Underdetermination and equivalence modulo p

Since the description of this blog states that it deals with "issues in logic" related to philosophy of science, I figure that, for the sake of truth in advertising, I should post something logical. (Though I don't feel particularly rushed: there are already a fair number of smart logicians actively participating in the blogosphere -- check my blogroll. For reasons I don't understand, the situation is different in philosophy of science. Any armchair anthropologists have an explanation?)

Underdetermination arguments occur in many quarters of philosophy: Descartes' demon is perhaps the most famous, but they have also played a leading role in discussions about scientific realism during the last few decades. In this post I want to characterize a particular sort of underdetermination using elementary logical notions. (This form of underdetermination either is -- or is closest to -- the Quine-Duhem variety, I'm not sure which at the moment.)

Consider two sets of sentences, A, B such that neither set is a logical consequence of the other. Now suppose there is a third set of sentences C such that:
If C then (A iff B).
That is, if we assume that C is true, then A and B are logically equivalent. (In all models where C is true, either both A and B are true, or both are false).
Then we say A and B are equivalent modulo C.

How does this relate to underdetermination? We can have two theories that are not logically equivalent (and thus are not 'the same theory'), but do become logically equivalent if we make some further assumptions (C above) -- and these further assumptions can be taken to be "auxiliary hypotheses" or "background knowledge" (or whatever one wishes to call the other claims a theory uses, besides its own, to make predictions). If we are committed to the truth of the background knowledge, then we cannot decide between the two theories.
(If this is a bit abstract, here's a toy example:
A = M and (if p then q) [assume M says nothing about p or q]
B = M and (if p then not-q)
C includes the sentence 'not-p';
so neither of A and B implies the other; if C is true then A and B are logically equivalent, while if C is false then A and B are inconsistent.)

My question: is anything philosophically interesting going on here? If we hold r to be true, do we really need to choose between (r or s) and (r or not-s)? I think not -- though they differ in logical content, they are not rivals (or are they?). At least, if we take r to be true, then they are definitely not rivals, though they might be considered rivals ‘on their own’. They certainly are genuine competitors when we hold r false -- though then they are no longer equivalent in any sense.

Comments:
1. This is not the usual sort of underdetermination situation. First, the notion of "empirical content" (or "empirical equivalence," i.e. identity of empirical content) does not appear, so the much-maligned observable/ unobservable distinction is never mentioned. Second, and more importantly, the two theories A and B are not incompatible: the Cartesian demon, on the other hand, is either deceiving us or not (at least on the usual interpretation). The demon-hypothesis is incompatible with the 'real-world' hypothesis. On the other hand, 'if p then q' and 'if p then not-q' are not inconsistent -- we need simply hold that 'not-p' is true.

2. How does this relate to a ‘real’ example, e.g. Bohmian mechanics? It is empirically equivalent to standard quantum mechanics as long as absolute position is undetectable -- but not if absolute position is detectable. In other words, the standard theory and the Bohmian theory are empirically equivalent modulo the claim that absolute position is undetectable. And that is structurally similar to the toy example above. (Of course, there is the difference in this case that the two theories are 'empirically equivalent modulo p,' not 'logically.')

3. Lastly, it is probably considerations akin if not identical to the above that prompted philosophers to move to ‘total theories’ (i.e. theories PLUS all their auxiliary assumptions) as the proper objects of epistemic evaluation. See e.g. (Leplin, Erkenntnis, 1997).

Thanks to the Philosophy of Biology blog

I'd like to thank the folks over at the group blog Philosophy of Biology for kindly advertising the existence of my blog on their site. If anyone reading this has not checked out that blog, I'd strongly recommend it. Not only did they recently have a very enlightening interchange on the topic of Elisabeth Lloyd's new book, "The Case of the Female Orgasm", but they just posted a link to a Naked Mole Rat-Cam.

Galileo:Scholastic natural philosophers :: Carnap:Quine

I'm pretty sure the following has been said before, but I don't know where: Frege (and Russell, et alii) did for the study of language what Galileo (et alii) did for the study of material nature. Galileo 'mathematized' new portions of the physical world -- previous students of nature thought that (most of) nature was too messy, imprecise, or chancey to be susceptible mathematical treatment: how could the unchanging, eternal realm of mathematics model the changing and temporal material world? Analogously, Frege and other founders of modern logic turned language into a mathematical object, by treating (e.g.) subject-predicate assertions in terms of functions and their arguments.

I bring this up because I've been looking at one of Quine's arguments against Carnapian analyticity in the 1940s. This argument appears in the long 1943 letter from Quine to Carnap in Creath's Dear Carnap, Dear Van, and in print in a 1947 article in the Journal of Symbolic Logic ("The Problem of Interpreting Modal Logic"):

The class of analytic statements is broader than that of logical truths, for it contains in addition such statements as 'No bachelor is married.' This example might be assimilated to the logical truths by considering it a definitional abbreviation of 'No unmarried man is unmarried,' which is indeed a logical truth; but I should prefer not to rest analyticity thus on an unrealistic fiction of there being standard definitions of extra-logical expressions in terms of a standard set of extra-logical primitives. What is rather in point, I think, is a relation of synonymy, or sameness of meaning, which holds between expressions of real language, though there be no standard hierarchy of definitions. (p.44, italics Quine's, boldface mine)

Quine puts the point somewhat differently in different places, but the basic idea is always that the 'rational reconstruction' of language, however it is carried out, is an 'unrealistic fiction.'

Now, I can ask the question: is Quine's charge that a 'hierarchy of definitions' is an 'unrealistic fiction' any different from Galileo's scholastic critics' charge that Galileo is somehow 'falsifying' nature by rendering it thoroughly mathematically? The answer to this question will turn on what Good Things the Galilean mathematizing strategy is able to achieve (explanatory power, new predictions, etc.), and whether these Good Things (or analogues of them) also appear in the case of a Carnapian language. It would also be useful to know of other scientific cases where the analogue of the Scholastic triumphed over the analogue of Galileo, i.e., someone tries to mathematize certain phenomena, but this mathematization is rejected for good reasons by workers in the field.