Logical pluralism and brother-in-law pluralism

In my philosophy of logic class this week, we discussed JC Beall and Greg Restall's version of logical pluralism. Our text was their 2000 Australasian Journal of Philosophy article, available on Restall's website here. I've been flipping through their fantastic book-length treatment (OUP, 2006) as well.

Here's their basic idea. The basic, accepted notion of logical consequence is adequately captured in the following:

(V) Consequence C is a logical consequence of premises P1, ... Pn = In every case in which P1, ... Pn is true, C is also true.

Beall and Restall further hold that the notion of case admits of a number of "precisifications" (2006, 88), that is, it can be 'spelled out' or 'fleshed out' in more than one way. [Note: I can't find the quotation now, but I think Beall and Restall said that 'case' is neither ambiguous nor vague (in the sense of having borderline examples). [CORRECTION (3/2/08): In their "Defending Logical Pluralism," Beall and Restall explicitly say that they think the concept of deductive consequence is ambiguous (p.3). This more or less vitiates the main point of this post. I say 'more or less' because there is a test accepted by linguists for distinguishing ambiguity from lack of specificity, and it's not clear that B&R's concept of 'case' passes the test; see my comment #6 in the comment thread.] Different spellings-out of 'case' give rise to different consequence relations (and thus different logics); as examples of cases, they give:
(i) Classical Tarskian models, (ii) possible worlds, (iii) constructions (which yield intuitionistic logic), and (iv) situations (which yield relevant logic).
Finally, because there are multiple ways of spelling out 'case', there is not one correct notion of consequence, since different consequence relations correspond to different ways of specifying the content of (V).

So if someone asks: "Does an arbitrary sentence p follow from a contradiction 'q and not-q'?", the pluralist answer is "Yes and No -- yes, it follows classically (when we take Tarskian models as cases), but no, it does not follow relevantly (when situations are the cases)." Similarly, the pluralist answers the question "Is 'p or not-p' a logical truth?" with "Yes and No -- yes, it is a classical logical truth (since it is true in all Tarskian models), but no, it is not an intuitionistic logical truth (since it is not true in all constructions)".

I find Beall and Restall's position attractive. But while thinking about it, I wondered about when, in general, pluralism is the right (or at least a reasonable) position to take. B&R's claim is the fact that 'case' can be precisified in more than one way -- the meaning of 'case' is somehow underspecified or indeterminate -- to justify being pluralists about 'case' and thereby via (V) about consequence. However, I wonder whether, if this rationale were accepted across the board, pluralism would be almost everywhere, and the appropriate answer to many, many questions would be "Yes and no".

Here's an example of what I mean. The meaning of the phrase 'my brother-in-law' is not completely specific; it is indeterminate between the brother of my spouse and the male spouse of my sibling. However, nobody is a "brother-in-law pluralist": When someone asks me "Is Leon your brother-in-law?", I shouldn't reply "Yes and No -- yes, he is the brother of my spouse, but no, he's not the male spouse of my sibling." And what holds for 'brother-in-law' holds for many, many other terms: lack of specificity is everywhere.

Hopefully the analogy is clear: 'case' and 'brother-in-law' can both be made (more) determinate in different ways. But if this underspecification in the notion of 'case' is all that is required to justify pluralism about consequence, then we should also be pluralists about 'brother-in-law', since there is underspecification there too.

How might someone sympathetic to logical pluralism (e.g. me) respond to this challenge? Well, we could find an example where pluralism seems like the right (or at least reasonable) attitude, and try to argue that 'case' is (more) like that example. For example, I think pluralism about the concept of 'thing' is reasonable: if someone holds out a deck of cards, and asks me "Are there 52 things here?", the right (or reasonable) answer should be "Yes and No -- yes, there are 52 cards, but no, there are far more than 52 molecules".

The question is then: What makes 'thing' different from 'brother-in-law'? And is 'case' (in Beall and Restall's use) more like 'thing' or 'brother-in-law'? The pluralist wants 'case' to be more like 'thing', but I haven't yet figured out how to draw a sharp line. Any ideas?



Which came first: logical truth or consequence?

This term, I am teaching a philosophy of logic class. We've twice run across the following sentiment:

(CPT) Logical consequence is prior to logical truth.

This sentiment is also expressed as 'The real subject matter of logic is the notion of consequence, not a special body of truths.' (References: We've seen this in Stephen Read's Thinking about Logic Ch.2, and in John Etchemendy's work (1988, p.74) too.)

Seeing (CPT) surprised me, since logical truth is (in most cases -- see below) definable in terms of logical consequence, and vice versa: If C is a consequence of P1 ... Pn, then 'If P1 and ... and Pn, then C' is a logical truth. And if T is a logical truth, then T is a consequence of the null set of premises. This is well-known: Beall and Restall, in their recent Logical Pluralism, make exactly this point.

So, in light of the interdefinability of logical truth and consequence, what would prompt someone to say consequence is somehow prior to logical truth? Stephen Read appeals to valid arguments that ineliminably use infinitely many premisses: A(0), A(1), ... Therefore, ∀x Ax. We can't turn this into a logical truth ('If A(0) and A(1) and ..., then ∀x Ax') in standard languages, because standard languages don't allow for infinitely long sentences. This seems like a fair point in favor of (CPT), but it does assume that (i) you accept arguments with infinitely many premises, and (ii) reject languages with infinitely long expressions. [Edit: as Shawn correctly notes in the comments, these two assumptions are fairly widely held. But I have always been a bit suspicious (perhaps for no good reason) about the idea of an argument with infinitely many premises.]

Here's another argument for (CPT), from extremely weak languages. Imagine we have a propositional language with sentence letters p, q, ..., and only two sentential connectives: 'and' and 'or' specified in the usual way. In this language, there are no logical truths (because we don't have 'If... then...' or anything equivalent), but there are still logical consequences: A is still a logical consequence of 'A and B', and 'A or B' is a logical consequence of A. So here is a case where we have logical consequence without logical truth.

But both of these arguments (Read's and mine) rely on somewhat unusual cases. Are there other reasons to accept (CPT) that do not appeal to unusual circumstances? Is there a big literature out there that I don't know about? And does anything really hinge upon whether we think logical truth is prior to consequence, vice versa, or neither?