Wednesday, May 28, 2008

Warmbrod on logical constants

I finished reading the article "Logical Constants" by Ken Warmbrod and what follows is a rough bunch of reflections on it. In the article he gives an analysis of the notion of logicality. He distinguishes two sorts of logicality: core and extended. Core logicality is the primary notion that we are after with logical studies and extended logicality is the cluster of notions that logicians also study but do not fall properly within the scope of core logicality. Core logicality is supposed to minimally capture the deductive procedures and consequence relation used in and needed by science. Warmbrod asks us to consider "the potential contribution of logical theory to a scientist's task of constructing and testing theories about the world. We first have to imagine a hypothetical scientist toying with some theory which she has heretofore understood only vaguely. She hopes to enlist formal logic as an aid in developing and evaluating her theory." This Warmbrod cashes out in three ways: clarifying the claims made by the theory, communicating the theory to other scientists, and enabling systematic testing of the theory. From these constraints Warmbrod builds up an account of logicality that is roughly truth-functions and the standard quantifiers.

There is a lot that strikes me as weird about this. First is the premiss that in order to get a grip on the general notion of logic we should look to the demands of science. Warmbrod may be intending this quite broadly but his examples are all of a physical or chemical kind. As far as the text as concerned, he seems to mean only empirical science. Earlier in the article he argued that all the intuitions one finds in the logical constant literature are not jointly satisfiable, but it is not clear why these particular intuitions or demands need be met. This move of appealing to science was reminiscent of Quine, but Warmbrod didn't, as far as I remember, invoke any of Quine's reasons. It is confusing to me how one would end up with this as a starting point for an account of the notion of logicality.

The three constraints put in place by Warmbrod's hypothetical scientist are likewise kind of odd. The first calls for axiomatization of a theory to tease out its consequences. This is well and good, but, as Wilson emphasizes repeatedly in his book, scientists don't go in for axiomatizations until late in the game, if then. It doesn't seem like this can bear all that much weight. The third requirement is apparently an instance of the hypothetico-deductive approach to theories and theory testing. At this point it sounds very Quinean, since this is one of the points where Quine gets logic to mesh with science. I don't want to get bogged down in the merits and flaws of the hypothetico-deductive approach, so I will jump to the second constraint.

The second constraint is perhaps the one that strikes me as the most wrongheaded. Logic and communication seem to run in different circles, so it would have been good for Warmbrod to motivate this one a little bit more. He doesn't though. One might emphasize communication as a way of trying to build in some appreciation for computability issues into the notion of logicality. This seems to be sort of what Warmbrod does. He says that quantifiers are helpful because they allow us to express things that would otherwise require an infinite disjunction or conjunction of sentences; we cannot write down such a sentence, so quantifiers aid in communication. Warmbrod suggests counting the universal quantifier as logical while excluding the infinite disjunction and conjunction. This, however, seems somewhat sloppy. There are lots of finite sentences that we will also will never be able to say or write, but these are not ruled out. These include sentences that have more atomic parts and connectives than there are atoms in the universe. The emphasis on communication would, one would think, bring with it a concern with being able to compute or manipulate formulae quickly, but that seems to be lacking. On purely logical grounds it seems weird to dismiss all of the infinite. Certain countably infinite sets support recursion, which sets Barwise has shown to be quite important. Even if an infinite set is not fully recursive, it may be recursively enumerable, which would seem to suffice for a broad notion of communication like that invoked by Warmbrod. Since the appeal to computability doesn't seem to support the second constraint in Warmbrod's article, I'm at a loss as to where it comes from.

I would think that if communication is at issue, one would reach for the most expressive bunch of logical notions available, or at least the more so the better. This does not seem to be a motivating factor in the paper though.

In his thesis, John MacFarlane cites Warmbrod's article as an instance of what he calls pragmatic demarcations of logic. He also cites Quine in this regard. Warmbrod's characterization seems less convincing than Quine's although the two are similar. As an aside, it seems like Carnap should be included in this camp as well. Although, on any of these accounts it is a bit unclear what, if anything, distinguishes the logical from the mathematical; perhaps the mere contingency of our practices so far?

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