Ridgeview Publishing Company

Vicious Infinite Regress ArgumentsAuthor(s): Romane ClarkReviewed work(s):Source: Philosophical Perspectives, Vol. 2, Epistemology (1988), pp. 369-380Published by: Ridgeview Publishing CompanyStable URL: http://www.jstor.org/stable/2214081 .Accessed: 11/12/2011 05:54

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

Ridgeview Publishing Company is collaborating with JSTOR to digitize, preserve and extend access toPhilosophical Perspectives.

http://www.jstor.org

Philosophical Perspectives, 2, Epistemology, 1988

VICIOUS INFINITE REGRESS ARGUMENTS'

Romane Clark Indiana University

Surprisingly, vicious infinite regress arguments seem to be ex- clusively philosophical arguments. They do not occur in the history or current practice of mathematics, or the law, for instance.2 They do not occur on editorial pages or in the Sunday Supplements. It seems natural then to conclude that these arguments are not arguments displaying some unique, universally valid pattern of for- mal inference. This conclusion is reinforced by the practice of logi- cians. In the way in which, say, Indirect Derivation is separately codified, and semantically justified, in standard logic texts and treatises,3 we lack in them any discussion, codification, or justifica- tion of "derivation by vicious infinite regress." This curious parochialism in turn suggests that regress arguments

do not so much constitute a general type of reasoning, whether valid or not, as share some special, philosophical, sort of content. But, think- ing about it, that, too, seems implausible. After all, instances of vicious infinite regress arguments range from remote causal, cosmological ones occurring in the history of metaphysics on down to justificatory, foundational ones which occur in contemporary epistemology. Where, we wonder, is the shared, common content to be found in applications as diverse as these? Given all this, it might be thought that what's special about regress

arguments is to be found elsewhere.4 Showing that some thesis im- plies a vicious infinite regress shows that there's something wrong with it all right. But what is wrong is not necessarily a matter of logical

370 / Romane Clark

form or special content. There are after all lots of bad arguments. But not all of these are deductively invalid. Quite to the contrary, some of these are not only deductively valid but even sound. Perhaps theses which imply a vicious infinite regress are like these. Circular arguments, arguments that beg the question, successions

of explanations that deteriorate into trivial repetitions of the same conditional justifications, each of these may involve reasoning which is logically valid. But each of these is nonetheless in other ways quite unacceptable. Each of these flouts one or more of the requirements which govern those underlying acts of informing or persuading which motivate the giving, and are ordinarily discharged in the making, of arguments and explanations. The requirements which are flouted by these arguments are pragmatic or dialectical requirements, not formal or semantical ones. (Thus, for instance, an argument which assumes what is to be

proved goes wrong, but not because the reasoning is not truth- preserving. It goes wrong because whatever need or point there might have been to motivate the production of the proof in the first place is merely transferred to, and so remains undischarged in, the conclusion's pleonastic occurrence as its own premise.) Theses which imply a vicious infinite regress might be thought to

be like these familiar informal fallacies. Valid regress arguments might be thought to show that the thesis which implies its regress also fails for dialectical reasons. Professor Rosenberg, for instance, thought so. Recently, in an introductory text, using as an illustration the thesis that an act is voluntary only if it is the result of an act of will, he wrote this:5

It must, in fact, be a voluntary act of will. And if this is so, we haven't been given an answer to our original question. We can only understand this answer if we already know what makes an act of will voluntary. But our question is, What makes an act of will voluntary? The only course open to us is to apply the theory again. When we do so, however, all we find is that we need yet another voluntary act of will. The question does not go away. That is the essence of the criticism. The question

does not go away. This is what makes the challenge dialectical rather than logical. It disqualifies the proposed answer as an answer for something qualifies as an answer to

Regress Arguments / 371

a question only if one can understand it without already knowing the answer to the question. The philosopher who offers this answer, therefore, violates a canon of rational practice. [Rosenberg's italics; my boldface.]

The main problem with all this, however, is that while the illustra- tion is meant to have general import, it is not at all clear how to generalize the account.6 More important, it is not clear that it can correctly be generalized. For one thing, we still lack in the example any explanation or

understanding of that puzzling parochialism which initiated our discussion. We don't yet understand why it is that regress arguments seem limited to philosophical theses. After all, arguments that beg the question, and circular explanations, have also been held to violate certain of the dialectical presuppositions which govern the contexts of their production. But these arguments and explanations are not limited in subject matter. Question-begging arguments and circular explanations are not limited to philosophy. Why should it be all that different for vicious infinite regress arguments?

More important is the fact that not all regresses even seem to have dialectical import. In soliloquy I ruminate on a philosophical thesis. I wonder perhaps whether each distinct event must have a distinct prior event as its cause. My silent concern is a metaphysical one. My concern is about the natural order of things. It is about an order which I suppose to be quite independent of my present thought, or of possible later discussions, or of the existence or nature of con- sciousness at all. Perhaps, rightly or wrongly, I conclude, on pain of a vicious infinite regress, that there must be some initial and in- itiating uncaused event, a philosophical "big-bang". If I reason in this way, the implication of my conclusion is not that

the original thesis is trivial, or insufficiently persuasive, or in yet some other way dialectically flawed. It is that the original thesis is false. I do not reject the thesis for dialectical reasons. Nor are dialectical constraints flouted, or operative, or even present in my coming to believe what I do. The process I go through, reasoning as I here do, is not a social process. Evidently, not all occurrences of vicious infinite regress arguments

are literally dialectical occurrences. And not all vicious infinite regress arguments involve unending processes (as distinct from infinite se- quences of elements.7) And not all are epistemological, involving

372 / Romane Clark

justification, or explanation, or the acquisition or exchange of information. These things being so, it is at the very least not obvious that a dialec-

tical characterization of vicious infinite regress arguments is a suffi- ciently general characterization of them. But there is by contrast an obvious, relevant, but absolutely general

fact about these arguments. The fact is that they are, all of them, cases of indirect argument. Philosophers who have reasoned in this way have in each instance provisionally adopted some target thesis in order to discredit it. They have in each instance attempted to show that the thesis implies something which could only obtain if the thesis itself were false. Vicious infinite regress arguments are, all of them, instances of a special kind of reductio ad absurdum. It is this obvious but general fact which helps us to understand why

it is that there is no need for a special form of derivation, no need for special codification or semantical justification to be separately laid down for vicious infinite regress arguments. None of these in fact exists. But none is separately required, for vicious infinite regress arguments are a special case, a species, of what already exists. They are thus already implicitly characterized by the generic standard for- mal representation, already at hand, of Indirect Derivation.3 Valid instances of Indirect Derivation, however, are crucially depen-

dent upon a demonstration of a contradiction. But not all (of the often enthymematic) occurrences of vicious infinite regress arguments familiar from the philosophical literature explicitly issue in one. It is true that these familiar philosophical arguments are each evident- ly a case of reductio ad absurdum. Each is aimed at the rejection of some target thesis. But as instances of a species of Indirect Deriva- tion in particular, these rejections must, in full formal representa- tion, be explicitly based upon some demonstration of an overt con- tradiction. What is special about valid infinite regress arguments as instances of species of reductio is the derivation of an infinite regress. But what is not clear is how, if at all, the regress in turn figures in delivering the required contradiction. The underlying intuition, of course, is that anything whose ex-

istence not merely implies but is exclusively dependent upon an in- finite succession of similar elements for which there is no indepen- dent existence proof does not after all exist. It does not flatly, categorically exist. It only conditionally does so. A foundationalist, for instance, may admit that a proposition which is only inferentially

Regress Arguments / 373

justified has a justification all right. What he denies, however, is that it is thereby justified. Justification may only be preserved, may only be conveyed but not conferred, by an inferential process.8 A propo- sition which is only inferentially justified is, a foundationalist may say, only conditionally justified. It is not flatly, unconditionally, categorically so. It is possible for a proposition to have a justification without being justified. One problem in trying to give consistent expression to this underly-

ing intuition is that we lack resources within standard logic for distinguishing implication and dependence. Even so it is possible to capture a bit of the motivating idea without all that. Let us say that something is conditionally F just in case there is something to which it stands in an F-preserving relation R which induces a partial order. That is, R must itself be, or imply the existence of, an asymmetrical and transitive relation which orders the entities which R relates.9 And R must preserve the property F in the sense that if an entity stands in that relation to entities which are F, then it, too, is F. (I.e., R is "upward F-preserving." Things may of course stand in a rela- tion which is F-preserving without being F.)

If this is the only way a thing comes to be F, if something is only conditionally F, then, with respect to F, it is downward dependent on its R-related heredity. By contrast, let us say that something is categorically F just in case

it is F but not only conditionally so. In these terms, vicious infinite regress arguments from the

philosophical literature are (often enthyematic) instances of a special version of Indirect Derivation. This is a version of Indirect Deriva- tion which is special only in exploiting an infinite regress in its deriva- tion of a contradiction. The typical infinite regress argument is an instance of a schematic

pattern which runs like this: Something in fact has some attribute, F. (This is a line already at

hand in the derivation, one which is not at the point of the reductio itself in question. We have independent of and serving as a premise for the present Indirect argument that something, a perhaps, is F. We have perhaps that a voluntary act has occurred; or that a cer- tain proposition is justified; or that an individual has, instances, a prop- erty. a's being F is accordingly a detached, finitely derived line of the derivation; it is an unconditional, categorical assertion. a's being F implies that a is categorically F.)

374 / Romane Clark

A target thesis, TF, about the nature or presence of F is assumed for the Indirect Derivation. (We assume perhaps that an occurrence of a voluntary act must be preceded by a voluntary act of will; or that a proposition is justified only if it is inferentially justified; or that whatever has a property must exemplify it, where exemplification is itself a relation and relations are multi-termed properties. Assum- ing things like these, we seek to derive a contradiction; to establish that, after all, the target thesis, TF, is not the case.) The unique thing about vicious infinite regress arguments, as the

special sort of Indirect Derivation which they are, consists we know in the derivation of a certain infinite regress from the thesis TF. The thesis, TF, to complete the proof, must be shown to imply that nothing is F unless there exists an infinite succession of elements, ones which stand in an "upward F-preserving relation," RF, and each of which is downward dependent upon its R-heredity. TF might for instance imply something of this form:

(x)[F(x) only if (3y)(Fy & R(x,y)], where R is some suitable asymmetrical and transitive relation. That is, it must be possible to demonstrate that the thesis together with an instance of F would, like the inset formula above, imply the ex- istence of an infinite succession of elements each member of which is only conditionally F. Since the target thesis implies that nothing is F unless a member of this sequence, whatever is F is at most only conditionally so. This conclusion, by a simple Separation of Cases, is inconsistent

with that earlier line, not itself in question, which implies that something is categorically F. Vicious infinite regress arguments of this form are thus special but valid instances of standard Indirect Derivation. This characterization of vicious infinite regress arguments is capable

of some generalization. Not every thesis, TF, which is subject to a valid vicious infinite regress argument, implies a simple linear in- finite succession. Some, like those laying down conditions for the justification of belief, may permit or require downward branching. Multiple propositions of higher rank may communally participate in the justification of some given proposition of lower rank; and some of higher rank may even participate in the justification of more than one proposition of lower rank.'0 (Justification may not necessarily consist in a simple tree, but in a possibly tangled hedge of further in-

Regress Arguments / 375

terlocking, justifications. On the other hand, sometimes even a tree may consist in just a single branch.) The important thing for the application of the regress argument

is just that the target thesis implies there to be embedded in the justificatory tangle an infinite tree which, relative to its initial propo- sition, can be isolated and explicitly defined, and which is such that any element at any node of higher rank on any arbitrary branch of the tree is RF related to at most one element of the next lower rank of that branch. (If there is none such, the element of "higher" rank is in fact an initial element occurring at the origin of the tree.) We know by a theorem due to Beth" that an infinite tree must have at least one unending, infinite branch. It is to such a branch that the regress argument can be applied and in terms of which its generaliza- tion can be stated.

'a is conditionally F' can be redefined "semantically" as stating that a occurs at a node of an infinite "Beth-branch," BF, of an infinite tree. The argument, now applied to such a tree, runs as before. We have

as before an earlier line of the derivation, finitely derived and not itself in question, that a is F. This is to say in the present, semantic idiom, that 'a is F' has an occurrence in a finite branch of the tree. As before, a target thesis, TF, is assumed for a reductio ad absur-

dum argument. The target thesis, TF, must as before make it demonstrable that

nothing is F unless it occurs at a node of the infinite Beth-branch, BF. It entails thus that whatever is F is only conditionally F. But this is inconsistent with a's being categorically F and occurring at some node of some distinct and finite branch.

It is possible, I think, plausibly to construe many of the vicious in- finite regress arguments familiar to us from the philosophical literature as enthymematic instances of a species of Indirect Deriva- tion like this. Construed in this way, the reasoning which underlies these arguments is perfectly cogent. But even so, it is not at all clear that these arguments are compelling. Everything turns now on the truth of their premises, and these are themselves often philosophically contentious. Vicious infinite regress arguments, like any argument, may perfectly well be valid but not sound. This is not to say, however, that such an infinite regress argument

is valid but not sound when the regress generated is "benign" rather than "vicious." If an infinite regress which is generated with respect to a target thesis, TF, does not yield an infinite succession ordered

376 / Romane Clark

by a relation which is upward F-preserving, the members of which are downward-dependent, then the regress is not vicious but benign. And if the regress is not vicious, then the argument is not only not a sound vicious infinite regress argument, it is not even a valid one. It is not a vicious infinite regress argument at all. (None of this is to deny that benign regresses may occur in valid arguments, and even in valid instances of (other forms of) Indirect Derivation.) Philosophical protagonists we know often fail to join issue. A

justificatory infinite regress may pull us in two directions. The Mr. Flip inside us may note that each proposition at any stage in an in- finite branch of an unending justificatory tree has its justification. Accordingly, since a simple induction establishes that every proposi- tion in the branch has a justification, an infinite series of them is not, it seems, vicious and so Foundationalism has not been established. But the Mr. Flop within us may in turn point out that at each stage of the infinite justificatory branch there is a proposition which is un- justified. There is always then some "last," unjustified proposition. But since earlier stages depend for their justification upon later ones, the regress does indeed after all seem to be vicious. Despite appearances, there is no clash of incompatibilities here.

There is no demonstration of the unsoundness of a vicious regress argument. For there are two distinct regresses involved in all this. Each protagonist correctly draws a conclusion appropriate to one of these. One of the regresses is benign; the other is indeed vicious. Flip takes the infinite justificatory branch to be a sequence of pro- positions each of which has the property of having a justification. It is a sequence generated by the thesis that a proposition has a justification only if there exists a distinct proposition to which it stands in some asymmetrical and transitive relation which preserves justification.

Flop takes the infinite justificatory branch to be a sequence of propositions no one of which has the property being justified. It is a sequence generated by the thesis that a proposition is justified on- ly if there exists a distinct justified proposition to which it stands in some asymmetrical and transitive relation which preserves justification.

Flip generates a benign regress, one in which no element of the sequence is downward dependent upon its heredity. The base case of its justifying induction is finitely and separately provable. It is easy to find or define a suitable justification-preserving relation (perhaps

Regress Arguments / 377

a qualified form of implication) such that for each proposition there provably exists another to which it stands in this relation. 12 All this is quite compatible with the tug we feel to our Flop side. For all of this, a proposition may indeed have, in the Flip sense, a justification and yet also be, in the Flop sense, unjustified. The presence of benign regress like this shows nothing at all about the existence, validity or soundness of a vicious infinite regress. Flop's regress is, by contrast, both infinite and vicious. Each propo-

sition of the sequence is only conditionally justified; each is downward-dependent upon its unending heredity. One can not establish inductively that the propositions of the sequence are justified. There is, and on Flop's interpretation of the target thesis there can be, no base case demonstration and no demonstration of the inductive step to establish this. There are, on that thesis, only conditionally justified propositions. The thesis is inconsistent with the existence of any categorically justified proposition at all. Flop's argu- ment is a valid infinite regress argument, one quite compatible with Flip's benign but accommodative sense of a proposition's having a justification. The existence of the latter does not show that former is unsound. Showing that a vicious infinite regress argument is invalid is typ-

ically a matter of showing that its target thesis, TF, in conjunction with an instance of F, does not after all imply the existence of an infinite sequence of downward-dependent entities. Showing that a valid vicious infinite regress argument, directed against some target thesis, TF, is unsound is typically a matter of showing that to assert that there is a categorical instance of F is either to assert something false or question-begging.

"No one is a human being unless the biological offspring of humans."'3 Is this true? In these days of exploding biological technologies one may feel a certain prudential tentativeness in the face of a general claim like this. But there are sharper reasons for rejecting the claim as well. The thesis is incompatible with there categorically being any humans like you or me at all. Under natural characterizations of the concept of being a biological offspring of humans, the underlying generating relation preserves the property of being a human and the thesis ensures that human beings are downward-dependent for their humanity upon (the infinite succes- sion of) their ancestors. This is to say that the thesis generates a vicious infinite regress.

378 / Romane Clark

One can accept the validity of the argument but maintain the thesis. One can reject the soundness of the argument but at a price. One must deny then that anyone manifests a separate, non-conditional but finitely determinable property of being a human. One must deny that an assertion of our palpable, actual human existence, is when properly understood, really a flat, categorical assertion of our human existence. Properly understood in these rather invidious terms, it will need to be said that we are, all of us, only "conditionally human."

Many of us, accepting as we do both evolutionary theory and the historical record as it comes to us, find the implication of an infinite succession of human generations too costly. The vicious infinite regress argument, we think, is not only valid but sound. The target thesis, naively plausible as it seems, is, we think, false. It would be nice of course to nail things down. It would be nice to pierce the mists of history and date man's emergence from the slime. It is not necessary however that it even be possible to do so. There may always remain a gray area of indeterminacy. But even so, we know enough about life before and after the initial records of human ex- istence to accept the argument and reject the thesis. In actual fact, knowing these things, we reject the thesis even without the formal argument.

All this returns us to our initial puzzle. At the beginning, we were struck by the fact that vicious infinite regress arguments seem to be largely or wholly philosophical arguments. We have since urged that these arguments are a species of a perfectly familiar and quite stan- dard form of inference, Indirect Derivation. What is special about the species is the way in which the contradiction necessary for the reductio is achieved. It depends upon the generation of a certain type of infinite succession of elements, one in which the existence of in- itial elements are dependent upon the unending succession of later ones. So there is after all a certain common, albeit formal, content which is unique to the diverse instances of vicious infinite regress arguments which have come to us. These infinite regresses, whether historical or contemporary, epistemological or metaphysical, silent- ly or socially produced, are each the basis for a special sort of univer- sal negative conclusion. Whatever can only be conditionally thus- and-so is not thus-and-so.

It is not necessary but it is natural that arguments of this sort tend to be philosophical arguments. Dealing in infinite totalities is already a fairly exotic sort of enterprise. (It is not, e.g., the common stuff

Regress Arguments / 379

of politics or the law.) Dealing in infinite sequences without the possibility of establishing a relevant base case for an inductive argu- ment, is a pretty restricted sort of task. (It is not, e.g., the typical mathematical sort of task.) Drawing a universal negative conclusion, where there can be no theory-independent fact of the matter, is not ordinarily a matter of common science, or common sense. It is understandable then that while there might, logically, be vicious

infinite regress arguments which are not philosophical, we do not find them and it remains plausible, I think, to construe the philosophical regress arguments we do know, as common instances of a quite special form of Indirect Derivation.

Notes

1. I have learned the most from, and have been most influenced on this topic by Day [2], Moser [5], and Sanford [8]. Their fine works contain in turn further references to other important discussions of this subject.

2. At least they do not seem to on casual investigation. It would be nice to have some authoritative study to support or discredit this, but I don't know of any.

3. See, e.g., Kalish & Montague [4], pp. 20-25. 4. Rosenberg takes this line in his [7] and John Passmore seems to in his

discussion of philosophical arguments. On this, see Day [2]. 5. [7], p. 62. 6. Day [2] and other authors have drawn a kind of "process/product"

distinction among regresses, contrasting infinite progressions of items with unending tasks.

In fact, of course, we lack anything like an interesting (let alone com- plete) list of "the canons of rational practice," or theoretical discussion of their basic properties. That is an important separate problem. Perhaps what can be said here is that it isn't at all obvious that the reason, on Rosenberg's analysis, that "the question doesn't go away" is not based in formal properties of the argument.

7. Ibid. 8. See, e.g., Harker [3], Moser [5], Chs. 1 and 4, and Post [6]. 9. Sanford [8], p. 110, notes that an ordering relation, like that of being

one inch taller, while not itself transitive may imply the presence of one that is, like that of simply being taller.

10. This is a point which Raymundo Morado stressed in an earlier discus- sion of this characterization of regress arguments.

11. [1], sec. 69, pp. 194-196. 12. Several authors have stressed the triviality of this justification relation,

see e.g. Harker, and Post, op. cit. Flip does ove'rall go wrong. He goes wrong in providing an account that is too true to be good; one which fails to be sufficiently discriminating in an interesting way. Every con-

380 / Romane Clark

tingent proposition has, in his sense, a justification. But he does not go wrong in developing a benign regress. His is not an instance of a valid but, since his regress is benign, unsound vicious infinite regress argu- ment. Rather, his is a valid instance of straightforward inductive argu- ment concerning a rather trivial property.

13. See Sanford's discussion of "Series that Do Not Terminate because They Trail off Gradually," [8], 113-115.

References

[1] Beth, E. W., The Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1968.

[2] Day, Timothy, Infinite Regress Arguments: Some Metaphysical and Epistemological Problems, Doctoral Dissertation, Indiana University, 1986.

[3] Harker, Jay E., "Can There Be an Infinite Regress of Justified Beliefs?," Australasian Journal of Philosophy, 62, no. 3, (September 1984) 255-264.

[4] Kalish & Montague, Logic: Techniques of Formal Reasoning, 2nd ed., Harcourt Brace Jovanovich, Inc., New York, 1980.

[5] Moser, Paul K., Empirical Justification, D. Reidel Publishing Co., Dordecht, 1985.

[6] Post, John F., "Infinite Regresses of Justification and Explanation," Philosophical Studies, 38, (1980) 31-52.

[7] Rosenberg, Jay F., The Practice of Philosophy, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1984.

[8] Sanford, David, "Infinite Regress Arguments," in Fetzer, Ch. 5, Principles of Philosophical Reasoning, 93-117.