexample, and you already have a pretty useful selection of tools for arguing about validity. I'm sure that additional descriptions of other types of inference about validity, invalidity, viability, and nonviability, could be added; this is just a fragment to show that it can be done.

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