## Saturday, 30 March 2013

### Principle of Sufficient Reason

(I have talked about this to some of you, but I thought it would be a good idea to get it out in writing).

The following is Alexander Pruss’ (2006) reconstruction of Peter van Inwagen’s (1983) argument against the Principle of Sufficient Reason (PSR). (Roughly the principle of sufficient reason says that anything that is the case has a sufficient explanation for why it is the case. Pruss and others have restricted the principle to exclude necessary truths, so that only contingent truths are the kinds of things that need a sufficient explanation).

(1) If the PSR holds, then every true contingent proposition has an explanation.
(2) No necessary proposition explains a contingent proposition.
(3) No contingent proposition explains itself.
(4) If a proposition explains a conjunction, then it explains every conjunct.
(5) A proposition q explains a proposition p only if q is true.
(6) There is a Big Conjunctive Contingent Fact (BCCF) that is the conjunction of all true contingent propositions, perhaps with logical redundancies removed, and the BCCF is contingent.

(7) Assume PSR for reductio.
(8) So, the BCCF has an explanation, q. (1, 6, 7)
(9) So, the proposition q is not necessary. (2, 6, 8)
(10) So, q is a contingent true proposition. (5, 8, 9)
(11) So, q is a conjunct in the BCCF (6, 10)
(12) So, q is self explanatory. (4, 8, 11)
(13) But q is not self explanatiory. (3, 10)
(14) Therefore, PSR is false (7, 12, 13)

Pruss argues that we have good reason to reject (2) and (3). Spinoza denied (6), since he thought that there were no contingent facts at all (and therefore no conjunction of them). The rest of the premises seem plausible.

My objection is as follows: (1)-(14) is formally invalid.
It is hard to point out whether the fallacious inference is (9) or (10). Here is why:

(1) claims that every contingently true proposition has an explanation. The inference to (10) (and plausibly to (9) as well) assumes that the explanation of a contingently true proposition must itself be a proposition. In order to amend this, and make the argument valid, we should rewrite (1) as follows:

(1*) If the PSR holds, then every true contingent proposition has an explanation that is itself a proposition.

With (1*), the argument above is valid. But it is not clear to me that we should accept (1*). Consider the following account of explanation of conjunctive facts:

(A) The explanation of a conjunctive fact is the plurality of its conjuncts.

(A) contradicts (1*), since a plurality of conjuncts is not itself a proposition. But I take (A) to be fairly plausible. If we are looking for the explanation of a proposition of the form p & q, and we give an explanation of p, and an explanation of q, it does not look like there is anything left to explain.

Any thoughts?

1. It is very interesting Damian, but could you explain what kind of entity "a plurality of conjuncts" is? Is it a set?

2. I'm not sure about (A). In defense of (A), you suggest that in explaining p&q, we need merely explain p and explain q, then our work is done. But I'm thinking this intuition may be better captured by (A*):

(A*) The explanation of a conjunctive fact is the plurality of the explanations of its conjuncts

With (A*), we would require you actually explain the conjuncts, but then no more work is necessary. However, I'm still unsure about (A*). We might want to weaken it to (A**):

(A**) An explanation of a conjunctive fact is the plurality of the explanations of its conjuncts

(A**) is the indefinite form of (A*). Why does this matter? Well, (A**), but not (A*), is compatible with the Conjunction Principle (CP):

(CP) 'A' and 'B' explains 'C' iff 'A and B' explains 'C'

Note that (CP) is merely a bi-conditional, and silent with respect to which side is more fundamental. For instance, one could holde that 'A and B' explains 'C' in virtue of the fact that 'A' and 'B' explains 'C'.

Should we accept (CP)? I'm not so sure, but it at least seems plausible on its face. But note that if van Inwagen accepts (CP), he gets a response to your objection. For he could then capture the intuition you were getting at by accepting (A**), and yet also hold (CP), which is compatible with (1*) (in fact, I'm pretty sure (CP) entails (1*))

3. @Ali: the main difference between a set and a plurality is that a set is one, but a plurality is many.

So, for instance, compare "the set of all philosophers" with "those philosophers". In the first case, we are referring to one thing, a set. In the second case, we are referring to many things, a plurality of things.

4. Sorry for the delayed response. Here goes.

Consider the following case. m and n are two necessarily true propositions. As it happens, m and n do not have an explanation for why they are true. Does their conjunction, ‘m&n’ have an explanation for its truth? I want to say ‘yes’. You may disagree with my intuition here, but I don’t think it is a crazy one to have, as far as intuitions go. I want to say that there are only two facts (rather than an infinite number of facts) that are unexplained in this case: m and n.

Here are some interesting consequences of saying ‘yes’. (A*) is false, since the explanation of ‘m&n’ is not the plurality of explanations of its conjuncts (there are no explanations of its conjuncts). Further, given that the plurality of explanations for the conjuncts is not available, we do not seem to have a lot of options for what the explanation of ‘m&n’ is. The one that jumps out to me is (A). Of course, if (A) is true, then (CP) is false. Is there any way to answer ‘yes’ to my question without it leading to a denial of (CP)? If not, then I think (CP) is in trouble.

5. Hey guys it's Carolyn, just came to see what is going on.

I'll just write a bit in this box and then go back to my paper.

I think (CP) is false for other reasons anyways.

(CP) 'A' and 'B' explains 'C' iff 'A and B' explains 'C'

Suppose that 'The dog is furry' and 'the dog is big' explains 'the dog furry and the dog is big'. ('A', 'B', 'C', respectively). If (CP) is true, then the result is 'the dog is furry and the dog is big' explains 'the dog is furry and the dog is big'. But 'the dog is furry and the dog is big' doesn't explain 'the furry and the dog is big'(assuming explanation is be irreflexive). Soooo yeah, I think (CP) is in trouble too.

6. I think preface-paradox type worries may put pressure on (A), (A*), and (PC).
Suppose we have a bunch of facts of the form pi: Damien, in his book, correctly asserts 'pi'. Suppose there are facts 'p1-p1000'. The explanation of each individual 'pi' is that Damien is smart and careful when writing books. But the fact that Damien asserts only truths (he gets nothing wrong!) in his book requires an explanation over and above the explanation for each individual asserted truth. Perhaps a fact checker removed all false assertions from the book, and this explains why the conjunction of Damiens assertions are true.
Of course, this will only save the argument for the PSR if there's something preface-paradoxy going on with the conjunction of all contingent facts.

7. Thanks for the comments, Dan. I'm having a hard time figuring out what to make out of your preface-paradox, but my main thought is this: the proposition -Damian asserts only truths in his book- has an explanation somewhat like (A) or (A*). But the sentence 'Damian asserts only truths in his book' is often used to conversationally implicate certain other propositions. These other propositions call for extra explanation. Here are two stabs at what that proposition might be.
1) the proposition that -Damian asserts only truths and what is asserted in Damian's book is up to Damian-. In order for this proposition to have a sufficient explanation, it is not enough to point out that each assertion Damian makes is true. We also need to point out that what is asserted in Damian's book is up to Damian. This rules out the fact-checker worry, I think. But it doesn't explain what's going on with the BCCF; so if we think that there is an analogy between the two cases, this isn't going to solve the problem.
2) The proposition that -Necessarily, for all x, (if x is asserted in Damian's book, then x is true)-. This is basically the strict conditional version of the standard claim. The necessity operator at play here is weaker than metaphysical necessity. It's probably something like personal necessity: what is necessarily the case given the current facts about Damian. This proposition needs more of an explanation than just what Damian in fact did. And this is probably the worry we have about the BCCF: It's not necessarily the case that every conjunct of the BCCF is true (assuming 'BCCF' is a rigid designator).
But necessarily true propositions are outside the domain of the PSR, so that seems fine to me.

I'm inclined to think that the propositions in (1) and (2) are both among those pragmatically implicated by the sentence 'Damian asserts only truths in his book'. It's possible that Dan's puzzle is a result of a confusion between these propositions. I don't have much of an argument for thinking that's the case, other than: it salvages some version of (A), which seems plausible.