EDIT: Dear people from 2022. Due to the unexpected recent interest in this post, I have edited it somewhat to improve the fairly awful layout it originally had (too many postscripts!!).

What is the PSR?

To quote the SEP, “The Principle of Sufficient Reason is a powerful and controversial philosophical principle stipulating that everything must have a reason, cause, or ground.” In simple terms, the PSR says that nothing is “just true” or “just exists”. This principle holds an important place in the history of philosophy, famously being promoted by Spinoza and Leibniz, and rejected by Hume, Kant and Russell.

Though the PSR is a centrepiece of Rationalist philosophy, our concern with the PSR in this post will primarily be in a more limited context, viz. the Leibnizian cosmological argument for the existence of God. In his excellent Blackwell Companion to Natural Theology article, Alexander Pruss states this argument as follows: ([..] comments mine)

(1)               Every contingent fact has an explanation. [The PSR.]

(2)               There is a contingent fact that includes all other contingent facts. [The Big Conjunctive Contingent Fact or BCCF.]

(3)               Therefore, there is an explanation of this fact.

(4)               This explanation must involve a necessary being.

(5)               This necessary being is God.

In that article, Pruss gives a number of arguments in favour of (1) (and, indeed Pruss has written a whole book on the PSR, which I have yet to get around to reading). These range from it being self-evident, to it being necessary for our scientific reasoning, to it being required to account for modality. Another recent defence of the PSR can be found in an interesting paper by Michael Della Rocca. Nevertheless, many philosophers have objected to the PSR, notably in contemporary philosophy Peter van Inwagen (1983) (for a summary and response to van Inwagen’s argument, see section 2.3.2 of the Pruss article). In this post I will present a short argument against the PSR I came up with when debating the Leibnizian cosmological argument.

An argument against the PSR

Some key concepts

Contingency and Necessity The PSR stated above talks about “contingent facts”. By fact we here just mean a true proposition, but what is contingency? For this argument, we shall be using ‘contingency’ in the sense of modal logic, the logic of possibility and impossibility. A contingent fact is a fact that could have been false, it was possible for it to be otherwise. Examples would seem to be facts like “it rained today” or “the Allies won WW2”. Examples of non-contingent (i.e. necessary) facts would seem to be facts like “1 + 1 = 2” or (if it is true) “God exists”.

Propositional Logic The other notion we will need for this argument is that of propositional logic. Propositional logic is one of the simplest logics, concerning only the abstract form of propositions (i.e. propositions as built up from atomic propositions via connectives like ‘and’, ‘or’, and ‘if … then …’). To quote Wikipedia:

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

A theorem of propositional logic is therefore a proposition which is true purely in virtue of its form, examples including “if p then p” or “p or not-p”. We shall use some of the notation from propositional logic in this article:

• “p & q” denotes “p and q” (note that such a proposition is called a conjunction, and p and q are then called the conjuncts)
• “p ∨ q” denotes “p or q”
• “¬p” denotes “not-p” (i.e. “p is false”)
• “p ↔ q” denotes “p if and only if q”, i.e. p and q are either both true or both false. If this is a theorem then we say that p and q are logically equivalent (in propositional logic).

The basic argument

The structure of this argument will be a reductio ad absurdum: we will present a bunch of ‘axioms’ of explanation, add the PSR and derive a contradiction.

First, the axioms:

A1. If p explains q and “p’ ↔ p” is a theorem of propositional logic, then p’ explains q.
A2. If p ∨ q explains r and q is false, then p explains r.
A3. If p is a contingent fact, then p does not explain p.
A4. If p is true and q is a contingent fact, then p & q is a contingent fact.
A5. There is a contingent fact b (a.k.a. the BCCF) such that all contingent facts are conjuncts of b.

Next, we follow Pruss and formulate the PSR thus:

PSR. If q is a contingent fact, then there exists a fact p such that p explains q.

Finally, we derive a contradiction:

1. There exists a fact p such that p explains b (A5,PSR)
2. “p ↔ [(p & b) ∨ (p & ¬b)]” is a theorem of propositional logic (easy to check)
3. (p & b) ∨ (p & ¬b) explains b (1,2,A1)
4. p & ¬b is false (A5)
5. p & b explains b (3,4,A2)
6. p & b is a contingent fact (A4)
7. p & b is a conjunct of b (6,A5)
8. b is a conjunct of p & b (by definition)
9. “p & b ↔ b” is a theorem of propositional logic (7,8)*
10. b does not explain b (A3,A5)
11. p & b does not explain b (9,10,A1)
12. p & b does and does not explain b (5,11,contradiction)

*This follows because it is a theorem of proposition logic that “(p & q) → p”, i.e. a conjunction implies its conjuncts. Hence, if two propositions are conjuncts of each other, then it will be a theorem of propositional logic that they imply each other.

Discussion of the axioms

It is easy to check that the derivation of the contradiction is valid, therefore either the PSR is false or one of the axioms must be false. We shall here motivate them, in order from least to most controversial.

The most straightforward seems to be A3. Part-and-parcel with contingency is the idea of contingent facts not explaining themselves. Furthermore if A3 is false then the Leibnizian cosmological argument is rather sunk.

Slightly less straightforward is A4. As we said above, by contingent we here mean could have been false. But if q could have been false, then it that same circumstance p & q would have been false, and A4 is demonstrated. However, it is noteworthy that there are other notions of contingency wherein this axiom does not hold.

On to the meat, A2 also seems very reasonable. Intuitively, if q is false then all of the explanatory work of p ∨ q must be coming from p, so p must explain r. Another way to argue this is to say that if p ∨ q explains r, then p explains r or q explains r, and if q is false then q does not explain r. Furthermore, we might think of an explanation of q by p as being something resembling a “reasonable” argument from true premises p to true conclusions q. But if p ∨ q is true and q is false then p must be true, and we can argue from p (in a trivial step of logic) to p ∨ q and thence to r via the argument by which p ∨ q explains r.

Finally, we get to A5. Of note first is that we can try to get away with laying this at the feet of the defender of the Leibnizian cosmological argument. After all, Pruss’ argument explicitly states that A5 is true. A5 is however possibly an issue. For example, (Grim, 1986) suggests that such a conjunction is actually “too big” to exist. These arguments are a bit too technical to get into here, but the fundamental idea is that there are “too many” contingent propositions (or even atomic contingent propositions) to form a mathematical set (just like there are “too many” sets to form a set of all sets). And if that is true, then surely you can’t have a conjunction if you can’t have a set.

Wait, did I forget about an axiom?

Arguments for A1, and weakening A1

A1 is the key idea of this argument. There are two avenues which motivate this axiom. First, we might say that if two propositions are logically equivalent in even the most basic of logics, then what that tells us is that they have the same content. They are essentially just rephrasings of the same idea. And therefore, they must explain the exact same things. Second, on the above picture of explanations as reasonable arguments, surely I give an argument that is just as reasonable if I start from p’ (which must be true if p is), derive p from it by propositional logic, and then follow the original argument to q. It seems hard to me to say what an explanation is if A1 is false.

Nevertheless, the main point of contention in responses I have received to this argument has been A1. In light of this, I thought I would zero in on exactly what I need out of A1 for this argument to work. We can break down A1 into the following weaker axioms:

A1a. If p explains q, then p & (q ∨ ¬q) explains q.
A1b. If p & (q ∨ r) explains s, then (p & q) ∨ (p & r) explains s.
A1c. If p & q explains r, then p & q & q explains r.
A1d. If p & q explains r and p is a conjunct of q, then q explains r.

These are the only equivalences that we rely on for this argument. A1d seems beyond reproach: subtracting repeated conjuncts surely can’t affect an explanation. A1b is a simple application of De Morgan’s laws: all that is happening is rearranging symbols, there is no change of meaning that is going on, nothing that could affect an explanation.

The only possible avenues of objection seems to be to A1a and A1c. Could conjoining a tautology or a repeated conjunct somehow spoil the explanation? It is hard to see how. To be sure, conjoining a fact to an explanation can spoil the explanation: for example, “John fell from a great height” explains why John died, but “John fell from a great height and John had a parachute” does not. In such a case, the conjoined fact conveys information that defeats the original explanatory link. But in the case of a tautology or a repeated conjunct, there is no defeating information, indeed there is no new information at all.

Perhaps the sheer irrelevance of the added conjunct spoils the explanation? But for A1a, q ∨ ¬q is not wholly irrelevant to the truth of q, and for A1c, the conjunct being added is not irrelevant, even if it is odd. Indeed, any explanation we actually make (i.e. one made in language, rather than abstract symbols) will have subtext. In “John fell from a great height”, a lot is left as going without saying, e.g. what falling is, that such falls are deadly to humans, that the past exists, etc. Surely the most basic of all such subtext is that the explanandum is either true or false. It is so basic that it always goes without saying, but surely it cannot harm the explanation to say it?

Finally, even if we did decide this isn’t how ‘explains’ works, we can easily adjust our notion of explanation. Let p ‘shmexplain’ q if it explains it except for any added q ∨ ¬q or any repeated conjuncts of p.* Then A1a and A1c would be true by definition for shmexplanation, and all of the other axioms seem equally true of shmexplanation. Finally, if p explains q then p shmexplains q so the PSR entails a PSR for shmexplanation. Thus A1a and A1c seem true for all intents and purposes of this argument.

*Formally, we would define

p shmexplains q if and only if  p = r & x, x is a finite (perhaps empty) conjunction with each conjunct either (q ∨ ¬q) or a conjunct of r, and r explains q

A further idea

Another idea that if p is a conjunct of q and q is a conjunct of p, then up to reordering p and q must just be the same proposition, since their set of conjuncts must be equal, even taking into account multiplicity of the conjuncts. Thus we have a further axiom:

A6. If p is a conjunct of q and q is a conjunct of p, then p = q.

But surely, we also have the following axiom:

A7. If p explains q and p = p’, then p’ explains q.

It would be very puzzling indeed if this axiom were false.

Two stronger versions of the argument

In light of these modifications to our axioms, we can present two stronger versions of our basic argument.  Both begin in the same way, showing that p & b explains b and that p & b is a conjunct of b.

1. There exists a fact p such that p explains b (A5,PSR)
2. p & (b ∨ ¬b) explains b (1,A1a)
3. (p & b) ∨ (p & ¬b) explains b (2,A1b)
4. p & ¬b is false (A5)
5. p & b explains b (3,4,A2)
6. p & b is a contingent fact (A4)
7. p & b is a conjunct of b (6,A5)
8. b does not explain b (A3,A5)

Note that thus far, we have used only A1a, A1b, A2, A3, A4, and A5. However, the two versions of the argument now diverge in how they show that p & b also does not explain b.

The first version, using  A1c and A1d, continues:

9. p & b & b does not explain b (7,8,A1d)
10. p & b does not explain b (9,A1c)
11. p & b does and does not explain b (5,10,contradiction)

The second version, using A6 and A7, continues:

9. b is a conjunct of p & b (definition of ‘conjunct’)
10. b = p & b (7,9,A6)
11. p & b does not explain b (8,10,A7)
12. p & b does and does not explain b (5,11,contradiction)

Possible responses

Non-modal contingency One option to avoid this argument (and also a more famous argument against the PSR from van Inwagen) is to use a different notion of contingency. Suppose we define a proposition to be “contingent” if it does not explain itself. Under this definition, A4 is false. Suppose that p explains itself and explains q, but q does not explain itself. Then q is contingent, but it seems plausible that p & q (via p) explains p & q. This definition is not without its drawbacks however. For it is no longer clear that the BCCF is contingent, which is crucial for a Pruss-style LCA to succeed. Indeed if the BCCF is not contingent then, barring worries about over-explanation, can the atheist not claim that the BCCF explains every contingent fact?

Modify the BCCF A key trick in the argument is using that p & b is a conjunct of b. But there are all sorts of ways one might restrict the BCCF to avoid this consequence–for example to a conjunction of atomic contingent facts. However, to pull this trick requires that one claim that A5 is false–that one of the arguments against the existence of the BCCF succeeds–and yet that some salvageable variant of the BCCF remains. That is a bit of a tightrope to walk. What is more, a key issue still remains. If b is not contingent, then the PSR does no work. If b is contingent, then so is p & b. If p & b can’t explain b, then the reductio succeeds. If p & b can explain b, then b can be contingently  explained. But if b can be contingently explained, it seems difficult for the proponent of the LCA to proceed. Surely the whole point of invoking a BCCF was to rule out contingent explanations?

Modify the PSR A final trick is to accept that this argument refutes the PSR as stated, but to offer an alternative PSR that will still allow a cosmological argument to run. Possible options might be an existential PSR:

(Existential PSR) For any beings x1, x2, … , if “x1 exists and x2 exists and …” is a contingent fact then “x1 exists and x2 exists and …” has an explanation. [Or existed/will exist, we don’t care about tense here.]

or Pruss (2003)‘s restricted PSR:

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

In either case, these PSRs no longer apply to the BCCF: for the former, because the BCCF is not existential, and for the latter because of arguments like van Inwagen’s or like the one in this post. Instead Pruss considers the Big Contingent Existential Proposition (BCEP), which we might state as a premise:

BCEP. There is a contingent proposition P that is the conjunction of all contingent propositions of the form “x exists(/existed/will exist)”.

If we run the above argument, we get that if there is a q such that q explains P then P & q is a contingent proposition that explains P. But then what? Unlike with the BCCF, there is no way to get a contradiction from this. Moreover, Pruss’ argument continues:

Now, the explanation of the existence of a concrete contingent being involves the causal efficacy of another concrete being.  Thus, the explanation of P must involve the causal efficacy of at least one concrete being.  Moreover, the beings whose causal efficacy is invoked in the explanation of P cannot all be contingent.  For then these beings by explaining P end up explaining their own existence.  However, neither the individual existence of a contingent being is self-explanatory nor is the existence of a bunch of contingent beings self-explanatory.  Thus, the explanation of P must involve the causal efficacy of at least one necessary being, a first cause.  QED

We can see from this that it does no harm to the argument to say that P is contingently explained–perhaps it is explained by a contingent fact about a necessary being–we must show that P is explained by a contingent being to disrupt this argument. And to do so will require more tools than merely the propositional logic trick of the above argument.

Perhaps the argument can still be salvaged. One objection is to argue that there can be no contingent facts about necessary beings–but Pruss will object to this by appealing to libertarian free will, and refuting that is no small task. Another is to argue that these PSRs entail the full PSR above. Indeed, in section 4.6. of the above paper Pruss considers such an objection levelled against the RPSR. Perhaps we can also do this for the existential PSR: let p be a contingent proposition about an entity x, e.g. “x went to the shops” (i.e. “x existed and went to the shops”). Perhaps we can define an entity y which is the-x-that-went-to-the-shops, and thereby reduce this to the strictly existential contingent fact “y existed”. This is perhaps a bit of a strange construction, but it doesn’t strike me as too outlandish. There are also other objections that might be made to such arguments, but I will leave that for perhaps another post.

UPDATE: Behold! the other post. 

Conclusions

I have here presented a short and fairly simple argument against a popular formulation of the PSR, which is relevant to the God question via the Leibnizian cosmological argument. The premises in this argument seem very reasonable, with the most contentious premises being ones borrowed from the Leibnizian argument itself. Thus this argument seems hard to avoid without undermining the Leibnizian argument in the process–however we have noted that there may be weakenings of the PSR that avoid this argument and yet may still allow a cosmological argument to run.