I Fought the Law, and the Law Won

Before the holidays I spent a couple of days visiting The Northern Institute of Philosophy (NIP) in Aberdeen. Ian Rumfitt had been invited up to give three talks on his new book project, The Boundary Stones of Thought. Great seeing friends again in Aberdeen, and a good opportunity to discuss the nature of logical laws. Perhaps some photos will appear on the NIP blog in the near future.

Rumfitt's project interests me a great deal. Although I disagree substantially with lots of what he has to say, I'm very happy to see a book-length attempt at getting to grips with the nature of logical laws. Even more so because Rumfitt's starting point - much like my own - is Dummett's The Logical Basis of Metaphysics [LBoM] (although, of course, the title is a reference which takes us even further back, to Frege's Grundgesetze). Here I'm only going to say something about Rumfitt's set-up, that is, the preliminaries before he presents the details of his theory. Maybe later, when the book is out, I'll look closer at the details.

Rumfitt's over-all aim is a defence of classical logic. Of course, this is ambiguous in a variety of ways. First, an easy defence of classical logic is to defuse concrete objections from the non-classicists camp, say, the intuitionist Dummett. Alternatively, however, a defence might be a wholesale justification of classical logic. Needless to say, this is a great deal harder to achieve (and, to understand). Second, what exactly is being defended is a bit unclear. It turns out that in Rumfitt's case the One True Logic is indeed a classical consequence relation, but, importantly, need not be associated with classical semantics. The hallmark of the latter, as Rumfitt understands it, is the principle of bivalence. So, in short, Rumfitt is willing to sacrifice bivalence to defend a classical consequence relation. Interestingly this separates Rumfitt from some other champions of classical logic, e.g., Timothy Williamson.

(An exception is the conditional. Rumfitt chooses to set this complication aside. For the purposes of the book he is only interested in the correct logic for conjunction, disjunction, negation, and quantifiers, admitting that the conditional most likely requires special (non-classical?) treatment. Of course, this already conflates a number of rival positions where the conditional is precisely the demarcating point.)

Rumfitt isn't going for the latter, wholesale justification of classical logic. He only promises to defend classical consequence against some of its most stubborn attacks - largely the attacks launched by Dummett and Crispin Wright. Instead, following Dummett, he sees the central problem of logical dispute as a sort of circularity charge (see for example LBoM, chs. 8-9). It is no good, it seems, to justify logical laws applying inferences that themselves are warranted by these very laws. The Holy Grail, then, is a framework in which such disputes can be settled, without illicit applications of laws under dispute. More, it ought to be a framework in which the semantics of the involved logical constants are stable across the logics under consideration. This is crucial if one wants the dispute to be more than a case of talking at cross-purposes. (Of course, if you think that the inference rules already determine the semantics of logical constants, this problem takes on an even more critical shape.)

Rumfitt wants to offer a framework where the semantic clauses are stable, but where the consequence relation is sensitive to how one defines the space of logical possibilities. A substantial part of his work is dedicated to defend a particular notion of logical possibility, but I won't go into that here. Rather, I'm interested in how this compares to another view: The logical pluralism of Beall and Restall. Formally speaking the two views, although different in aim, share some features. Both give a sort of all-purpose template for logical consequence, which fluctuates with the choice of model-input. Beall and Restall call them cases while Rumfitt calls them logical possibilities, but the idea is similar. Consequence is schematic. (See Beall and Restall's Logical Pluralism, OUP 2006, p. 29.)

Here is Rumfitt's general account of consequence:

• Some premisses A_1, ..., A_n R-relate to a conclusion B if and only if, for any possibility x in S, if A_1, ..., A_n are all true at x then B is true at x too.

As Beall and Restall, Rumfitt requires consequence relations to have the Tarski properties (reflexivity, transitivity, monotonicity), and to be truth-preserving, although perhaps designated-value-preserving would be more appropriate. Rumfitt also insists that consequence-relations are single-conclusion (rather than multiple-conclusion). His reasons are more or less those stated in his 2000 paper ''Yes' and 'No'' and the more recent 'Knowledge by Deduction' (2008), so I won't bother to report them, although I don't finding them convincing. As it happens, Beall and Restall also opt for this constraint, but I suspect that they're not wedded to it. (For Restall's view on Rumfitt on multiple conclusion, see here.)

That is not to say that the two approaches to consequence relations are the same. It all boils down to the type of structures you allow in your schematic consequence relation. For example, Beall and Restall considers not only classical and intuitionistic logic (via classical models and constructions), but also relevant logic (via situations). Rumfitt on the other hand, perhaps following the spectrum in Dummett's LBoM, appears to limit his scope to classical, intuitionistic, and quantum logic. The algebraic framework he proposes allows one to suggest different structure for the space of logical possibilities. Which logical consequence relation one gets depends on the choice of structure.

I'll bypass the objection that these frameworks are not sufficiently encompassing. (Why rule out other substructural candidates?) The genuine challenge is to explain why there is sameness of content for the logical connectives across the rivals. Despite the fact that the semantic clauses are the same, the type of structures allowed obviously make a difference to the resulting logic. Take for example:

For every v,w, v_w(¬A) = 1 iff for every w' s.t. wRw', v_w'(A) = 0.

Is there sameness of content for the negation independent of the structure of R? If there is, well then there is reason for thinking that the classical and intuitionistic negation has the same meaning.  Similarly, the semantic clauses of many-valued logics are formulated with MIN and MAX functions, with obvious flexibility depending on the set of truth values. Same content? The problem, put bluntly, is that only formal ingenuity stands between us and the conclusion that any two logical connectives have the same meaning.

I'll return with more once I have the details from the book.