Introduction

In the last post, we considered an argument against the Principle of Sufficient Reason, formulated by Alexander Pruss in the Blackwell Companion to Natural Theology as:

(PSR) Every contingent fact has an explanation.

We saw that this principle has intuitive appeal, and various arguments have been offered in favour of it, e.g. the above Pruss article, Della Rocca (2010) and Koons and Pruss (2020). Nevertheless, in both van Inwagen (1983) and in my previous post it was argued (for different reasons) that the “Big Conjunctive Contingent Fact” (BCCF), the conjunction of all contingent facts, constitutes a counterexample to this principle.

So, we have a situation here where we have an intuitive conjecture or principle, and a handful of ‘proofs’ for that principle, and yet we also have a counterexample. And just to pile on, we have other potential counterexamples too, like libertarian free willings, or truly random events. What does one do in such a situation? Well, this exact such situation (in a mathematical context) is the subject of Imre Lakatos’ excellent book Proofs and Refutations. In that book, he examined (in the form of a highly entertaining dialogue between teacher and students) various strategies that one can take in such a situation.

A note on terminology: Lakatos makes a useful distinction between local counterexamples–which refute the lemmas of the offered proof–and global counterexamples–which refute the conjecture. Some counterexamples are local but not global, some are global and local, and some are global but not local (which reveal the proof to be invalid). In this case, the BCCF is a global counterexample.

Strategies for facing a counterexample

The first strategy is total surrender. The principle has a counterexample, it is false. We should pack up the PSR and go home. Needless to say, Lakatos notes that such a strategy is a bit too hasty, and throws out the baby with the bath-water. “Columbus did not reach India but he discovered something quite interesting.”

A second strategy is the method of monster-barring. Let us let Lakatos’ character Delta speak for themselves:

DELTA: But why accept the counterexample? We proved our conjecture–now it is a theorem. I admit that it clashes with this so-called ‘counterexample’. One of them has to give way. But why should the theorem give way, when it has been proved? It is the ‘criticism’ that should retreat. It is fake criticism. … It is a monster, a pathological case, not a counterexample.

We can see this attitude in some responses to the above arguments against the PSR. For example, Tomaszewski (2016) (following a Cantorian argument from Grim (1984)) argues against van Inwagen’s argument by arguing  that the BCCF is not a valid proposition. This strategy is not without its issues however. First, the monster might be a thing we’d like to keep around. For example, Pruss uses the BCCF in his Blackwell Companion article to argue for the existence of God. Grim argues that with the death of propositions like the BCCF also comes the death of thinking of possible worlds as sets of propositions. Second, as Lakatos shows us in his dialogue, monster-barring can get out of hand as more counterexamples arise.

A third strategy is the method of monster-adjustment. Here, instead of slaying our counterexample, we tame it instead. We show that it is really an example after all. This is the strategy of Pruss’ Blackwell Companion article. Van Inwagen’s argument for the BCCF being a counterexample rested on explanations entailing their explananda. By refusing that requirement for explanation, Pruss argued that the BCCF (and indeed, our other potential counterexamples like libertarian free willings or random events) is no counterexample. This strategy too has its issues. For one thing, we may not believe this adjustment of the monster. For another, as we saw in my previous post, the monster may have more teeth than expected.

We turn then, to the fourth strategy that Lakatos discusses…

Exception-barring, and restricted PSRs

The fourth strategy is that of exception-barring. Let us allow Lakatos’ character Beta to speak for themselves:

[BETA:] There are certainly three types of propositions: true ones, hopelessly false ones and hopefully false ones. This last type can be improved into true propositions by adding a restrictive clause which states the exceptions. I never ‘attribute to formulae an undetermined domain of validity. In reality most of the formulae are true only if certain conditions are fulfilled. By determining those conditions and, of course, pinning down precisely the meaning of the terms I use, I make all uncertainty disappear.’ [Quoting Cauchy (1821)]

This strategy has two approaches. Either we give the original principle with a list of exceptions, which can seem very ad hoc, or we state some restriction into a safe domain, where we are most sure the principle holds.

It is in the context of this strategy that restricted PSRs arise. Let us consider three of those here, two from the literature and also one from a discussion I had concerning my previous post.

Restriction to purely contingent facts

There is a bit of an odd property of the BCCF: not only does it have all contingent facts as conjuncts, it seems to have all the necessary ones too. For consider any contingent fact p and necessary fact q. Then p & q is a contingent fact, and hence is a conjunct of the BCCF. But q is a conjunct of p & q, and a conjunct of a conjunct is a conjunct. Hence q is a conjunct of the BCCF. (I’m pretty sure this detail was first noticed in the literature by Oppy, but I’m not sure of the reference.)

It was pointed out to me that this same detail was used in my previous post arguing against the PSR (and indeed, in Tomaszewski’s argument against the BCCF). We might think there is something dubious about contingent facts like p & q with necessary conjuncts. Let us suppose for a moment then that we can rule such propositions out, and define a purely contingent fact which will have no necessary conjuncts or anything like that. Then we can define a PSR:

(PC-PSR) Every purely contingent fact has an explanation.

The proponent of cosmological arguments can be happy with such a restriction. In place of the BCCF, they can just use the conjunction of all purely contingent facts, or all atomic contingent facts, or such like.

But there is an issue: how are we going to define purely contingent facts? Let us try a natural approach. Define a proposition to be strictly possible if it is neither necessarily true nor necessarily false. (Contingent facts are just the true strictly possible propositions, so it seems more natural to focus on the latter.) We might recursively define a proposition p to be purely strictly possible if and only if

• p is atomic (has no ¬, & or ORs) and is strictly possible
• p = ¬q and q is purely strictly possible.
• p = q & r and q, r are both purely strictly possible.
• p = q OR r and q, r are both purely strictly possible.

But this has a couple issues. First, the recursion gets stuck for infinite propositions like the conjunction of all purely contingent facts. More importantly, it seems to be false! Let q = “it rained on me today”. This is surely purely strictly possible. But then so is ¬q by the first bullet point, and therefore so is q & ¬q by the second bullet point–but that is clearly necessarily false.

This issue may be insoluble. Let q be as before and let p = “I got wet”. Then both p OR q and p OR ¬q are contingent facts, and they seem purely contingent if anything is. Hence the conjunction of all purely contingent facts has as a conjunct:
(p OR q) & (p OR ¬q). But we know from De Morgan’s laws that this expresses the same fact as p & (q OR ¬q), which has a necessary conjunct.

Perhaps we could try to sidestep all these issues by restricting the PSR all the way to:

(A-PSR) Every fact which is a conjunction of atomic contingent facts has an explanation.

But now this principle seems a bit ad hoc. It may be true that when we were first thinking about contingent facts we might not have been thinking about facts with necessary conjuncts–so the first restriction was fair game–but where does this new restriction come from? Should we believe this new principle, which is not the one our intuition suggested? Let us put this aside for now, we shall take it up again when we discuss exception-barring in general.

The Restricted PSR

As mentioned in the last post, Pruss (2003) offers the restricted PSR:

(R-PSR) If p is a true proposition and possibly p has an explanation, then p actually has an explanation.

This offers the simple diagnosis of our counterexamples: they are all unexplainable in principle. Perhaps it is not unreasonable then to restrict the PSR to those propositions which can be explained. Pruss then offers a cosmological argument from the Big Contingent Existential Proposition (BCEP), the contingent proposition that is the conjunction of all contingent propositions of the form “x exists(/existed/will exist)”.

Pruss argues that the BCEP is explainable: there is a possible world in which each actually existent contingent entity x is caused to exist by some other contingent entity (e.g. for each such x there is a genie which has the power explicitly to create an x). Hence it is actually explained, and therefore must be explained by the causal activity of a necessary being (since for it to be contingently explained in the actual world would be circular).

There are a number of things to be said here. Pruss anticipates a major objection drawn from Kripke–that the origin of a being is essential to it and couldn’t possibly be otherwise–which involves modifying the BCEP in a way we shall not dwell on here.

First, we can ask if this restriction is actually a restriction. We can argue that the R-PSR entails the PSR–and so fails to avoid the counterexamples after all. Pruss anticipates this objection thus:

Let r be the false proposition that Bill Clinton died in 1990.  Let p be any true proposition.  Let q be the disjunction p or r.  Observe that it is possible that q have an explanation: In a possible world where Bill Clinton dies of a heart-attack in 1990, q is explained by Bill Clinton’s heart-attack.  Thus, by the RPSR, q actually has an explanation.  But an explanation of a disjunction one of whose disjuncts is false will have to explain the true disjunct.  Thus, an explanation of q will have to explain p.  Thus, p has an explanation.  Since p was an arbitrary true proposition, it follows that all true propositions have explanations.

Pruss offers two lines of response to this argument. The first, is to invoke the (undefined, but intuitively familiar) notion of a proposition being true in virtue of another proposition. We then further restrict the PSR to:

(R-PSR2)  Suppose p is an explainable true proposition that is not true in virtue of an unexplainable proposition.  Then, p has an explanation.

This avoids the above entailment of the PSR, but as Pruss notes it is a bit unnaturally weak. He offers a third R-PSR (where p>=q denotes p being true in virtue of q or of p and q being identical and true):

(R-PSR3)  Suppose p is an explainable true proposition and that there is a q with p >= q such that for any r such that q>=r we have r explainable.  Then, p has an explanation.

Are you scratching your head at this point? I am, and it seems even Pruss is, saying “at this point things may be getting too complicated for one’s intuitions.”

A different approach Pruss offers is that a proposition like “Napoleon lost at Waterloo OR Clinton died in 1990” is explained simply by “Napoleon lost at Waterloo”, without having to explain why Napoleon lost. That is, the explanation of a disjunction with a false disjunct need not explain the true disjunct. Pruss doesn’t give much argument for this, and frankly it strikes me as implausible. If I asked (in say 2009) “why did one of the Williams sisters win Wimbledon this year?” (i.e. “Why is ‘Venus Williams won Wimbledon OR Serena Williams won Wimbledon’ true?”) and you replied “Because Venus Williams won Wimbledon”, you wouldn’t really have answered my question. To answer my question, you’d need to say something like “Because the final was Williams v Williams” or “Because they’re the best players in the world right now”.

There is a further objection that Pruss doesn’t mention in his paper. There feels like there is a bit of a bait-and-switch going on with the explainability of the BECP. That is, Pruss’ argument relies on the BECP listing all the actual contingent entities, then embedding the BECP in a world where there are lots more. But explaining the BECP in such a world feels, well, a bit like cheating. The spirit of the BECP has dramatically changed between these two different contexts. That is, when we agreed to restrict our PSR to explainable propositions, the notion of explainability was narrower than just “is explained in some possible world”.

We already saw this in the above objection that all propositions are one false disjunct away from being explainable, and Pruss therefore trying ever more sophisticated attempts to refine this notion.  In that case, the issue seemed to be that the “spirit” of p OR q (for q false but possibly true) was p in this world and yet it was explained via q in the possible world where it was explained. Likewise then, the “spirit” of the BECP in the actual world is surely the universal “for all existing contingent beings, those beings exist”. The BECP is essentially the extension of this universal in the actual world. But in the world which Pruss invokes to explain the BECP, it is no longer the extension of this universal–there are other contingent beings not listed in the BECP.

Put another way, one might assert that the BECP is true in virtue of this universal being true (since the BECP is essentially just spelling out this universal), and that this universal might well be unexplainable. So then the BECP doesn’t obviously fall under Pruss’ R-PSR2 (and who can tell whether it falls under the R-PSR3).

Existential PSRs

Another strand of restriction is to restrict the PSR simply to existential statements. Consider the following from Rasmussen (2010) (edited only for formatting limitations):

(Def4) Contingent state of existence = def A possible state of affairs of certain
contingent individuals, the c’s, existing.
(Causal4) Necessarily: (Normally, for all contingent state of existence S, possibly (S’s
obtaining, or the obtaining of a duplicate of S, is causally explained)).

Or in simpler terms (ignoring the Kripkean worry about duplicates)

(E-PSR) Every normal contingent state of existence is possibly causally explained.

Note how restricted this is! This principle not only restricts to existential statements, it furthermore weakens explanation to possible explanation and furthermore allows for exceptions by the term “normal”, which they clarify means that “one may infer that its existence can be caused—unless one has a reason for thinking that the object in question is an exception.”

Rasmussen then develops a cosmological argument from the existence of a state M which is a maximal contingent state of existence (i.e. it entails every contingent state of existence compatible with it). He argues that we have no reason to think that such a state is an exception to (Causal4) (and so likewise our E-PSR). Then if M is possibly explained, it is not explained by the causal activity of any contingent being on pain of circularity, so there must be a necessary being in order to causally explain it.

In a sense, there is not much that needs to be said here. Rasmussen doesn’t spend too much time defending the premise that M is not an exception to the E-PSR. Perhaps we do not need to argue that M is an exception to defeat the claim that it is not. I might suggest that the maximality of M makes it strange enough to not be a “normal” state of existence. We shall treat this type of thing a bit more systematically later.

A review of exception-barring, and the method of proof and refutations

OK, so that was a lot of text, where do we stand? Well, we saw a key theme in these exception-barring restrictions. The restrictions started off intuitive and non-ad hoc, but they quickly had to become much more technical, or much less intuitive, or both. We often end up with a strange-looking principle, that no longer includes many examples that motivated the original principle (e.g. the A-PSR and E-PSR do not say anything about “A Williams sister won Wimbledon in 2009”, but we don’t really think for a moment that this lacks an explanation) or that looks like someone dumped a load of symbolic logic all over it. Or in the case of the E-PSR, has lost all of its boldness and asserts a principle with the caveat “except for the exceptions”.

Finally, Lakatos notes two global concerns with exception-barring. First, how can you be sure your restriction has caught all of the exceptions? Second, and relatedly, where has your proof been in all of this? We have forgotten, in all this talk of PSRs and restricted PSRs, why we ever thought the PSR was true to begin with.

Lakatos answers both worries with the method of lemma-incorporation. When we are confronted with a counterexample, we must return to our proof. We may find then, that the proof does not apply to the counterexample, i.e. the counterexample is also a counterexample to one of the lemmas. When this happens, we incorporate the lemma as a condition. In so doing, we restrict our principle to the one that our arguments prove. In so doing, we ensure that our restrictions are never ad hoc, that it includes all of the examples that motivate our arguments, and that we have an argument that we have caught all of the exceptions. This is also called the method of proof and refutations.

The many proofs of the PSR

We turn, for the first time so far, to the argument for the PSR. This post is already very long, so let us summarise the broad strokes. Pruss has argued for the PSR on the grounds that (I’m sure there are more than these, I’ve still not gotten round to Pruss’ PSR book):

1. It is self-evident.
2. Without the principle, we face global scepticism.
3. Without the principle, we face scientific/empirical scepticism.
4. Without the principle, we can’t justify inference to the best explanation.
5. If it were false, we should expect to see widespread violations
6. We need the principle to account for the nature of modality (I’m going to ignore this one, because I don’t understand it).
7. We need it to run various philosophical arguments.

Della Rocca (2010) argues for the PSR by arguing that in philosophy we frequently make what he terms explicability arguments. “In such an argument, a certain state of affairs is said not to obtain simply because its obtaining would be inexplicable, a so-called brute fact.” He motivates this by appealing to how we argue that a balanced scale should be level, Parfit’s argument that a split-brain case must create 0 or 2 persons, how we argue that various properties (e.g. dispositions, consciousness) supervene upon or reduce to others, and to the Early Modern argument against Aristotelian forms. (Frankly, I would argue that none of the examples Della Rocca gives are really explicability arguments–they strike me as mostly either symmetry or parsimony arguments–but that would take us a bit afield.)

Della Rocca then turns to the question of existence:

What is it in virtue of which this thing exists? In the same way, just as the consciousness of a given mental state must be explicable, so too, perhaps, the existence of a given thing must be explicable. We must have an account of what it is in virtue of which that thing exists, just as we must have an account of what it is in virtue of which a given mental state is conscious. If we take this path, then we advance an explicability argument: the existence of each thing that exists must be explicable, just as the consciousness of each conscious mental state must be explicable, […] to insist that there be an explanation for the existence of each existing thing is simply to insist on the PSR itself, as I stated it at the outset of this paper.

A proof-analysis in light of counterexamples

So we have our proofs, and for this post we shall not attempt to criticise these proofs directly, but the counterexamples have revealed to us they do not prove the full PSR. We must examine then how the counterexamples challenge our proofs. What is it that these proofs really prove?

I think proofs (1) and (5) immediately suggest us upon a particular train of thought here. These prove, respectively

(PSR-1) Every proposition that we have an intuitive feel for has an explanation.

(PSR-5) Every (or almost every) proposition that occurs in everyday life has an explanation.

Arguably, these are the same PSR. Likewise, (7) and Della Rocca’s arguments lead towards a PSR:

(PSR-7) Every (or almost every) proposition that occurs in common philosophical discourse has an explanation.

Though here I think we can be sceptical of even that. After all, we can be sceptical of how much of common philosophical discourse is in fact meaningful. Perhaps a lot of philosophy is a lot of searching for explanations that aren’t there. Many of Della Rocca’s examples are after all analyses of concepts that may not have analysis.

Finally, we have arguments (2), (3), and (4). Let us abbreviate these as arguments for the following PSR:

(Epistemic-PSR) Every epistemologically necessary proposition has an explanation.

Where by “epistemologically necessary” we mean those propositions which, were they to be brute, would conjure sceptical scenarios (either globally or for empirical knowledge). What sorts of propositions are these? Well, they are things “I have sense-impression such-and-such” (e.g. “I have a sense-impression of my hand in front of me”). If these were brute and uncaused, then I would seem to lack any knowledge from my senses. I surely have knowledge about my hand from my sense-impressions only if my hand is causing those sense-impressions.

Cosmological arguments defeated?

By looking at these various PSRs, we can be unsurprised that the BCCF caused us all these issues. For we have no intuitive feel for it, it never turns up in everyday life, it is not particularly central to philosophical discourse, and it is not epistemologically necessary. Therefore, we should not be oversurprised if the PSR doesn’t apply to the BCCF.

But hang on, the sceptic of cosmological arguments hardly needs to stop there. For is the same not true of the conjunction of all atomic contingent facts, of the BCEP, and of Rasmussen’s maximal contingent states of affairs? When we restrict the PSR by lemma-incorporation, we find that we end up with PSRs far too weak to run a cosmological argument!

The method of proofs and refutations

The above defeat of cosmological arguments is however too hasty. For Lakatos reveals yet another method: the method of proofs and refutations. We saw that none of the basic arguments proved a PSR strong enough for a theistic purpose–and that the counterexamples required that PSR proponents not overreach beyond what they have proved–but this leaves open the idea that a new proof will expand the domain wide enough to argue for a God.  For example, Koons and Pruss (2020) argues that the (Epistemic-PSR) must not only be true, it must be robustly true (i.e. true in all nearby possible worlds in all epistemically possible scenarios) and knowable a priori. They argue that this in turn entails a PSR for natural facts, which in turn entails the existence of a supernatural being.

I shall leave a full discussion of Koons and Pruss’ argument for another post. But even supposing it fails, what is to say that some clever philosopher might not fix it? How can the atheist show that no PSR strong enough to entail God is true?

One option is straightforward: they can prove that God doesn’t exist. But at that point, they’ve already won the debate. Absent such a counterexample (or should I say counter-non-example?) we shall need some sort of counter-principle, that either rules out such a PSR or rules out us ever knowing such a PSR. For example, for the latter we might turn to the argument of Kant. To quote Kreines (2008)

Kant argues that the faculty of reason always demands that we seek underlying conditions, and ultimately that we seek a complete series of conditions, or an unconditioned ground: ‘the proper principle of reason in general … is to find the unconditioned for conditioned cognitions of the understanding’ (A307/B364). But Kant argues that we can never achieve theoretical knowledge of anything unconditioned. So the faculty of reason demands that we seek knowledge of something of which we cannot achieve knowledge. This means that reason’s principle must be restricted to a special status: it is legitimate only as ‘regulative’ or guiding principle for our theoretical inquiry. But Kant argues that we are naturally tempted to instead accept it as an objective principle guaranteeing that, for anything conditioned, there must really be some complete series of conditions, or some unconditioned ground. […] And this tempting principle is a version of the principle of sufficient reason: for anything that is not a sufficient reason for itself, or for anything conditioned, there must be a complete series of conditions that provides for it a sufficient reason. Kant can explain in these terms, for example, why we are naturally tempted by rationalist arguments from the existence of anything conditioned to the existence of God as an ultimate ground or an original being. But he can also argue that such rationalist arguments must be rejected: to assert knowledge of the principle grounding such arguments, or of such conclusions, is to violate our epistemic limits — for these limits prevent all knowledge of anything unconditioned.

That is, Kant notes that the PSR emerges from transcendental reasoning on the presuppositions of human knowledge–as we have indeed seen, with perhaps the most promising arguments for a strong PSR all being based heavily in epistemology. But Kant argues that this imposes a limitation on the strength of any PSR so inferred. Such a PSR can only act as a regulation on human reason, delivering conclusions about “conditioned things”, it cannot leap beyond its confines and deliver conclusions about the ultimate unconditioned ground of all being.

Of course, Kantian epistemology is not a small thing to invoke. I am not sure that it is at all popular in contemporary philosophy. Nevertheless, there is a dialectic at work here. The proponent of cosmological arguments is putting forth proofs of PSRs they hope can prove God. The atheist is putting forth refutations, counterexamples to those strong PSRs and to the lemmas in the proofs. But the bold atheist can go further. They can conjecture:

There is no (knowable) PSR that is strong enough to prove that God exists.

Now it is the atheist that must engage in proofs and refutations–the refutations being the strong PSRs (e.g. the R-PSR) and the proofs being either arguments for atheism or things like the Kantian argument above. Even if we reject many of the premises of the Kantian argument, the germ of it–that the PSR is too bold for the arguments that support it–is something that has some intuitive appeal.

Conclusions

In this post, I have hoped to achieve a couple of things. First, I hope to have piqued any reader’s interest in Proofs and Refutations, which is an absolutely fantastic book. Second, I hope to have shown how the framework of Proofs and Refutations can illuminate the conversation regarding the PSR and its restrictions. Third, by framing the restricted PSRs as examples in exception-barring I hope to have demonstrated the fundamental drawbacks of those PSRs. Fourth, I hope to have shown that the better strategy of lemma-incorporation results in PSRs that the atheist can quite happily accept, that the PSRs often used in cosmological arguments are much stronger than what the arguments for them typically support. Finally however, I hope to have indicated that the atheist cannot be complacent here–new arguments may arise (or have already arisen) that yield strong enough PSRs. To be truly able to rest, the atheist must take up arms themselves and prove that no such strong PSR is true (or is knowable).