A Response to Almeida and Judisch


Alexander R. Pruss

Department of Philosophy

Georgetown University

Washington, DC 20057




Richard M. Gale

Department of Philosophy

University of Pittsburgh

Pittsburgh, PA 15260





Our new cosmological argument for the existence of God weakens the usual Principle of Sufficient Reason premise that every contingent true proposition has an explanation to a weaker principle (WPSR) that every such proposition could have an explanation.  Almeida and Judisch have criticized the premises of our argument for leading to a contradiction.  We show that their argument fails, but along the way we are led to clarify the nature of the conclusion of our argument.  Moreover, we discuss an argument against us based on a principle of alternate explanation incompatible with our WPSR, and show that this argument fails.


Our new cosmological argument has generated some lively and fruitful critical discussion.[1] Jerome Gellman, in a friendly addition to it, developed an ingenious argument to narrow the gap between the God of traditional theism and the creator whose existence is proven by our argument.  Whereas our argument proves the existence of only a very powerful, intelligent and good necessarily existent creator, leaving it open whether it is omnipotent, omniscient and omnibenevolent, no less essentially all these things, Gellman proves the essential omnipotence of this being.[2]  Graham Oppy criticized our argument for being question-begging[3], and Kevin Davey and Robert Clifton developed a set theoretical objection and also tried to show that for some of our premises there are incompatible propositions that are intuitively just as plausible, thus resulting in a conceptual stalemate.[4]  We have responded to all of these criticisms.[5] Michael Almeida and Neal Judisch have presented in this Journal a new challenge to our argument which attempts a reductio ad absurdum of it.[6] Their objection will prove to be a very fruitful one, because it can be neutralized only by giving a much more in depth account of our explanatory creator than our original argument offered. Before we attempt to respond to this reductio, a brief sketch of our argument is needed.

Our new cosmological argument has a significant advantage over traditional cosmological arguments in that the latter employ the Strong Principle of Sufficient Reason which requires that every true contingent proposition actually have an explanation, our argument needs only the Weak Principle of Sufficient Reason which requires that every true proposition possibly have an explanation:

(WPSR)   For any contingently true proposition, it is logically or conceptually possible that it has an explanation.[7]

A possible world is a maximal conjunction of compossible abstract propositions (a BCF in our abbreviational terminology), with repetitions and other logical redundancies eliminated so as to avoid to set-theoretic paradoxes. Each world is individuated by its Big Conjunctive Contingent Fact (BCCF), which is the conjunction of all of the  contingent conjuncts in its BCCF. The argument makes use of the following two principles: The S5 system of modal logic, whose key axiom is that if a proposition is possibly necessary, it is necessary; and The principle of the Existential Commitment of Explanation –- if  q explains p, then  q and p.  Let “@” be a name rigidly designating the actual world so that it is true that @ is actual but it is possible that @ is not the actual world.

(1)   @ is the actual world and p is the BCCF of @.   By stipulation

By appeal to WPSR and using the semantics of possible worlds, it follows that

(2)   There is a possible world w1 that has the proposition that ($q)(q explains p) as one of its conjuncts.

By existential instantiation it follows from (2) that

(3)   The proposition that q explains p is a conjunct of w1.

By appeal to the Principle of the Existential Commitment of Explanation it follows that

(4)   Propositions q and p are conjuncts in w1.

Given both S5 and that worlds share their necessary conjuncts in common, it follows that words are individuated by their BCCF.

(5)   For any worlds, w and w1w=w1 if and only if w’s BCCF is a conjunct in w1’s BCCF.

The proof of this can be found in our original paper, and Almeida and Judisch do not question it.  From (1), (4) and (5) it follows that

(6)   @=w1.

(7)   q explains p, q and p are all conjuncts of @.  From (3), (4) and (6)

Thus there actually is a true explanation of @’s BCCF.

It is then argued that this explanation must be a personal explanation in terms of some person’s free intentional action. This person cannot be contingent, since then a proposition reporting its  existence will be a conjunct in p but no person can bring it about that it exists.  Thus, this person must necessarily exist. Appeal to various teleological considerations must be made to show that this being probably is very intelligent, powerful and good.

Almeida and Judisch present two different reductio ad absurdum rebuttals of our argument. Using the premises of our argument, the first deduces that @ has q and ~q as conjuncts, the second that @ contains q and some other proposition, r,  that also explain p and is logically incompatible with q. The former begins with the fact that q is contingent.  Almeida and Judisch then reason as follows:

Since q is contingent we know, too, that M~q [it is logically possible that ~q].  It is possible that q is not the explanation of the actual world’s Big Conjunctive Contingent Fact.  But if M~q then there is some world w2 such that w2 contains p and ~q and the fact that q does not explain p. (59)

From the latter conclusion, absurdity ensues.  For if w2 contains p, then, by our argument, w2=@, and thus the actual world, @, has both q and ~q as conjuncts!     

It is far from obvious that the conjunction of the premises of our argument with M~q entails this absurdity.  Almeida and Judisch are aware of this and present an indirect proof of this entailment. It begins with the assumption for the sake of an indirect proof that it is false that M(~q and p and ~(q explains p)).  Then, “every world w* which contained p would also contain q and the proposition that q explains p.  But that is just to say that Lq (that it is logically necessary that q) contrary to our assumption…”(63n15). Hence, M(~q and p and ~(q explains p)). This reductio of our argument fails, because the claim that every world containing p contains q and q explains p means only that

(8)   L(p É (q and (q explains p)))

whereas the authors seek the stronger conclusion Lq.  This stronger conclusion would follow only if in addition to (8) we had Lp.  But if we had Lp, then p would not be a Big Conjunctive Contingent Fact.

           Fortunately, we need not confine the defense of our argument against the reductio based on the attempt to deduce M(~q and p and ~(q explains p)) from M~q to a rearguard action in which we wait for someone to produce a deduction and then show where it fails. For we are in a position to prove that the premises of our argument entail the impossibility of this deduction.   Recall premise (1)’s stipulation that p is the BCCF of @. Now, our argument proves that q explains p in @, and moreover (q explains p) is a contingent proposition since it is actually true and entails the contingent proposition p by the Existential Commitment of Explanation. Thus, (q explains p) is a conjunct of p. Since a conjunction (like a set) is identified by its conjuncts (or members), it follows that it is necessary that a conjunction (or set) has the conjuncts (or members) that it has. And since (q explains p) is a conjunct in p, there is no possible world that has p and ~(q explains p) as conjuncts, and thus ~M(~q and p and ~(q explains p)).

           At this point one might note that since theism entails that every contingent proposition has an explanation in terms of the free actions of God and of any free creatures, an argument proving that the WPSR fails to hold in some world would also show that theism is false in that world.  It would, thus, indeed be a remarkable achievement to show that the WPSR is incoherent or false, but this achievement Almeida and Judisch have not achieved in the above argument.

           It is the aim of the second reductio of Almeida and Judisch to prove that the premises of our argument entail that @ has r and q as conjuncts, where r is a possible explanation for the actual BCCF incompatible with the explanation q:

Since q is a contingent explanation of the BCCF of [the actual world, @], we know that the actual BCCF might have had some other explanation r, such that ~M(r & q). (63)

They then make use of the premises of our argument to show that if it is M(r explains p), then @ contains r as a conjunct. But since it has been established by our argument that @ also contains the incompatible proposition q, the desired logical absurdity has been derived.

           Almeida and Judisch appear to claim that just from the fact that q is a contingent explanation, it follows that there might have been some other explanation.  But this is a non sequitur.  To see this, observe that the claim that q is a contingent explanation of p is ambiguous between two claims, neither of which is strong enough to show that there might have been some other explanation.  These two claims are:

(9)   (q explains p) and it is contingent that q

(10)  it is contingent that (q explains p).

It is clear that (9) is insufficient to show that there might have been some other possible explanation incompatible with q.  After all, all that (9) claims is that (q explains p) is a conjunct of the actual world and that there are some possible worlds of which q is not a conjunct.  It in no way follows from this that there are some possible worlds of which (r explains p) is a conjunct for some r incompatible with q.  But if (9) is insufficient to show the possibility of an incompatible explanation, neither is (10) sufficient to show it, because in fact (9) entails (10).  To see this, observe that if (9) holds then (q explains p) is a possible proposition.  By the Existential Commitment of Explanation, (q explains p) entails the proposition q which is contingent by (9), and hence (q explains p) is not a necessary proposition.  Thus, (q explains p) is possible but not necessary, and hence it is contingent, if (9) holds.

           It is thus obvious that Almeida and Judisch’s reductio must be employing a stronger version of the principle of sufficient than our

(WPSR)   For any contingently true proposition, it is logically or conceptually possible that it has an explanation.

Their reductio requires instead:

(WPSR2) For any contingently true proposition, it is logically or conceptually possible that it has an explanation, a, and it is logically or conceptually possible that it have another explanation, b, such that a and b are logically incompatible.[8]

           We grant that the conjunction of (WPSR2) with the premises of our argument does entail the logical absurdity that @ has the incompatible conjuncts q and r. This is check-mate for our new cosmological argument only if (WPSR2) is plausible and moreover more plausible than our (WPSR). It will be argued that it is neither.

           That it is less plausible follows from the principle of modal conservatism—that the weaker of two propositions is more likely to be possible than is the stronger one, in which a proposition a is weaker than a proposition b just in case b entails a but a does not entail b. By this principle (WPSR) is more likely to be possible, and thus is more plausible than (WPSR2), since (WPSR) is entailed by but does not entail (WPSR2). Thus, if we had to choose between accepting (WPSR) or accepting (WPSR2), the epistemically reasonable thing to do is to accept (WPSR). And, as a result, the epistemically reasonable thing to do when confronted with a choice between accepting our (WPSR)-based cosmological argument and accepting the (WPSR2)-based reductio of it is to accept the former.

           Almeida and Judisch might try to make (WPSR2) look plausible by appealing to ordinary cases of true contingent propositions, such as that Smith’s briefcase is on his desk is explained by the proposition that he left it there but also possibly explained by the incompatible proposition that his wife left it there. But it would be a hasty generalization indeed to generalize, as does (WPSR2), from these ordinary cases of true contingent propositions; for a world’s BCCF is a very strange bird, as is seen in the fact that unlike the former propositions, if it possibly has an explanation, then it actually has one.

Alternatively, they might argue for a more modest conclusion: not that (WPSR2), when taken in full generality, is plausible, but that it is plausible that the specific case of there being a possible explanation of p, the BCCF of the actual world, satisfies (WPSR2). In correspondence, Almeida has suggested that it is possible that p be explained in terms the free intentional free action of Odin or of some other god or great genius. However, as the second part of our argument shows, p can be explained only by the action of a necessarily existing person.  Pagan gods and geniuses are not conceptually suitable candidates for having necessary existence. Furthermore, it is implausible that there would be more than one necessarily existing intelligent being.  It must be borne in mind that the proposition, q, that explains p is existentially quantified, saying only that there necessarily exists a very powerful, intelligent and good being who freely brings it about that p. Being general, q leaves it open for us to fill in the details of the exact nature of this being in many different ways and what its reason for creation might be. But we must be careful not to confuse the epistemic possibility that there are different necessary explainers of p with the logical possibility that there are.  There are exactly ten epistemic possibilities for the 10100th digit of p, namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, but there is only one logical possibility.

But we need not confine ourselves to the negative task of showing why attempts to make (WPSR2) look plausible fail, since we are in a position to present a positive argument for its implausibility. This argument is based on the fact that when (WPSR2) is conjoined with a compossible conjunction of propositions, namely the premises of our argument, a contradiction is entailed whereas no contradiction is entailed by the compossible conjunction alone. The second reductio has painted itself into a corner. For it must employ (WPSR2). But (WPSR2) entails (WPSR). And, thereby, if the reductio argument succeeds, it applies to itself, becoming impaled on its own sword.

There is an alternative principle of explanation to (WPSR2) that will enable the Almeida-Judisch reductio to avoid self-immolation, since this principle does not entail (WPSR). This Principle of Alternate Explanation holds that,

(PAE)    necessarily, for any contingent proposition p that has an explanation q it is possible for there to be an alternate explanation r of p which is incompatible with p

(PAE) says both more and less than (WPSR) does; for, on the one hand it does not require that every true contingent proposition possibly has an explanation, but, on the other, requires that every true contingent proposition that actually has an explanation also actually has another explanation that is incompatible with the former one. Now the conjunction of (PAE) and (WPSR) entails (WPSR2).  Since, as we have just seen, (WPSR2) is incoherent, it follows that (PAE) and (WPSR) are logically incompatible.  We must thus choose between (PAE) and (WPSR). 

One consideration in favor of the (WPSR) is that there are controversial philosophical doctrines about which (WPSR) can remain neutral but (PAE) cannot, thus rendering it more likely that (PAE) is false than that (WPSR) is.  For example, the (PAE) is incompatible with the Kripkean doctrine that there exist entities whose origin is essential to them.  For suppose that it is essential to me that I come from a union of sperm A and egg B.  Then that I exist is explained by A and B having come together.  Moreover, by the Kripkean doctrine, this is an essential property of me: in every world in which I exist, it is true that A and B came together.  Therefore, there is no world in which my existence is explained by a proposition incompatible with the one based on A and B coming together, which is a counter-example to (PAE) but not to (WPSR).

Yet another counter-example concerns explanations of the one and only basis of propositions that report the obtaining of a moral obligation. One such explanation of moral obligation propositions is the divine command one according to which their sole basis can be found in their being commanded by God. Another is the contractarian account of them as arising from the consent of people. There also is the natural law account that finds their basis in our having received a certain kind of nature.  If any one of these three accounts of the only possible source of moral obligations is true, and well one of them might be, there is a counter-example to (PAE) but not (WPSR).

Furthermore, the (WPSR) is prima facie compatible with being generalized to non-contingent propositions: they might, for instance, be all taken to be self-explanatory.  However, the PAE is clearly false of necessary propositions, since presumably these are to be explained in terms of necessary propositions, and no possible proposition is incompatible with a necessary proposition.  The falsity of the (PAE) in the case of necessary propositions cautions one not to think that there is something about the nature of explanations that ensures that there are always alternate explanations.  Because of all of the above considerations, the WPSR is to be preferred to the PAE.

           At this point in the discussion, Almeida and Judisch might want to avail themselves of a radical move: hold onto (WPSR2), or even just the weaker (WPSR), but restrict it in such a way that one could deny that it is possible for a world’s BCCF to have an explanation. This is the sort of move a Hempel, Salmon or Grünbaum would make when confronted with our new cosmological argument. But someone who denies the possibility of a world’s BCCF having an explanation is in effect denying the possibility that theism is true, for it is a necessary conceptual truth that if theism is true, than there is an explanation for the actual world’s BCCF. Again, we are confronted with a question about the relative degree of plausibility of rival propositions. Is it more plausible to think that theism is possible than to think that it is impossible for there to be an explanation of a world’s BCCF? Most people, including atheists, have a prima facie modal intuition in favor of the former. But our prima facie modal intuitions, although conversation beginners, are not conversation enders. However, plainly, the onus is on the denier of these intuitions to show why they cannot be trusted. And this is something that neither Almeida and Judisch nor anyone else that we know of has succeeded in doing.[9]

[1].   “A New Cosmological Argument”, Religious Studies 35 (1999), 461–476.

[2].   “Prospects for a Sound Stage-3 of Cosmological Arguments,” Religious Studies 36 (2000), 195–201.

[3].   “On ‘A New Cosmological Argument’,” Religious Studies 36 (2000), 345–353.

[4].   “Insufficient Reason in the ‘New Cosmological Argument’,” Religious Studies 36 (2000)

[5].   “A Response to Oppy and to Davey and Clifton,” Religious Studies 38 (2002) 89-99.

[6].   “A New Cosmological Argument Undone,” International Journal for the Philosophy of Religion 51 (2002), 55-64. References to this article are included in the body of the paper.

[7].   Our original formulation did not restrict this to contingent propositions, but following Kevin Davey and Robert Clifton we now make the restriction.  The notion of explanation for necessary propositions is an obscure one, requiring further research into mathematical explanation, and we do not need to go there.

[8].   That these alternative explanations are incompatible is required by the second reductio, for if it were just a case of explanatory or causal overdetermination, no absurdity could be deduced.

[9].   We are grateful to Michael Almeida for interesting discussions of these issues.