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).


Anonymous said...

Here's the small 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.)"

I can't see how these two theories are inconsistent or incompatible in the absence of C. We can drop M, since it plays no role in the incompatibility of the two theories: both assume M. So how are (p -> q) and (p -> ~q) inconsistent in the absence of C? In the absence of C the theories together entail that ~p. And they are perfectly consistent. If the sentences are theorems of the theories respectively then they are still consistent and entail that []~p. For instance, let p =
(H&~H). So we don't have a case of theories inconsistent in the absence of C. But matters are worse. If you did have two theories T and T' that were inconsistent, there could be no additional theory C that would render T and T' consistent. What could be added to P & ~P that would make it consistent? Nothing. Certainly to make T and T' consistent you'd have to *subtract* something from the theories (not add something to them).

Greg Frost-Arnold said...


I've been meaning to look at that paper of yours ever since I saw it mentioned on your blog -- since I study the logical empiricists, I'm always interested to hear someone argue for (what sounds to me like) one of Carnap's pet theses. I'm curious whether you've read a recent paper by Chris Brink and Ingrid Rewitzky in the Dec 2002 issue of the Journal of Philosophical Logic, entitled "Three Dual Ontologies." Here's their abstract:
Abstract  In this paper we give an example of intertranslatability between an ontology of individuals (nominalism), an ontology of properties (realism), and an ontology of facts (factualism). We demonstrate that these three ontologies are dual to each other, meaning that each ontology can be translated into, and recaptured from, each of the others. The aim of the enterprise is to raise the possibility that, at least in some settings, there may be no need for considerations of ontological primacy. Whether the world is made up of things, or properties, or facts, may be no more than a matter of how we look at it.

I have only skimmed through it, so I can't say much about it, but I thought it might be useful for your project.


I'm glad to see someone from San Antonio here -- a lot of my family lives there, and I have many fond memories of summers spent in my grandparents' pool in San Antone.

I think I agree with what you've said -- I made the point that A and B (in the toy example) are not inconsistent in the original post, under comment #1.

So what's left to say?/ Why bother writing this post? First, I'm not 100% sure two sentences have to be logically inconsistent in order for them to be rivals. A and B don't imply each other -- might that be a strong enough difference? (I could be completely wrong about that; I'm really not sure.) Second, and more importantly, "actual" underdetermination arguments might not suffer this defect of my toy example, in which case this 'equivalence modulo p' perspective might still be helpful/ useful in understanding the structure of underdetermination arguments. (The word "actual" is in quotes, because many philosophers of science think that all or virtually all the so-called cases of underdetermination in science are the fantasies of armchair philosophers.) But as I mentioned in the original post, I suspect there may be nothing useful left.

Anonymous said...

It might be interesting to note, relative to underdetermination and rival theories, that it is inevitable that your axioms are underdetermined by whatever empirical evidence you consider relevant. But that can't mean that two scientific theories with different axioms are not rivals. Put M as before and let T include M + Classical logic + S5. Let T' include M + classical logic + S2. T and T' are no doubt rival theories (the licensed inferences--including counterfactual inferences--varies between them), and yet your choice between them is certainly underdetermined.