Logic and Experimental Data

In the presence of lots of philosophers of methods, it's hard not to spend some time thinking about formal methods in philosophy. For a long time I've wanted to write something about what formal methods in philosophy are, and why we should keep on using them. But, since I've yet to formulate my thoughts on this, I'll compromise and give you some related thoughts about logic and experimental data. More precisely, experimental data and the model- vs proof-theory divide.

Although frequently labelled as one, I'm not at all the sort of hardline proof-theorist who finds no room in his philosophy of logic for model-theoretic techniques. (Does the hardliner exist? Yes - check out this paper by Tennant.) Quite the contrary, I believe the two approaches complement each other, both being equally valuable in formal methods broadly construed. Nevertheless, for particular purposes in one's philosophy of logic, preferring one over the other is sometimes right. Lots of philosophers share the suspicion that inference rules are somehow integral to acquiring and possessing logical concepts. And, accordingly, a lot of energy has been put into exploring the link between this idea and entitlement to infer. (Examples are Christopher Peacocke, Paul Boghossian, and Crispin Wright.)

Yet, in the spirit of experimental philosophy one might wonder whether or not the critical claim about concept-aquisition is empirical (and, further, experiment-susceptible). I started thinking harder about this issue after an introductory talk by Ruth Byrne, a leading expert on the mental model theory in psychology. The mental model theory is a theory of reasoning in psychology which is accompanied by numerous experimental results. Roughly, the theory says that human capacity to imagine mental scenarios or possibilities is what engenders basic inferences. The experimental work involves Wason task studies, studies on counterfactual reasoning, and on how children develop mental models. For some information about the programme, see the Mental Models Website. Actually, I noticed that the psychological theory is not entirely unconnected to its formal counterpart, model theory: In the SEP entry for  model theory, Wilfrid Hodgson mentions mental models as cognitive theory support for the importance of model theory in logic.

Correspondingly, proof-theory has its own psychological counterpart. In fact, the mental model theory is a response to the idea that inference is a result of a rule-based mental capacity which relates to formal calculi. Byrne calls it the formal rule theory. As its model-theoretic rival, the formal rule theory offers experimental support for why reasoning as a psychological process is a largely rule-based affair. One source for this view is The Psychology of Proof (1994) by L. J. Rips.

You don't have to subscribe to experimental philosophy in its most rampant forms in order to see the value of connecting the above experimental work with the philosophy of logic. In fact, my hunch is that philosophers - although experimentally crippled (aka "don't try this at home") - can contribute to the experimental work with logical expertise. Actually, philosophers of logic have hypotheses about actual human reasoning that are primed for testing, and which aren't likely to be tested by psychologists who don't know discussions about, say, vagueness and semantic paradoxes. Earlier I've mentioned one admirable adventure into experimental research by Dave Ripley, and another by Jeff Pelletier.

My plan is to start out by checking out The Rational Imagination (2005, MIT Press) by Ruth Byrne.

Studentships, Birkbeck College, University of London

PhD studentships in philosophy of logic and mathematics available at Birkbeck College, University of London. More information here. The studentships are associated with Øystein Linnebo's project 'Plurals, Predicates, and Paradox'.

HT: Logic and Rational Interaction

CFP: Conference on the Foundations of Logical Consequence

June 11-15, 2010

I'm very happy to announce that there is a call for paper out for the 1st Arché Conference on the Foundations of Logical Consequence. This is the first major event on the FLC project's output schedule. Hopefully, some of my readers will want to contribute a paper to make this a great event. There will be both open sessions for submitted papers, and designated sessions for graduate speakers.

The list of invited speakers are as follows:

• JC Beall (UConn)
• Josh Dever (University of Texas at Austin)
• Hartry Field (NYU)
• Michael Glanzberg (UC Davis)
• Hannes Leitgeb (Bristol)
• Vann McGee (MIT)
• Agustin Rayo (MIT)
• Dag Westerståhl (Gothenburg)
• Robbie Williams (Leeds)

Any questions? Get in touch with me or flc.conf@googlemail.com.

More information about our 3rd FLC Workshop will follow soon.

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.