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?

Boghossian on ('metaphysical') analyticity

I've been thinking recently about an objection Paul Boghossian (and many others) make against the Tractarian/ Carnapian conception of an analytic truth, viz. a sentence that is true solely in virtue of the meaning of the sentence. (Boghossian calls this kind of analyticity 'metaphysical analyticity,' which I think is potentially misleading, given the staunch anti-metaphysical tastes of the logical empiricists. Oh well.)

Boghossian considers the notion of metaphysical analyticity untenable. Why? He asks a rhetorical question:
"Isn't it in general true---indeed, isn't it a truism---that for any statement S,

S is true iff for some p, S means that p and p?

How could the mere fact that S means that p make it the case that S is true?" (Boghossian 1996, "Analyticity Reconsidered," Nous [p.364]

Boghossian is not alone in this view: the basic idea can be found in Quine's "Carnap and Logical Truth," and is developed by Gilbert Harman, Elliott Sober, and Margolis & Laurence. How should we interpret this rhetorical question? Boghossian appears to be claiming that the truth of a sentence of the form 'S means that p' is never a sufficient condition for the the truth of a sentence of the form 'S is true'---that appears to be intended force of the rhetorical question in the quotation immediately above. And that is certainly one reasonable way of cashing out the notion of the truth of a sentence being `fixed exclusively by its meaning.'

If we do understand Boghossian's view in this way, then I think his claim is either misleading or incorrect. Consider a standard material biconditional of the form

(1) p iff q

If such a biconditional is true, we usually say that q is a necessary and sufficient condition for p. But as we teach undergraduates in Introduction to Logic classes, if this biconditional is true, then (within the classical propositional calculus) so is

(2) p iff [q and (r only if r)]

(Any other logical truth of the classical propositional calculus could be substituted for r only if r.) If we simply read off the surface structure of sentence-schema (2), one might think that q was no longer sufficient for the truth of p--because there appears to be a second condition that has to be met in order for p to be the case, namely that r only if r. Of course, strictly speaking, this is true: every sentence of the propositional calculus presupposes the truth of all the logical truths of the propositional calculus. However, it seems seriously misleading to me to say that the truth of q is not a sufficient condition for the truth of p in our original biconditional--for that is not the way we standardly understand sufficient conditions.

Hopefully the direct parallel with Boghossian's claim is clear. I certainly agree that his 'truism' quoted above is true. However, when a logical truth--which, as Carnap and Quine agree is a paradigmatic case of analytic truth (if there are any)--is substituted for p in his schema, then that instance of the truism will have (almost) exactly the form of the second biconditional (2). Then, in the usual sense of 'sufficient condition,' we will have a case in which (contra Boghossian) an instance of 'S means that p' is sufficient for 'S is true.' To say otherwise, we would have to give up either classical logic (specifically, the idea that (2) follows from (1)) or the usual understanding of sufficient conditions.

However, one could object that neither classical logic nor our standard view of sufficient conditions is sacrosanct. I think there are reasonable replies to these objections (telegraphically: for whatever non-classical logic you choose, you can substitute some other logical truth for 'r only if r' in (2) above, and the point carries); but I'll leave matters here since this post is too long already.

Comments and criticism from any angle are very welcome, but what I personally go back adn forth on with the above argument is whether it's a 'cheap point' or not... superficial logic-chopping, or genuine insight?

Can a widespread local realist be a global anti-realist?--and the preface paradox

I have been thinking about whether there might be something like the preface paradox in the scientific realism debates. There is now a distinction being drawn between local (or 'retail') realism, in which one argues for the truth of particular scientific theories (e.g. quantum mechanics) or the existence of particular scientific entities (e.g. quarks), on the one hand, and global (or 'wholesale') realism, in which one argues for the approximate truth (or referential success) of mature scientific theories in general.

What I'm wondering is whether it can be justified and/or rational to be an everywhere local realist (so QM is approximately true, and general relativity is approximately true, and population genetics is (approximately) true, etc.), but still be a global anti-realist -- say, because you place a lot of weight on the pessimistic induction on the history of science. Or, on the other hand, whether everywhere local realism really pushes us towards global realism.

I'm currently guessing that one CAN be an everywhere local realist without being a global one, for the following two reasons.
(1) One standard response to the preface paradox seems perhaps even more applicable here than in the preface case: while the author assigns a high probability to each individual assertion in her book, the probability of (p & q & r & ...) will be low.
(2) Also, although if A is true and B is true, then 'A and B' must be true, it seems to me that even if A is approximately true and B is approximately true, then 'A and B' need not be approximately true, for A and B could be contradictory (for example, the prima facie conflict between quantum mechanics and general relativity).

I'm happy to hear any reasons for the opposite view, viz. that widespread local realism pushes us towards global realism.

Azzouni and existential commitment in science

Last week Jody Azzouni was here to give a pair of talks: one about scientific theories, another about his view that English is inconsistent in a pretty radical way: Every sentence is both true and false. They were both a lot of fun, and Jody is a great interlocutor -- he kept both presentations relatively short and to the point to there'd be more time for questions and clarifications. I also have a soft spot for arguments defending unpopular ideas -- though I usually side with the orthodoxy, incredible ideas are often a bit more interesting to think about.

In the philosophy of science talk, Jody was building on his work on what he calls "thick epistemic access." His argument was that we should not (contra Quinean orthodoxy) have existential committment to all the posits of our current best scientific theory, but rather only those posits to which we have thick epistemic access. (See e.g. Kenny's post here for a quick but accurate description of thick v. thin v. ultrathin posits.)

I was wondering, however, whether the Quinean orthodoxy could be undermined in a more direct way that does not involve developing a whole epistemological apparatus to distinguish when we really do have strong evidence that such-and-such thing exists. (Such a question is certainly philosophically interesting and worthwhile, but it is likely to be complex and contentious in places.) Rather, I thought a simpler argument against the Quinean orthodoxy could go as follows:
Science is rife with idealizations -- some of which are ineliminable/ indispensible. But no one should be committed to such idealizations, since they are (almost by definition) deliberate and conscious falsifications in our theoretical account of the world. So existential commitment does not follow our best theories as well as Quine would like.

I realize that (1) there may sometimes be a legitimate question about whether a given bit of a theory is an idealization or not, but that just shows the term 'idealization' is vague -- all parties agree there is some idealization in science, even if they don't agree on every case. Also, (2) most examples of idealizations are not entities, but rather inaccurate properties (e.g., treating some body that we know to exist, like a point particle: we give an inaccurate description of the thing's dimensions). So maybe pointing out the widespread use of idealization will not create widespread problems for the Quinean orthodoxy.

Why no 'scientific wisdom'?

Why do people often talk about 'scientific knowledge,' but we virtually never hear of scientific wisdom? Is there something about the content or practices of science that precludes them from counting as wisdom? After all, if ‘wisdom’ means something in the neighborhood of 'deep, important, or fundamental knowledge,' it seems (to me at least) that science should be a paradigm case of wisdom.

Is this merely a linguistic quirk that bears no relation to the relationship between the nature of science and the nature of wisdom? Or does the fact that we rarely—if ever—speak of 'scientific wisdom' reveal something important? Here's a reason for thinking the latter.

Science provides instrumental reasons for action only, not categorical ones: If you want to build a nuclear bomb, then the atomic theory of matter will be an extremely useful tool in designing the weapon. It is no part of physics—or any other of the (paradigmatic?) (natural?) sciences—to say whether you should build a bomb or not. Science does not inform us as to what should be valued for its own sake, and not merely as a means to some further end (though e.g. sociology could inform us what is in fact valued). The information provided by science only helps us acquire those things we already value. This is closely related, if not identical, to the old saw that science tells us about facts, and says nothing about values. Wisdom, in contrast, is thought to involve knowledge of what should be valued for its own sake (as well as how best to achieve those ends). That is, wisdom can offer categorical reasons, whereas science provides instrumental ones only.

The literature on the relationship between science and values is both vast and contentious. However, I do not know of anyone who says that the content of theories in natural science includes claims about what is valuable for its own sake. (I could be oblivious and/or uninformed about this; I'm no expert in this sub-field.) It may be that the scientific ethos includes e.g. valuing truth over personal gain, but that’s not part of the general theory of relativity.

So now we have a reason why no one says 'scientific wisdom'—science is silent on what is valuable for its own sake, whereas wisdom requires this information.

proxy bleg for a textbook

A post by request: one of my colleagues will be teaching a course for philosophy majors called "Contemporary Philosophy" focusing on what is current in the discipline now. Does anyone know of any good textbooks/anthologies that would work well for such a course?

antimeta in the house

One of my favorite bloggers, Kenny of Antimeta, was in Vegas last weekend and gave an interesting talk on philosophy of mathematics to our department. His basic aim was to find criteria that separated probabilistic proofs from other proofs (including, hopefully, proof sketches and computer-aided proofs). I'm not going to discuss that directly here.

I'm interested in a related claim Kenny made: that in mathematics, a theorem will be accepted only if the proof does not (he put it variously) appeal to authority/ depend on the reliability of other people/ rely on the testimony of others. That is, for a specialist in the field, they should be able to start out as serious skeptics of the theorem's truth, but end up at the close of the proof as believers. The contrast with experimental science is pretty clear: even specialists in a sub-field of experimental science have to trust (to some degree) the experimental reports of their fellow-workers, or the field would grind to a halt.

Question: Is there such a thing as mathematical fraud, of the sort we hear about periodically in experimental science? If not, that fact looks like evidence for Kenny's distinction being important and robust (since fraud is much harder in the absence of trust).

Comment 1: Some of the posters on FOM endorse Kenny's idea to the extreme: someone suggested that Fermat's Last Theorem will not really be proved until it is written in a way that average mathematics PhDs (whoever that is) can work through it themselves. I don't think Kenny wants to say anything nearly that strong, but the fact that such a strong position exists is a sign that the sentiment Kenny claims to discern really is there in the mathematics community.

Comment 2: At the end of the talk, Kenny suggested that philosophy may be closer to mathematics than experimental science in this regard. He may be right, but one thing that distinguishes philosophy from math in this regard is that in philosophy far more than in mathematics, one person's modus ponens is another person's modus tollens. This is just a direct result of mathematical axioms' being widely accepted throughout the mathematical community, whereas philosophers will challenge any premise, no matter how obvious or fruitful.

Yet another way to think about Quine's critique of Carnap

Several of the blog entries here have been about the Quine-Carnap debate over the status of analytic truth. Generally, I don't feel the force of Quine's arguments as they are usually presented, either because his interpretation of Carnap is unfair or inaccurate, or the arguments just aren't that persuasive. Multiple commentators on the Quine-Carnap debate have suggested that the two are 'talking past each other,' at least to some degree. So, I am constantly trying to find a way to make Quine's view make sense to me, AND simultaneously really disagree with Carnap. This seems like installment 19 or so in that endeavor.

Carnap and Quine agree that language can be studied at various levels of abstraction. Using Carnap's taxonomy, we start at the level of pragmatics, where we study how individual speakers use expressions under particular circumstances. This level contains the most detail: speakers, their circumstances, plus the meanings of the words for particular speakers under particular circumstances. At the next, more abstract level, we have semantics, which abstracts away from particular speakers and particular circumstances. And at the highest level, we have syntax, which abstracts away the meanings of words, leaving just the symbols, the way they are put together, and which strings follow from others.

In each transition from pragmatics to semantics to syntax, some information about language is omitted/ discarded. (Like the move from Euclidean geometry to neutral geometry, which drops the parallel postulate.) Now, we can conceive of Quine's indeterminacy of meaning thesis (the radical translation thought experiment) as critiquing Carnap in the following way: Carnap is importing or introducing new information at the semantic level, because the semantic facts Carnap includes in a semantically-characterized language [a "semantic system"] cannot be 'read off' even the information contained at the pragmatic level. The analogy in the geometry case shows why this is clearly an unacceptable maneuver. It would be: thinking that there exists some claim that could be proved in neutral geometry (= Euclid's first four postulates only) but couldn't be proved in Euclidean geometry.

This may not be Quine's actual worry; his concern may stem from the fact that applied semantics (or whatever branch of language study) underdetermines pure semantics (or whatever). However, Carnap is perfectly happy to accept that claim: Creath says this is why Carnap's copy of Word and Object Ch.2 (Indeterminacy of Translation) has no marginalia. [But how does the geometry analogy fare here? Would Carnap admit that applied geometry underdetermines pure geometry? My guess is yes; and that that's not so bad...

Is arithmetic empirical?

One of the questions I've been wanting to think about (in part because of my interest in the Quine-Carnap relationship) but haven't really got around to yet is: Is there any important sense in which arithmetic is empirical? I know there is some good literature on the subject, but I've thus far only perused it without really digging into it.

For me, one consideration that makes me think it might not be crazy to think of arithmetic as empirical is what happened with geometry and general relativity. If Einstein can show that the space in which we live is non-Euclidean, isn't it at least imaginable that some future scientist will show us that the 'true' arithmetic of our physical world is non-classical (which I suppose means: it does not obey the Peano axioms). [There could still be a mathematical structure that obeys classical arithmetic, just as Euclidean space is still a mathematical object that obeys all five of Euclid's axioms.]

However, I've always had a hard time imagining what possible observation could cast doubt on classical arithmetic. In last week's Science, there's a report that at least might merit consideration as a candidate. Researchers found that if you add one photon to a light beam and then take one away, you observe a different end-state than if you reverse the order of operations, i.e., first remove one and then add one. In other words, x + 1 - 1 does not equal x - 1 + 1. Even stranger, the authors find that "under certain conditions, the removal of a photon from a light field can lead to an increase in the mean number of photons in that light field," that is, (roughly) that x-1>x. The summary and background for non-specialists is here, and the full technical report is here (both behind subscription walls).

Now, this effect depends on the failure of commutation relations ubiquitous in quantum mechanics, so it is quite possible that this in no sense makes arithmetic look empirical. But I'm not 100% sure about that. Any thoughts?