Infinite Regresses of Justification and of ExplanationAuthor(s): John F. PostReviewed work(s):Source: Philosophical Studies: An International Journal for Philosophy in the AnalyticTradition, Vol. 38, No. 1 (Jul., 1980), pp. 31-52Published by: SpringerStable URL: http://www.jstor.org/stable/4319392 .Accessed: 12/09/2012 18:50

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JOHN F. POST

INFINITE REGRESSES OF JUSTIFICATION AND

OF EXPLANATION

(Received 23 January, 1979)

According to William Alston, the weakest link in the regress argument for foundationalism is the rejection of infinite regresses of justification.' I am not so sure;some other links look as weak. But Alston is right that the reasons typically given for rejecting regress leave much to be desired. What follows is (1) an argument against infinite justificational regresses that is free of problems in the arguments to date; and (2) an application of this result to show that for a wide variety of concepts of explanation, including some according to which an explanation is not a justification, an infinite regress of explanations is also impossible, for reasons that have the additional effect of undermining leading versions of the Principle of Sufficient Reason.

In exploring (1) we shall see that there are logical or conceptual grounds, contained in any plausible concept of rational justification, for rejecting infinite justificational regresses. This contrasts with arguments that make the pathology of regress either a practical matter, such as the finiteness of our faculties, or pragmatic, such as the circumstances in which it would be appropriate to request or give a justification.2 If, instead or in addition, it is conceptually impossible for there to be such a regress, then it makes no difference whether we consider an infinite intellect, who could actually justify every statement in the regress, or a finite one, who would only be able to justify any particular statement on request. In either case there cannot be a regress in which every statement is justified by prior statements. Thus my argument takes into account the objection that only an unreasonable thesis about justification would require us to reject the regress, namely that a person must actually justify every statement in the regress, as opposed to being able to justify any particular one on request.3

Rejoicing by foundationalists would be premature. No foundationalist moral follows from the rejection of justificational regresses unless the remaining links in the regress argument are sound. In Section II I enumerate these links and sketch objections to the most important one. Whether the ob-

Philosophical Studies 38 (1980) 31-52. 0031-8116/80/0381-0031$02.20 Copyright ? 1980 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

32 JOHN F. POST

jections can be met is an issue too complex to settle here (indeed epistemolo- gists seem generally to have underestimated its complexity). Instead I conclude the paper with a couple of morals that can be drawn far more readily. Both extend beyond epistemology, into metaphysics. One moral (Section III) is that there cannot be an infinite regress of explanations, for a very diverse family of concepts of explanation, including some according to which explanations are not justifications or even arguments. Again the reasons are not practical, such as the finiteness of our faculties, but logical or conceptual, entailed by the very notions of explanation involved. Even for an infinite intellect, regresses of such explanations must end. The other moral (Section IV) is that leading versions of the Principle of Sufficient Reason are either demonstrably false or question-begging in their intended applications in Cosmological Arguments for God.

What conditions should inferential justification satisfy? In particular, what is it for person P to be inferentially justified at time t in believing statement Y on the basis of statement X, or for X to justify Y for P at t? No answer appears as yet to have achieved consensus. Fortunately none is required for our purpose. Most of the lists of conditions proposed in the literature include something like the following: at time t, (a) P believes Y (dispositionally or occurrently); (b) P is justified in believing X; (c) P believes that X adequately supports Y; (d) P is justified in believing X adequately supports Y; (e) P believes Y because he believes both X and that X adequately supports Y; and (f) there is no defeater; that is, no statement Z such that P is justified in believing both Z and that (X & Z) does not adequately support Y.4 For our purpose, conditions (a)-(f) may be refined or augmented in many ways, according to one's views about inferential justification. For example, let X, Y, Z be sets of statements; or replace (d) by 'X adequately supports Y', or (f) by 'P is justified in believing there are no defeaters', and so on. Adopt any plausible revision you like. Then construe 'X justifies Y' wherever it occurs below in terms of your revision. My argument against infinite justificational regresses would still work, with only minor modifications.

Suppose, contrary to what is to be shown, that for some person P at a time t, and some statement XO,

INFINITE REGRESSES OF JUSTIFICATION 33

(1) ',Xn justifies Xn -1, ..., X1 justifies XO,

where no Xi in the regress is justified by any set of Xi<i (to prevent circularity). (1) is a non-circular, justification-saturated regress (for P at t), meaning that every statement in the regress is justified by an earlier statement, and none is justified by any set of later statements (for P at t). Thus the question of whether there can be an infinite justificational regress is to be construed here as the question of whether there can be a non-circular, justification-saturated regress.

If anything counts as an inferential justification relation, logical implica- tion does, in a sense to be specified, provided it satisfies appropriate relevance and non-circularity requirements. Let us say a statement X properly entails a statement Y iff X semantically entails Y, where the entail- ment is relevant and non-circular on any appropriate account. Thus if anything counts as an inferential justification relation, proper entailment does, in the sense that where X and Y are statements rather than sets of statements,

(2) If X properly entails Y, then Y is justified for P if X is - provided P knows that the proper entailment holds and would believe Y in light of it if he believed X

Next we shall see that if there could be even one justification-saturated regress - like (1) - then we could justify any logically contingent statement whatsoever. The point is not new, nor is my argument for it entirely new. But the argument will plug some old holes and help us to see what is new, the implications for regresses of explanation.

Let XO be a logically contingent statement, and adopt some (alphabetical) ordering of the infinitely many statements of P's language. Then construct the entailment-saturated regress

(3) ., Xn , ... I X, XO,

where Xi (i >0) is the (alphabetically) first statement such that (i) Xi properly entails Xi-,; (ii) Xi is not entailed by any Xj<1; and (iii) Xi is not justified for P on the basis of any set of XA < i. Also, assume that for each Xi, P knows (or could come to know) that Xi properly entails Xi- 1 . And assume that P would believe Xi-, in light of this entailment if P believed Xi.

The construction of (3) presupposes that at every step of the regress there is some statement XA satisfying conditions (i)-(iii). This is one of those old

34 JOHN F. POST

holes to be plugged. But first note that deductive non-circularity in (3), which in the intended sense is guaranteed by (i) and (ii), does not entail justifica- tional non-circularity, which is guaranteed by (iii). Even though no Xi is entailed by any X,<i, it does not follow that no Xi is justified by any Xj<1, or by any set of X1i<, either of which would induce justificational circularity. A person P might justify some Xi in (3) on the basis both of Xi+, (deductively) and of some set of Xi<i (non-deductively). Hence we must require that Xi is not justified for P on the basis of any set of Xj<i, not merely that Xi is not entailed by any Xi<i. The need for this sort of requirement has been overlooked - another of those holes - in some attempts to show that a non- circular justificational regress could be constructed for any statement.6

James Cornman requires the regress to contain no empirical statement Xi that is logically equivalent to one of its own evidential ancestors Xj>i.7 But even though this excludes one sort of justificational circularity, it allows others, in which Xi is justified ('inductively', say) by some set of descendants

Xj<i in a regress where no statement is equivalent to one of its own ancestors

Xj>i (or indeed to any set of them). Thus constructing a regress that satisfies Cornman's requirement (a 'Comman regress') does not in general count as constructing a regress that is non-circular in the required full sense of (iii). Even though a Cornman regress could be constructed for any contingent statement, it would not follow that a justificational regress, non-circular in the required full sense of (iii), could be constructed for any contingent statement. Hence there is a hole in Comman's attempt to show that a non- circular justificational regress could be constructed for any contingent state- ment XO.

To plug these holes, let us begin with regress (3) and the presupposition that at every step of the regress there is a statement Xi satisfying conditions (i)-(iii). It helps to consider an example of (3). To construct one modulo some appropriate ordering of the statements of P's language, let Z be contingent and use modus ponens as follows:

(4) ...,X&(XD(Y&(YDZ))), Y&(YDZ),Z,

where Z does not entail Y, (Y & (Y D Z)) does not entail X, and so on. This sort of infinitely iterated application of modus ponens guarantees that for every statement Xi-1 in (4), there is a statement Xi that satisfies (i) and (ii). Satisfaction of (ii) can easily be checked. As for (i), misgivings on the score of relevance can be met either by requiring that Z and Y share some non-

INFINITE REGRESSES OF JUSTIFICATION 35

logical terms in common, that Y and X do, and so on; or more strongly by requiring that any such term in Z appear in Y, any in Y appear in X, and so on. Modus ponens is only one entailment-form with which to construct instances of (3) that satisfy (i) and (ii); there are many others. Furthermore, there are complex or 'mixed' instances of (3) in which the form by which Xi entails Xi-1 is distinct from that by which Xi+1 entails Xi. Whatever the entailment-forms, we see that for any logically contingent XO we can construct an instance of (3), such as (4), that satisfies (i) and (ii).

What about condition (iii)? Again, consider (4). Statement Z will justify for P a set JO of statements, possibly null and possibly infinite, but definitely not universal. For either Z justifies every statement whatsoever for P or it does not. If it does, then since the negation of a statement is also a statement, every statement plus its negation is justified by Z, which is intolerable for rational justification (recall also that Z is contingent). Thus we can be sure there is some Y that is not justified by Z - some Y not in JO. In particular, let Y be the (alphabetically) first statement not in JO such that any non- logical term in Z is in Y, and Z does not entail Y; thus Y satisfies not only (iii) but (i) and (ii). Next, the set {Z, (Y & (Y D Z))} will justify for P a set J1 of statements. As with JO, we can be sure that there are statements not in J1. Let X be the first statement not in J1 such that any non-logical term in Y is in X, and (Y& (Y D Z)) does not entail X; thus X satisfies not only (iii) but (i) and (ii). In this way we see that at every step of the regress, there will be a next statement that satisfies (i)-(iii), hence that the regress is non-circular in the required full sense of (iii). It follows that

(A) For any contingent XO one can construct an instance of (3) that satisfies (i)-(iii), out of which one can construct a non-circular, justification-saturated regress like (1).

For X1 properly entails XO, so that by (2), XO is justified for P if X1 is. P would be entitled to claim that X1 is justified, hence that X1 justifies XO, by arguing again from (2) that X1 is juw+ified if X2 is, and claiming that X2 is justified (see further the second paragrap>h below). If challenged about X2, P could appeal again to (2) and claim that X3 is justified, and so on. For each statement in (3) P could, if challenged, cite its predecessor as justification. But then Xo is justified, since everything in Xo's justificational ancestry (3) is justified (again see the second paragraph below). Moreover, that ancestry is non-circular, since by (iii) no Xi in (3) is justified by any set of Xi<i. Note

36 JOHN F. POST

that Xo's negation could be justified too (for example, substitute 'not-Z' for 'Z' in (4) plus the instructions for constructing (4)).

Clearly, being able arbitrarily to justify any contingent statement what- soever, hence any plus its negation, is unacceptable. So we must conclude

(B) There can be no justification-saturated regresses.

For if we allow even one, circular or not, then we have no plausible way of disallowing non-circular justification-saturated regresses like (3), that satisfy (i)-(iii). Since for any contingent XO we can construct such a regress, we could, in light of (2), arbitrarily justify any contingency at all, which is absurd. Generally the complaint has been that a saturated regress would justify nothing. The real trouble is the reverse, an absurd embarrassment of riches. Of course we may go on to say that really the regress justifies nothing after all, provided we understand the basic reason why: if it justified something, then for any contingent XO another regress could be constructed to justify XO.

Foundationalists sometimes reject regress on the grounds that no matter how far back in the regress we go, we always find a statement that is only mediately justified if at all. From this it is supposed to follow that the terminal statement XO is not justified.8 But this clearly is question-begging. It presupposes not only the sound principle that XO is justified only if everything in its justificational ancestry is justified, but also that XO is justified only if something in its ancestry is immediately justified. The latter is just the characteristic thesis of foundationalism. It might be true, but it cannot be assumed in the regress argument for foundationalism.

Foundationalists (and others) also operate on another sound principle, namely that XO is justified if and only if everything in its justificational ancestry is either mediately or immediately justified. (Non-foundationalists can accept the principle because it does not entail that there are immediately justified statements.) So long as the ancestry is finite we can trust our intui- tions in applying the principle to help decide when XO is justified and when it is not. But the structure of the regress argument for foundationalism requires at one point that we assume, for the sake of argument, that the ancestry is infinite. As so often with infinity, our intuitions need tutoring. To begin with, the principle last mentioned entails that XO is justified if everything in its ancestry is mediately justified. For if everything in Xo's

INFINITE REGRESSES OF JUSTIFICATION 37

ancestry is mediately justified, then a fortiori everything is either mediately or immediately justified, from which it follows by the principle that XO is justified. Next, when the regress is finite, there is a first statement X, at which we must face the question of whether X,, is mediately or immediately justified. Since in the present context we are considering only non-circular regresses, we would have to answer that X, is immediately justified if at all. But when the regress is infinite, there is no first statement, and we never have to face the music. Every statement is mediately justified. Hence every state- ment is either mediately or immediately justified, so XO is justified.

The situation is analogous to what happens when the usual notion of deductive proof is applied (uncritically) to infinite sequences of formulas. A proof becomes a sequence, finite or infinite, in which every formula is either an axiom or entailed by earlier formulas in the sequence. But it is easy to construct for an arbitrarily chosen formula X an infinite sequence in which every formula is entailed by earlier ones (cf. (4)), so that every formula is either an axiom or entailed by earlier formulas. Hence on this notion of proof, we could prove anything.9 The moral seems to be that a proof-sequence must contain at least one unproved prover. Of course we cannot infer that a justificational regress must contain an unjustified justifier, nor an immediately justified one. Nevertheless this foundationalist intuition, that there must be immediately justified justifiers, doubtless owes much to deductive models of justification descended from Aristotle's theory of demonstration.

To conclude this section, note that the argument for (B), the impossibility of justification-saturated regresses, depends not on pragmatic considerations, but conceptual, entailed by features of justification presupposed in the fore- going. In this connection, what has been presupposed (in addition to (a)-(f) or their plausible revisions) is that if anything counts as an inferential justification relation, proper entailment does, in the weak sense of (2); that XO is justified if (and only if) everything in its ancestry is (we can even require that everything in the ancestry be either mediately or immediately justified); and that we should not be able to justify arbitrarily any contingent statement we like (hence any plus its negation). These features are present in any plausible concept of rational justification. Thus even for an infinite intellect there cannot be an infinite regress of justifications.

38 JOHN F. POST

II

Granting the conceptual impossibility of an infinite justificational regress, what are the implications for foundationalism? Certainly no foundationalist moral follows without further assumptions. What has been shown is only that in any sequence R of justifications, there must be at least one statement Xn

not justified by earlier statements in R; that is, R cannot be justification- saturated. Little can be inferred about Xn from this fact or its proof. Xn might be unjustified, immediately justified, self-justified, justified by later statements in R, or justified by statements Y outside R altogether. In this last case, of course, any sequence R' that includes Y and Xn cannot be justifica- tion-saturated either, and must contain at least one Xn' not justified by earlier statements in R'. But Xn ' might be justified by statements that occur in R' later than Xn', or by a third sequence R", and so on, provided the branching sequences R, R', R", ..., do not form a justification-saturated regress themselves.

Foundationalism follows only if the non-foundationalist altematives among the above are excluded. Notoriously, this is not so easy as it may look, especially with regard to the alternative in which Xn is justified by later state- ments in R. Such a case would count as an instance of what I have called 'justificational circularity' (in which some Xi is justified by a set of Xf<d). But the term 'circularity'hides an important distinction. Let R be (1), so that

Xn justifies Xn-1, ..., X1 justifies XO, and suppose XO justifies Xn. Then Xn is justified by a later statement in R, namely XO. At this point, foundationalists (and some others) typically infer that XO justifies XO, and charge R with circularity in this literal sense of a closed loop, en route to concluding that so far as R is concerned, whether XO is justified is therefore left completely open.1 0

But this inference and this notion of cicularity charged to R presuppose that justification is transitive (i.e., if X justifies Y and Y justifies Z, then X justifies Z). Suppose there are justification relations that are non-transitive (whether there are any I consider below). Then even if Xn is justified by XO (or by other statements occurring in R later than Xn), it would not follow that XO justifies XO or that R contains a circle or closed loop in the intended literal sense. Instead, non-transitive relations of justification obtaining among Xn, ..., XO could allow some of Xn, ..., XO to support each other without circularity in any such literal sense, and in such a way that

INFINITE REGRESSES OF JUSTIFICATION 39

XO would be justified. Thus uncritical talk of justificational circularity hides the distinction between literal circles of justification, which presuppose transitivity, and those 'circles' that consist of kinds of mutual support that do not give rise to transitivity.

Generally, regress arguments for foundationalism concern themselves only with circles and not with 'circles', except perhaps to claim they can make no sense of 'circles' that are not circles. Transitivity is presupposed in either case. The presupposition lurks also in the occasional foundationalist gibe that that coherentist rejects small circles but accepts big ones."1 The gibe misses the point that the big circles are meant to be 'circles' of mutual justification for which transitivity fails, so that the big circles cannot be reduced to little ones by applying transitivity.

It follows that the alternatives presented by regress arguments typically are not exhaustive. The altematives usually presented are that any (branch of a) justificational regress either terminates in immediately justified beliefs, terminates in unjustified beliefs, contains a circle, or continues infinitely. Too often, whether it contains a 'circle' is neglected. Excluding circles of justification seems easy. Excluding 'circles' is not at all easy, unless justifica- tion is obviously transitive. Is it?

Attacks on the transitivity of justification have taken several forms. I sketch them not in order to establish non-transitivity - the argument would be too long and complex - but to show that transitivity is not at all obvious. Indeed a number of philosophers hold the opposite: it is non-transitivity that is obvious (for some important types of justification). One form of attack uses direct counter-examples: concrete instances of statements X, Y, Z such that for P at t, X justifies Y, and Y justifies Z, but X does not justify Z. 12

Both to deflect these counter-examples and to clarify what the foundationalist may actually have meant all along, one should consider the modified transitivity principle,

(MT) If X justifies Y, and Y justifies Z, and (X & Y) justifies Z, then X justifies Z. 13

But (MT) appears vulnerable to a second form of attack (and possibly also to the first). Suppose, as is often the case, that Xjustifies Y only if X confers a high degree of probability upon Y. Then (MT) seems in trouble, since the probability conferred upon Z by Xmay be less than the required degree, even though X does confer the required degree upon Y, Y confers it upon Z, and

40 JOHN F. POST

(X & Y) confers it upon Z. 14 A third form of attack is to exhibit a theory that best explains certain phenomena, within which theory some statements support or justify each other. Done properly, this would be to present a cir- cumstance in which X justifies Y, Y justifies X, and (X & Y) justifies X. If (MT) held, X would justify itself, which all parties agree is to be rejected.' 5

The final form of attack to be considered here focuses on those justifica- tions that consist in inference to the best explanation. So far as I know, this attack on transitivity is new. Suppose X justifies Y for P at t just because hypothesis Y is the best explanation of (the evidence described by) X according to P at t - Y provides a better explanation than any alternative available to P at t. Suppose further that Y justifies Z in exactly the same sense, as does (X & Y). Then the antecedent of (MT) is satisfied. What about the consequent? If X justified Z in the intended sense, then Z would be the best explanation of X. But by hypothesis Y is the best explanation of X. Presumably there can be only one best explanation of X. Hence Z cannot be the best explanation of X, so X cannot justify Z in the intended sense. (MT) fails when justification is taken as inference to the best explanation; so too for simple, unmodified transitivity.16

This argument may seem too true to be good. But rather than analyze it here, I content myself with noting that like the other forms of attack, it at least succeeds in raising serious questions as to whether all types of inferential justification are transitive. Perhaps in the last analysis they are, but regress arguments for foundationalism can hardly take the point for granted. Their weakest link, it would now appear, is the rejection of 'circles' of non-transitive justification, rather than rejection of infinite regress.

III

I turn now to the first of a couple of morals that can more readily be drawn from the impossibility of a saturated regress of justifications: there cannot be a saturated regress of explanations, for a very large family of concepts of explanation. To begin with, let us say an explanation affords a justification, or is justification-affording (J-A), according to a given concept of explanation, if and only if

(5) If X explains Y for P at t, then X justifies Y for P at t.

INFINITE REGRESSES OF JUSTIFICATION 41

We need to be very clear about what (5) commits us to. The answer is: very little. It does not commit us to saying, even in the case of J-A explanations, that X's explaining Y is a justification of Y by X. The reason is that X might justify Y for P on grounds independent of the explanation of Y by X. Even if the grounds are not independent, as is most often the case, still the explana- tion need not be a justification, but only afford or provide materials for the construction of a justification."7 Thus (5) does not commit us to the view, even for those concepts for which it holds, that explanations are arguments, though such explanations may happen to be the most intuitive instances of J-A explanation, at least according to those who view explanations as arguments. All (5) implies is that for any concept of explanation which is or can be reconstructed as a relation between statements, if X explains Y for P at t, then there is some justification or other of Y by X forP at t (i.e., some way or other in which conditions (a)-(f) or their plausible revisions are satisfied). Nor does our focussing here on J-A concepts of explanation imply that any other concept is somehow inferior, for example as a model of scientific explanation.

Clearly (5) is quite latitudinarian. Nevertheless (5) can be used to show the impossibility of a saturated regress of explanations, for a very wide variety of concepts of explanation. For suppose, contrary to what is to be shown, that for some person P at a time t,

(6) ..., Xn explains Xn -1, ..., X1 explains XO.

(6) would be an explanation-saturated regress for P at t, meaning that every statement in the regress is explained for P at t by an earlier statement. Provided the explanations in (6) are all justification-affording, it would follow from (6) by (5) that every statement in the regress is also justified for P at t by an earlier statement. That is, it would follow by (5) that for P at t,

(7) ..., Xn justifies Xn _1, ..., X1 justifies XO.

But (7) is just the discredited (1), a justification-saturated regress, which we have shown to be conceptually impossible. Therefore

(C) An explanation-saturated regress is conceptually impossible, for any regress of J-A explanations.1 8

The family of explanatory concepts to which result (C) applies is very large and very diverse, as one would expect in light of how little (5) requires. The

42 JOHN F. POST

family includes not only deductive-nomological concepts, but also deductive- statistical, inductive-statistical, functional, plus a variety of non-empirical concepts, such as those often used in Principles of Sufficient Reason. It also includes concepts according to which explanation is given only in answer to why-questions actually asked in appropriate circumstances, with due regard to emphasis problems and any other context-dependent features (provided, of course, (5) remains satisfied)."9

One notable concept not in the family is Wesley Salmon's statistical- relevance (S-R) conception of explanation. On the S-R view the explanation of Y by X consists of (among other things) pointing out a statistical relevance between the events or regularities mentioned in X and those mentioned in Y. But the relevance is not in general such as to warrant claiming there is some justification or other of Y by X, though often there will be one. For example, the fact that a particular person contracted leukemia may be explained by the fact that (a) he was two kilometers from an atomic explosion, and (b) the probability of leukemia is causally related to the distance. But (a) and (b) hardly justifj the statement that this person contracted leukemia, since the probability of his doing so, though causally related to his distance from the blast, is very small (much less than 1/100), hence far too small to warrant claiming the statement is inductively justified by (a) and (b).20 Therefore the S-R concept does not satisfy (5); that is, S-R explanations are not in general J-A. So we cannot use (C) or the argument for (C) to show there can be no saturated regress of S-R explanations. Perhaps there cannot be, but some other argument is required.

Nevertheless there are important subclasses of S-R explanations for which (C) does hold. Salmon recognizes that "deduction of a restricted law from a more general law constitutes a paradigm of a certain type of explanation", and that for the S-R account to be successful, it must be able to "handle cases of this sort"21. Salmon handles them by having the justificatory deductive relations exhibit the physical relationship between the restricted and the more general regularities expressed by the two laws. It is this physical relation, not the deductive justification, that has explanatory significance. In other words, "an explanation may sometimes provide materials out of which an argument, deductive or inductive, can be constructed", even though explanations are not arguments.22

Obviously, explanation of restricted regularities by more general ones is only one type of case in which a justificatory argument can be constructed

INFINITE REGRESSES OF JUSTIFICATION 43

exhibiting the physical, causal relations that do have explanatory significance. Often a certain particular event e cannot but occur, given other relevant events plus physical regularities; that is, determinism obtains in the case of e. Let us grant that the physical, causal relations are what have the explanatory significance for P at t. Nevertheless such explanations would provide materials out of which an argument can be constructed that justifies, for P at t, the belief that e occurs. Even where determinism of this sort fails, the probability of e's occurrence often is high enough to warrant justified belief, in contrast to the leukemia example.

In all such cases, an S-R explanation provides materials out of which a justification, deductive or inductive, can be constructed. Hence in all such cases, if X S-R explains Y for P at t, then X justifies Y for P at t, even though S-R explanations are not arguments. In other words, in such cases even S-R explanations are justification-affording. What such cases have in common is that in light of the relevant events, causal relations, processes and interactions mentioned in X, the probability of the events or regularities mentioned in Y is high enough for a justification of Y by X to be constructed from the materials of the S-R explanation. Let us call such S-R explanations high probability S-R explanations. Since they satisfy (5), and thus are J-A expla- nations, it follows by (C) that

(D) A saturated regress of high probability S-R explanations is im- possible.

(D) is of considerable interest, even though there are events and regularities that do not have high probability S-R explanations. The reason is that for centuries philosophical discussions of explanatory regresses typically have presupposed some variety of determinism for the physical events or regularities to be explained (most notably, perhaps, in connection with Principles of Sufficient Reason). Deterministic presuppositions were not shaken fundamentally until the advent of quantum theory, and hardly over- night even then; indeed thriving pockets of resistance survive. Thus philosophers who discuss explanatory regresses, and who do not think of explanations as arguments, typically are discussing what in effect are high probability S-R explanations: regresses in which events and regularities are explained for P at t by reference to, and have a high probability in light of, their causal relations with other events and regularities. Therefore even when such philosophers do not construe explanations as arguments, the

44 JOHN F. POST

explanations they have in mind typically are such as to provide materials out of which justificatory arguments can be constructed. In light of (D), we know that a saturated regress of such explanations is impossible, whatever certain philosophers might have thought.

Explanatory regresses are one thing, causal regresses another. The latter are regresses of causally related non-linguistic events and regularities. Even though there can be no explanation-saturated regresses for any person P at t (assuming we are speaking of J-A explanation), it does not follow that there cannot be, for example, an infinite series of physical events each caused by a temporal predecessor. (Such a series need not be temporally infinite, but might occupy a finite temporal interval.)23 The causally related events could occur without P's knowledge, hence without P's having an explanation of them; or even with his knowledge yet without explanation.

But suppose for the sake of argument that P is an omniscient God. Suppose further that the type of causation involved in the series of events is such that if event ei causes e1, then there is an explanation of e1 by reference to ei (that is, a statement Xi to the effect that ei occurs explains

Xi). Being omniscient, God would know the explanation (i.e., that Xi explains Xi). Suppose finally that God and the type of explanation involved are such that

(8) If X explains Y for God, then there is a justification of Y by X for God (cf. (5)).

With all these suppositions, it would follow that if there were an infinite series of events each caused by a predecessor, there would be a justification- saturated regress in which each Xi is justified for God by some earlier Xi. But as seen, a justification-saturated regress is impossible even for an infinite intellect.

So there could be no infinite series of events each caused by its temporal predecessor, provided all the above suppositions are true. But they are all highly problematic. For example, the type(s) of causation involved in the series might very well not be such that there is an explanation of e1 by reference to ei; or if there is such an explanation, it fails to satisfy (8), just as S-R explanations fail to satisfy (5) unless they are high probability S-R explanations. But even though there are so many problematic suppositions in this argument, we may conjecture that certain philosophers at times have accepted them, and that this might help explain why they sometimes thought

INFINITE REGRESSES OF JUSTIFICATION 45

they could show the impossibility of an infinite series of events each caused by a temporal predecessor.

So far the explanatory regresses we have considered are monotonic, consis- ting of only one type of explanation. But there seem also to be mixed explanatory regresses, where the type of explanation by which Xn explains Xn-I for P at t can be very different from the type by which Xn+l explains Xn for P at t. One can imagine deductive, inductive, functional, and S-R explanations strung together in various combinations, perhaps together with other types. So long as each type in the regress is J-A, a saturated mixed regress is as impossible as a saturated monotonic regress, since either sort of explanatory regress would, by (5), give rise to a discredited justification- saturated regress like (7).

Since there can be no saturated regresses of J-A explanations, mixed or monotonic, it follows that in any J-A explanatory regress there is at least one statement Xn not explained by an earlier statement. Could Xn be explained either by itself, or by later statements? On most concepts of expla- nation, asymmetry holds:

(9) If X explains Y for P at t, then Y does not explain X for P at t.24

On any such concept, no statement Xn explains itself. But could Xn be explained by later statements in the regress? Let us say

that

(10) Explanation is hierarchical according to a given concept iff in any regress of such explanations, no Xi is explained by any set of Xj<i (for P at t).

Hierarchy includes asymmetry. For consider a unit regress of explanation, in which X explains Y for P at t; then by (10), X is not explained by Y for P at t. Thus it follows that

(E) For hierarchical concepts of J-A explanation, any regress of explanation contains at least one statement Xn explained neither by earlier statements, nor by itself, nor by any set of later state- ments.

That is, Xn would be an ultimate explainer - a statement that explains but has no explanation. The point is not that we would not or could not know

46 JOHN F. POST

the explanation of X, (though that would be true so far as it goes). Rather, there would be none to be known, even for an infinite intelligence.

I suspect that most concepts of explanation are hierarchical. Explanation has intimate ties with concepts of causation, most of which appear to be hierarchical. In addition, it is worth noting that a foundationalist is committed to holding that any concept of J-A explanation is hierarchical. For according to the foundationalist, justification is hierarchical. To say it is hierarchical is the same as saying, as he does, that there can be no justifica- tional circularity (compare (10) with the second paragraph of Section II). But (5) holds for any concept of J-A explanation, by definition. If (5) holds, and if justification is hierarchical, then explanation is too, as can easily be checked. As seen, many, perhaps mlost concepts of explanation satisfy (5), including high probability S-R explanations. Therefore to the extent that our concepts of explanation were formed or refined under the influence of foundationalist intuitions - until recently the dominant epistemology in westem philosophy - we might well expect them to be hierarchical. But I do not know to what extent our concepts of explanation were formed under this influence, although I suspect that the influence on philosophers' concepts (e.g., Spinoza's) may have been considerable.

IV

I tum now to the second of the two metaphysical morals, the implications for Principles of Sufficient Reason. Even when explanation is not hierarchical according to a given concept, nevertheless

(F) Not every statement can have a J-A explanation (for P at t).

For suppose every statement did have a J-A explanation, for P at t, by some statement or other. Then statement X0 would be J-A explained by some statement X1, which in tum would have a J-A explanation, say by X2, and so on. It would follow that for P at t,

(11 ) ...,X, explains X1, X1 explains X0.

But (11) is just the discredited (6), a saturated regress of J-A explanations, which is impossible.

If not every statement has a J-A explanation, perhaps every statement in a certain class does - say, every truth, or every contingent truth, or every

INFINITE REGRESSES OF JUSTIFICATION 47

truth that merely asserts the existence of a thing. These are the leading ver- sions of the Principle of Sufficient Reason (PSR), and figure in various Cos- mological Arguments. But whether every statement in a certain class has a J-A explanation depends crucially upon what concept of J-A explanation is used. Let p be a property of statements (e.g., being true, contingent, existence-asserting, etc.). Call k hereditary with respect to explanation in a given sense iff when X explains Y in that sense, and Y has 0, then X has ?. We can prove

(G) Heredity Theorem: Given any concept of J-A explanation for which 4 is hereditary, not every 4-statement has such an expla- nation.

For if XO has such an explanation, say by X1, and if XO has X, then Xl has 4. But if every ?-statement has such an explanation, then X1 has such an explanation, say by X2; and so on to infinity. This would give rise to a saturated regress of J-A explanations, which by (C) is impossible.

For example, consider the property of being true, and suppose we are using a J-A explanation relation for which truth is hereditary (as is the case in Cosmological Arguments). From (G) we obtain

(H) Not every truth has a J-A explanation in any sense for which truth is hereditary.

Thus PSR is false in any fonn which implies that every truth (and/or every fact, state of affairs, etc.) has an explanation in any such sense. Since there are versions that do, result (H) is of some interest.

Other versions claim less. Often they assert only that every logically contin- gent truth has an explanation. However, the notions of explanation presup- posed by PSR require not only that the explanans be true, but that it logically entail the explanandum - the sufficient reasons mentioned in PSR are supposed to be logically sufficient. But if Y is contingent and entailed by a true X, then X is also contingent. So being contingently true is hereditary with respect to the notions of explanation presupposed by PSR. Therefore by (G),

(I) PSR is false in any form which implies that every contingent truth has an explanation (in the presupposed sense).

A defender of PSR might object that what gets explained, and what has

48 JOHN F. POST

explanatory significance, are not true statements (contingent or otherwise) but things and/or events, phenomena, regularities, states of affairs, etc. But the objection can be met by recalling the discussion of high probability S-R explanation that immediately followed (D): even when philosophers who defend some version of PSR do not construe explanations as (justificatory) arguments from statements to statements, nevertheless the explanations they have in mind are such as to provide materials out of which justificatory arguments can be constructed. For example, suppose one says, in the spirit of PSR, that the explanation (for P at t) of the existence of some contingently existing thing a lies in the causal efficacy of some other such thing b. Even though it is the things and their causal relations that are to have explanatory significance, still the explanation would provide materials out of which a justification of Y' by X' could be constructed, where Y' states that a exists, and X' states that b exists and is the cause of a (or perhaps that b together with other things form the cause of a). Since the sufficient reasons are supposed to be logically sufficient, in effect we are dealing with a high probability S-R explanation. As noted above, (5) holds for such explanations, so they are J-A. Hence we are free to apply the Heredity Theorem (G) via (H) and (I), where p is a property of the statements X' and Y, to show that these (material mode) versions of PSR are false.

The final version of PSR I shall consider here asserts merely that every existing thing has an explanation for its existence, either in the existence and efficacy of another thing or in the necessity of its own nature. Let us call this version 'Existence' PSR. Recently the claim has been made that Existence PSR has not been disproved (unlike other versions) and that it is all that is needed, by way of a PSR, for a valid Cosmological Argument for God.25 Existence PSR implies only that every truth that asserts the existence of a thing has an explanation. Thus (H) and (I) are inapplicable. But Existence PSR implies that every existing thing a has an explanation either in the existence of something else b, where b causes a to exist, or in its own existence, where a exists by the necessity of its own nature; that is, where a is self-caused. This implies in tum that every truth of the form 'a exists' has an explanation either by the truth 'b exists', where it is under- stood that b #a and b causes a to exist, or by the truth 'a exists', where it is understood that a is self-caused, or exists necessarily.26 Here we see that the property of being a truth of the form 'x exists' is hereditary with respect to this sort of explanation. By (G), then, not every truth of that form - not

INFINITE REGRESSES OF JUSTIFICATION 49

every truth that asserts the existence of a thing - has an explanation of this sort, contrary to Existence PSR.

A defender of PSR might object that we have misconstrued what it means for a's existence to be explained by the necessity of a's own nature: it does not imply that 'a exists' explains 'a exists', even where it is understood that a is self-caused, or exists by the necessity of its own nature; for this would violate the asymmetry of most concepts of explanation (cf. (9)).

In reply, let us ignore the fact that asymmetry is already in jeopardy when defenders of PSR talk of self-caused beings. Instead, even granting the objec- tion, we can use the Heredity Theorem to show that Existence PSR, even if not refuted, would be question-begging in its intended application. Call a being that exists by the necessity of its own nature a 'necessary being'. Suppose there is no necessary being and that Existence PSR is true. It would follow that every existing thing a has an explanation in the e..'_^nce of something else b, where b causes a to exist. Hence every truth of the form 'a exists' would have an explanation by the truth 'b exists', where it is understood that b $ a and b causes a to exist. Once again, having the form 'x exists' is hereditary with respect to this sort of explanation. Therefore

(J) Not every existing thing has an explanation in the existence of something else (for the notion of explanation presupposed by Existence PSR).

But Existence PSR says that every existing thing has an explanation for its existence either in the existence of something else or in the necessity of its own nature. That is, every existing thing either has an explanation in the existence of something else or in a necessary being. By (J), then, it follows that

(K) Existence PSR is true only if there is a necessary being.

(K) tells us that in assuming the truth of Existence PSR, the theist thereby presupposes there is at least one necessary being. But whether there is a necessary being was one of the points at issue, at least according to many opponents of Cosmological Arguments. This question is begged, when the theist assumes Existence PSR. The theist also explicitly denies that the world is a necessary being. In the context of his denial, the truth of Existence PSR presupposes, in light of (K), that there is a being distinct from the world. Hence the theist is presupposing, unwittingly, that there is a being distinct

50 JOHN F. POST

from the world, when he assumes Existence PSR in order to show there must be one. Once again, appealing to Existence PSR begs the question it was intended to answer.

Perhaps there is some version of PSR that is neither refutable by the Heredity Theorem nor question-begging in its intended application, though I doubt it. Of course even if there is such a version, it could still be false (e.g., by quantum indeterminacy); and there remain other links in Cosmological Arguments for God that look as weak.27

Vanderbilt University

NOTES

1 In 'Two types of foundationalism', Journal of Philosophy 73 (1976), p. 173, Note 10. I am indebted to the referee for valuable criticism. Section III below benefited most. I am indebted also to the following for helpful discussion on various occasions: Clement Dore, Charles Davis, Gary Gutting, John Hooker, Janet Kourany, Stephen Levy, David Pomerantz, Scott Shuger, Jeffrey Tlumak, and Red Ulrich. 2 E.G., Keith Lehrer, Knowledge, (Clarendon Press, Oxford, 1974), pp. 15-16, 155- 56; and I. T. Oakley, 'An argument for scepticism concerning justified beliefs', American Philosophical Quarterly 13 (1976), pp. 226--27. Oakley also presents an argument that does not appeal to the pragmatic, but I believe it fails for reasons mentioned in Note 6, below. 3 The objection is due to Bruce Aune, 'Remarks on an argument by Chisholm', Philosophical Studies 23 (1972), p. 329. Aune's objection, suitably applied, also appears to undercut the idea that a person cannot believe an infinitely complex proposition. This idea is a key assumption in the argument against regress advanced by Richard Foley, 'Inferential justification and the infinite regress', American Philosophical Quarterly 15 (1978), pp. 311-316, esp. p. 313. See also Note 13, below. I Cf. David B. Annis, 'Epistemic justification', Philosophia 6 (1976), pp. 259-66; and R. A. Fumerton, 'Inferential justification and empiricism', Journal of Philosophy 73 (1976), pp. 557-69, esp. p. 564, plus G. Harman's counter to Annis on pp. 570-71 with regard to what in effect are my (c)-(e). 5 We may wish to add that X's entailing Y is brought to P's attention; cf. F. Schick, 'Three logics of belief, in M. Swain (ed.), Induction, Acceptance, and Rational Belief (Reidel, Dordrecht, 1970), p. 25. Note that (2) is a weak entailment principle. Strong entailment principles, to the effect that a set of rational beliefs is closed under entail- ment, are controversial. H. Kyburg implicates strong principles in the so-called lottery paradox, in 'Conjunctivitis', in M. Swain (ed.), op. cit., pp. 55-59. 6 E.G., Oakley, op. cit., makes this mistake twice, on p. 227. John Poilock apparently avoids the mistake of supposing deductive non-circularity to be sufficient, but he overlooks the obligation to guarantee that at every step of the regress there is some statement that satisfies the (justificational) non-circularity condition. See his: Knowl- edge and Justification (Princeton University Press, Princeton, 1974), pp. 27-28, esp. footnote 5. 7 In 'Foundational versus nonfoundational theories of empirical justification', American Philosophical Quarterly 17 (1977), pp. 287-97. See clause (4) of Ni on p. 289, plus series El on p. 290. I am deeply indebted to James Comman for comments here, and regret all the more his tragic death.

IN FINITE REGRESSES OF JUSTIFICATION 51

Alston notes this sort of move, op. cit., p. 173, item D. 9 Cf. Bas C. van Fraassen, 'Theoretical eintities: The five ways', Philosophia 4 (1974), pp. 95-109, esp. 99-100 and footnote 7, where the idea is attributed to Richmond Thomason. Thomason reports (in conversation) that the idea occurred to him in connection with the Gentzen-style consistency proof for formal number theory presented by Elliott Mendelson, Introduction to Mathematical Logic (Van Nostrand, Princeton, N.J., 1964),pp. 258-70; unwrittingly,Mendelson defmes a proof (on p. 261) so that any wf turns out to be provable. But a less esoteric illustration occurs in I. Copi, Introduction to Logic, 5th ed. (Macmillan, New York, 1978), p. 311: by failing to men- tion that his proof-sequences must be finite, Copi's definition of a formal proof renders any statement Z provable (hence any plus its negation), since an instance of (4) can be constructed for any Z in his 'system'; so his rules were complete all along, but I pray none of this will inspire yet another edition in his regress of editions. 10 Cf. Alston, op. cit., p. 173, item C. l Cf. R. A. Fumerton, op. cit., p. 559. 12 Cf. Peter D. Klein, 'Knowledge, causality, and defeasibility', Journal of Philosophy 73 (1976), pp. 806-807. 13 Due to Klein, ibid., p. 807. Foley's requirement B, op. cit., p. 314, implies that justi- fication is transitive, approximately in the sense of (MT). Hence transitivity is presup- posed in his argument against infinite justificational regresses. '4 Klein, ibid., p. 806, raises this probability objection against unmodified transitivity but makes no mention of it in connection with (MT). Lehrer gives a probabilistic argu- ment against a very close relative of (MT) that might well be effective against (MT), in 'Justification, explanation, and induction', in M. Swain, op. cit., pp. 122-123. '5 Robert A. Jaeger, 'Implication and evidence', Journal of Philosophy 72 (1975), pp. 475-485, esp. pp. 482-84, constructs such an argument against unmodified tran- sitivity. If it works at all, I see no reason why it would not also work against (MT). 16 Lehrer raises doubts about the transitivity of explanation, though not best explana- tion, in 'Justification, explanation, and induction', in Swain, op. cit., pp. 112-113. 17 Cf. Wesley C. Salmon, 'Why ask, 'why?'? An inquiry concerning scientific explanation', Proceedings and Addresses of the APA 51 (1978), pp. 683-705, espec. 700. This paper contains references to Salmon's and other work relevant to my discussion here. 10 (C) is just an instance of a more general result. Let R be any relation such that if XR Y (for P at t), then X justifies Y for P at t (cf. (5)). The above argument then shows there can be no R-saturated regress. 19 Cf. Bas C. van Fraassen, 'The pragmatics of explanation', American Philosophical Quarterly 14 (1977), pp. 143-50, which contains further references. 20 Cf. Salmon, op. cit., pp. 688-89. 21 Ibid., p. 700. 22 Ibid., p. 700. 23 Cf. C. Misner, K. Thome, and J. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973), pp. 813-14; and Adolf Griinbaum, Modern Science and Zeno's Paradoxes (Wesleyan University Press, Middletown 1967), pp. 83-86. 24 Cf. Salmon, op. cit., pp. 6 86-87: "Explanations demand an asymmetry not present in inferences.... The asymmetry... is inherited from the asymmetry of causation". 25 By William L. Rowe, The Cosmological Argument (Princeton Unversity Press, Princeton, 1975), pp. 73, 112-13. 26 Cf. the notion of 'causally sufficient condition in the context at hand', plus the related notion (in effect) of 'explanatorily sufficient in the context at hand', on pages 107 and 110-11, respectively, of Brian Skyrms, 'The explication of "X knows that p" ', reprinted in M. Roth and L. Galis (eds.), Knowing (Random m-'TS :w York, 1970). Roughly, 'b exists' is causally (explanatorily) sufficient for 'a exist. , the context at hand iff there are statements describing relevant conditions such that their conjunction

52 JOHN F. POST

with 'b exists' is causally (explanatorily) sufficient for 'a exists'. In the case of explanation, we require that these additional statements be known to P at t, in the context. It can be shown that the Heredity Theorem still applies. 27 Research for this paper was supported in part by the Vanderbilt University Research Council.