Williamson's Metaphysical Modal Epistemology and Vacuism about Counterpossibles

Timothy Williamson has argued that our capacity for metaphysical modal judgement comes along with our capacity for counterfactual judgement. This passage gives a flavour of his view:

Humans evolved under no pressure to do philosophy. Presumably, survival and reproduction in the Stone Age depended little on philosophical prowess, dialectical skill being no more effective then than now as a seduction technique and in any case dependent on a hearer already equipped to recognize it. Any cognitive capacity we have for philosophy is a more or less accidental byproduct of other developments. Nor are psychological dispositions that are non-cognitive outside philosophy likely suddenly to become cognitive within it. We should expect cognitive capacities used in philosophy to be cases of general cognitive capacities used in ordinary life, perhaps trained, developed, and systematically applied in various special ways, just as the cognitive capacities that we use in mathematics and natural science are rooted in more primitive cognitive capacities to perceive, imagine, correlate, reason, discuss… In particular, a plausible non-skeptical epistemology of metaphysical modality should subsume our capacity to discriminate metaphysical possibilities from metaphysical impossibilities under more general cognitive capacities used in general life. I will argue that the ordinary cognitive capacity to handle counterfactual carries with it the cognitive capacity to handle metaphysical modality. (Williamson, The Philosophy of Philosophy (2007), p. 136. Found in Section 3 of the SEP article 'The Epistemology of Modality'.)

In order to argue for this, Williamson takes a schematic semantic story about counterfactual conditionals:

Where “A>B” express “If it were that A, it would be that B”, (CC) gives the truth conditions for subjunctive conditionals: A subjunctive conditional “A>C” is true at a possible world w just in case either (i) A is true at no possible world or (ii) some possible world at which both A and C are true is more similar to w than any possible world at which both A and ¬C are true.(Formulation from Sec 3 of the SEP article.)

On the basis of this, he proves the following equivalences:

(NEC) □A if and only if (¬A>⊥)
It is necessary that A if and only if were ¬A true, a contradiction would follow.

(POS) ◊A if and only if ¬(A>⊥)
It is possible that A if and only if it is not the case that were A true, a contradiction would follow.

(Renderings and spellings out from the SEP article. The box and diamond are metaphysical modal operators, and the falsum - which looks like an upside-down 'T' - represents a contradiction.)

But this only works because the schematic theory of counterfactuals Williamson adopts is understood as working against a background of a notion possible worlds, where Williamson understands this as the notion of metaphysically possible worlds. This theory deems true all counterfactuals with metaphysically impossible antecedents, and this is crucial to his demonstation of the equivalences on the basis of his assumed schematic theory.

There are lots of reasons not to adopt a theory of counterfactuals which uses the notion of metaphysical possibility in this way. One reason to move away from a theory like this is if you think that there are counterpossibles - counterfactual conditionals with metaphysically impossible antecedents - which have their truth-values non-vacuously. But you might be agnostic about that. For instance, you might be happy with metaphysical modal distinctions but doubt that there are any clear cases where countepossibles have truth-values non-vacuously. In that case you might see no good reason for the backdrop of worlds or scenarios in a theory of counterfactuals to be exactly the metaphysically possible ones. Or, you might be skeptical about the very distinction between metaphysical possibility and impossibility, in which case you won't want to understand the backdrop in a way that involves that distinction. And it seems that you can get most, or even all, of the theoretical benefits of a Stalnaker-Lewis approach to counterfactuals without using that distinction. It doesn't really seem to play a starring role in the theories. 

If a semantic theory for counterfactuals which does not draw on any bright line between metaphysically possible and metaphysically impossible scenarios is as good or better than the theory that Williamson uses to prove his equivalences, that seriously undermines the equivalences, and in turn Williamson's story about how we get metaphysical modal knowledge.

(And note that this turns on Williamson's understanding his chosen theory so that 'possible world' means 'metaphysically possible world' - even accepting an identically worded theory, but where 'possible world' is understood in a way which does not involve the distinction between metaphysical possibility and impossibility, would make Williamson's equivalences unavailable.)

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.