Ever since his Philosophy of Logic was published in 1970 (2nd ed. 1986), W. V. Quine's view on logic has been under attack by all sorts of logical deviants: pluralists, intuitionists, relevant logicians, dialethists, fuzzy logicians, etc. Anyone who has felt the revisionist urge in logic should be rightly provoked by Quine's monism for classical logic. Since I have a devillish inclination towards proof-theoretical semantics, I find myself at least associated with the multitude of revisionists listed above. However, associated perhaps not by conviction, but by a common enemy. Let me give you a short and revealing example of Quine's rhetorics on the issue:
Classical quantification theory enjoys an extraordinary combination of depth and simplicity, beauty and utility. It is bright within and bold in its boundaries. Deviations from it are likely, in contrast, to look rather arbitrary” [Quine, 'Existence and quantification' (1969), pp. 112-113, in Ontological Relativity and Other Essays].
Now, of course, Quine's position has more content than the above quote admits, and there are in particular two arguments he employs to convince his readers of the superiority of classical logic. Firstly, there is the argument from completeness, that classical logic is, in a sense, the strongest logic which allows for a completeness result. This claim, however, is not even close to uncontroversial. See for instance L. Tharp's, 'Which logic is the right logic?, Synthese, 31:1–21, (1975). Moreover, if you resist the distribution of work presupposed by the distinction between provability and model-theoretical consequence, the result will not have the importance attributed to it by Quine. See for instance Etchemendy's The Concept of Logical Consequence. Secondly, there is an argument from translation, that any attempt at revising classical logic simply changes the subject (see Quine's Philosophy of Logic, ch. 7). This argument rests heavily on a Quinean theory of radical translation, so it is sufficient to say that the success of the former depends on the success of the latter. Additionally, there are substantial worries about whether or not Quine simply begs the question on the revisionists.
These are interesting debates in their own right, but it is not my intention to enter into them here. Rather, the idea is to briefly compare Quine's position to one of the chief contemporary proponents of classicism: Timothy Williamson. Williamson, as Quine, has made it his business to advocate the supremacy of classical logic, albeit with somewhat different motivations. Williamson's classicism is first of all connected to his epistemicism for vagueness. Whereas Quine's core worry was with quantification and ontology, Williamson focus is bivalence. But importantly, that is not to say that he is a classicist only for classical semantics. The following passage indicates the scope of his classicism:
Conditional proof, argument by cases and reductio ad absurdum [these are all invalidated by supervaluationism] play a vital role in systems of natural deduction, the formal systems closest to our informal deductions. [...] Thus supervaluationists invalidate our natural mode of deductive thinking.
In other words, Williamson outright states that any theory deviating from the above classical deductions invalidate our natural mode of deductive thinking. Now, why is it, the revisionist asks, that these deductions are considered integral to our natural mode of reasoning? Sure, instances of these deductions are applied frequently, but that is something quite different from saying that they are valid without further ado. Is Williamson going to provide us with an argument for the rational role of these deductions?
Perhaps not. Perhaps his classicist position is more or less an assumption. Indeed, there are some reasons to consider it as such. In his article 'Imaginiation, Stipulation and Vagueness', Philosophical Issues 8, (1997), he starts off with the following claims: "Humans are better at logic than at philosophy. When philosophical considerations leads someone to propose a revision of basic logic, the philosophy is more likely to be at fault than the logic" (p. 215). But perhaps you want to object that by 'basic logic', Williamson doesn't necessarily mean 'classical logic'. No, not necessarily - but that's what he's actually doing. In a following footnote, he says: "Michael Tye plausibly suggests that humans are also better at non-classical logic than at philosophy. Indeed they are, when they use a classical metalogic." (ibid., n. 2). If you are not puzzled by these claims, then I propose that you are either (1) Williamson himself or (2) the ghost of Quine.
To say the least, these claims raise a series of questions. How are we to understand 'logic' in this context. Surely, Williamson must have our natural, informal reasoning in mind, but that doesn't sit well with the footnote about metalogic. But if the claim is about formal logic, then what has this got to do with our natural mode of deductive thinking? (Of course, I is tempting to ask how we are to understand 'philosophy' in this context as well, though I think that is pushing it.)
More on this later.