From an e-mail by Spencer Case:
. . . by my lights, parsimony might be a consideration that puts the burden of proof on the theist. Theories that multiply entities unnecessarily are less likely to be true and the theist's theory postulates an entity. Now, it may be that the theist will say that we need God as a first cause or something like that-- that could be enough to absolve him of the burden. But in the absence of other reasons for believing in God (known to the interlocutors), the burden of proof would be on the theist.
Let's think about this. I doubt the usefulness in philosophy of burden-of-proof considerations, at least when we are discussing such big questions as God, freedom, and immortality. I also doubt the usefulness in philosophy of considerations of parsimony. What is parsimony anyway?
Parsimony or Occam's Razor is a principle of theoretical economy that states or rather enjoins:
OR. Do not multiply entities beyond necessity.
It is sometimes formulated in Latin: Entia non sunt multiplicanda praeter necessitatem. The principle or rather injunction is presumably to be interpreted qualitatively rather than quantitatively, thus:
OR*. Do not multiply TYPES or CATEGORIES of entity beyond necessity.
Thus it is not individual entities that are not to be multiplied, but types or kinds or categories of entity. To illustrate. Some criticized David Lewis' extreme modal realism on the ground that it proliferates concreta: there are not only all the actual concreta , there are all those merely possible ones as well. He responded quite plausibly to the proliferation charge by pointing out that the Razor applies to categories of entity, not individual entities, and that category-wise his ontology is sparse indeed.
'Multiply' is a picturesque way of saying posit. (Obviously, there are as many categories of entity as there are, and one cannot cause them to 'multiply.') And let's not forget the crucial qualification: beyond necessity. That means: beyond what is needed for purposes of adequate explanation of the data that are to be explained. Hence:
OR** Do not posit types of entity in excess of what is needed for purposes of explanation.
So the principle enjoins us to refrain from positing more types of entity than we need to explain the phenomena that need to be explained. It is obvious that (OR**) does not tell us to prefer theory T1 over theory T2 if T1 posits fewer types of entity than T2. What it tells us is to prefer T1 over T2 if T1 posits fewer types of entity AND accounts adequately for all the data. So there is a trade-off between positing and accounting.
Spencer tells us that "Theories that multiply entities unnecessarily are less likely to be true . . . ." I don't think this is right. Theories that posit entities or types of entity beyond the needs of explanation are uneconomical and to be rejected for this reason. We prefer simpler theories to save cognitive labor, not because simplicity is the mark of truth (simplex sigillum veri) or even because simpler theories are more likely to be true. Now it may be that simpler theories are more likely to be true -- how would one show this? -- but this is no part of the principle of parsimony as I understand it. It is a principle of Denkoekonomie.
The defeasible presumption in favor of parsimonious explanations is very much like the defeasible presumption of innocence (POI) in the law. The accused is presumed innocent until proven guilty even if the probability of his being innocent is low or even near or at zero. The presumption of innocence does not vary with the probability of innocence, and is in fact logically consistent wth guilt. And of course the presumption of innocence does not entail innocence.
POI is a procedural rule: we proceed in the law as if the accused is innocent even if it is evident that he isn't. (Suppose 100 reputable winesses observe a man in a non-self-defense situation bludgeoning a woman to death. There still has to be a trial, the accused will enjoy the POI, and the prosecutors will bear the burden of proof. It's just that this trial will be very short.) Similarly, the principle of parsimony is a principle of procedural presumption. (See N. Rescher, Presumption and the Practices of Tentative Cognition, Cambridge UP, 2006. p. 124 ff.)
Suppose everything could be explained just as well without God as with God. Then we would have no reason to posit God as playing an explanatory role. But it wouldn't follow that God doesn't exist, or even that it is unlikely that God exists. All that would follow is that we would have no reason to posit God as an explanation of the existence, order, intelligibility of the universe: The 'God hypothesis' would not be rationally motivated.
Now one point I want to make is that Parsimony is a fairly useless and trivial injunction. After all, who wants to posit types of entity in excess of what is needed for purposes of explanation? The real question is what is needed for explanation. Parsimony gives us no help with this question. I would argue that God is needed to explain the existence and the intelligibility of the universe. Now that is a meaty set of issues that cannot be resolved by brandishing the Razor. We all agree about the Razor. What we don't agree about is what is necessary for an adequate explanation of what needs explaining.
And so it would be a cheap shot for an atheist to claim that theists violate Parsimony by positing God. Spencer of course understands this. For again, the issue is whether the posit is necessary for explanation.
Burden of Proof
Who bears the burden? Theist or atheist? The question is senseless or else has a trivial answer: both bear it. For it is not evident that God exists, nor it is evident that God does not exist. Neither side can invoke a defeasible presumption.
But there is a defeasible presumption in favor of the reliability of memory as a source of knowledge; so it does make sense to place the burden of proof on one who denies it.
Finally, does parsimony put the burden of proof on the theist as Spencer claims? No and for two reasons. First, Parsimony is a trivial injunction that, by itself, cannot decide between theism and atheism. Second, it is either senseless or trivial to ask where the BOP lies in the atheism-theism dispute.
UPDATE (10/28): Spencer Case e-mails: "I think you should make clear to your readers that your post attributes views to me that I do not hold. The part you quote from me is given in a context that is meant to show how my view of burden of proof would apply to a particular dialectical situation where an atheist thought parsimony mattered for the reason I stated. I wasn't actually subscribing to that view of parsimony. My account of what philosophical burden of proof amounts to was the main point of my comments.