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.
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).