Maverick Philosopher

Infinite Regresses: Vicious and Benign

A reader asks: 

   Are all infinite regresses (regressions?) vicious? Why the pejorative
   label? Of the many things I don't understand, this must be near the
   top of my list, and it's an ignorance that dates back to my undergrad
   Intro to Philosophy days. When I first read the Thomistic cosmological
   proofs, I found myself wondering why Aquinas had such trouble
   countenancing the possibility that, as the lady says, "it's turtles
   all the way down."

   Without a first, there can't be a second... so what? It doesn't follow
   that there must be a first element to a series. What makes a
   temporally infinite series (of moments, causes/effects, etc.)
   impossible?

No, not all infinite regresses are vicious. Some are, if not 'virtuous,' at least innocuous or benign. The term 'benign' is standardly used. The truth regress is an example of a benign infinite regress. Let p be any true proposition. And let 'T' stand for the operator 'It is true that ( ).' Clearly, p entails T(p). For example, *Snow is white* entails *It is true that snow is white.*  The operation is iterable. So T(p) entails T(T(p)). And so on, ad infinitum or ad indefinitum if you  prefer. The resulting infinite series is  unproblematic. Whether you call this a progression or a regression, it doesn't cause any conceptual trouble.  Nor does it matter whether you think that infinity is potential only, or hold to actual infinities.  Either way, the truth regress is a nice clear example of an infinite regress that is benign.

So some infinite regresses are benign.

Setting aside the lady and her turtles, suppose, contrary to current cosmology, that the universe has an infinite past, and that each phase of the universe is caused by an earlier phase. Suppose further that there is nothing problematic in the notion of an actual (as opposed to potential) infinity, and that there is a good answer to the question of how, given the actual infinity of the past, we ever arrived at the present moment. Granting all that, the infinite regress of causes is benign.

But note that one cannot explain why the universe exists by saying that it always existed. For even if there is no time at which it did not exist, there remains the question why it exists at all. The universe is contingent: it might not have existed. So even if it exists at every time with earlier phases causing later phases,  that does not explain why it exists at all.

To say that the universe always existed is to say that it has no temporal beginning, no temporally first cause. But this gives no  answer to the question why this temporally beginningless universe exists in the first place. 

Here is where the theist invokes God. God is the ontologically, not temporally,  first cause.  Now if Mill asks, "But what causes God?" the answer is that God is a necessary being. If God were a contingent being, then a vicious infinite regress would arise. For one cannot get an ultimate  explanation of U if one invokes a contingent G. And if there were an  infinite regress of Gs, the whole series would be without ultimate explanation.  Thts is true whether the regress is potentially infinite or actually infinite.

If we compare the truth regress with the regress just mentioned, we can perhaps see what makes the latter vicious.  The viciousness consists in the failure to satisfy the need for an explanation.  P if and only it is true that p.  No one will take either side of this biconditional as explaining the other.  But explanation comes in when you ask why the universe U exists.  If you say that U exists because G caused it to exist, then you can reasonably ask: what caused G?  The classical answer is that G is causa sui, i.e., a necessary being.  The buck stops here.  If, on the other hand, you say that G is contingent, then it cannot be causa sui, in which case the regress is up and running.  Because the explanatory demand cannot be satisfied by embarking upon the regress, the regress is said to be vicious.

To answer the reader's  question, there is perhaps nothing vicious about a temporally infinite regress of empirical causes. But that gives us no explanation of why a temporally infinite universe exists in the first place.

Related articles

The Modified Leibniz Question, Maitzen's Critique of its Meaningfulness, and My Response

On Infinitely Regressive Explanations of the Universe's Existence

If Life Has a Meaning, Then it Cannot be Subjective

Causation, Existence, and the Modified Leibniz Question

On Infinitely Regressive Explanations of the Universe's Existence

We’ve never chatted. I’m Tom Belt, a friend of Alan Rhoda. I believe you know Alan.

Yes, in fact I was thinking about him just the other day in connection with his espousal of presentism.

I’ve always appreciated being challenged when I drop by your blog. I’m wondering if you’d be willing to help me understand something.

I'll do my best.

I’ve been exploring Hartshorne’s Modal/Ontological Argument with a friend, Jeff. Basically Jeff wants to agree that some manner of ‘necessity’ needs to be posited in order to explain the existence of the universe. So he agrees that CH's "Something exists" entails "Something exists necessarily." But he then argues that both ‘an infinite regress of created beings’ and ‘a single, necessary being’ equally fit the bill. Both are equally possible and both have the same explanatory value. So his point is, “Look, parsimony is the only thing that gets us a single, necessary being; there's no obvious metaphysical advantage that a necessary being has over an infinite regress of created beings. Either might be the case, and parsimony is all we have to adjudicate the choice between them.”  But something seems wrong here.

There is indeed something wrong here. 

But first let's lay out Jeff's suggestion -- or a plausible candidate for that office -- a bit more clearly.  To make things hard on the theist we begin by assuming that the universe has an actually infinite past.  Hence it always existed.  Let us also assume that the each total state of the  universe at a time  is (deterministically) caused to exist by an earlier such state of the universe.  A third assumption is that the universe is nothing over and above the sum of its states.  The third assumption implies that if each state has a causal explanation in terms of earlier states (in accordance with the laws of nature), then all of the states have an explanation, in which case the universe itself has a causal explanation.  This in turn implies that there is no need to posit anything external to the universe, such as God, to explain why the universe exists.  The idea, then, is that the universe exists because it causes itself to exist in that later states are caused to exist by earlier states, there being no earliest, uncaused, state.  We thereby explain why the universe exists via an infinite regress of universe-immanent causes thereby obviating the need for a transcendent cause.

If this could be made to work, then we would have a nice neat self-contained universe whose existence was not a brute fact but also not dependent on anything external to the universe.

The five or so assumptions behind this reasoning can all be questioned.  But even if they are all true, the argument is still no good for a fairly obvious reason.  The whole collection of states, despite its being beginningless and endless, is contingent: it might not have existed at all.  The fact that U always existed, if it is a fact, does not entail that U must exist.  If I want to know why this universe of ours exists as opposed to there being some other universe or no universe at all, it does no good to tell me that it always existed.  For what I want to know it why it exists AT ALL.  I am not asking about its temporal duration but about  its very existence.  Why it exists at all is a legitimate question since there is no necessity that there be a universe in the first place.

So Jeff is wrong when he says that both a single necessary being and and infinitely regressive series of contingent causes "have the same explanatory value."  The latter has no explanatory value at all.  And this for the reason that it is contingent.

I mentioned to him Hartshorne’s point that the only conceivable way to posit the non-existence of a necessary being is to hold such a being’s existence to be impossible. A necessary being can only exist or not exist necessarily. So I told him he’s free to say “I can’t figure out which is in fact the case, an infinite regress of contingent beings or a single necessary being,” but that once he settles upon the latter for reasons of parsimony, what this moves amounts to is settling for the necessity of one option over the impossibility of the other, since the (modal) possibility of an infinite regress of contingent beings entails the impossibility of a single necessary being. But he’s not buying.

First of all, considerations of parsimony come into play only when we are comparing two theories which are both explanatorily adequate.  In that case Occam's Razor enjoins us to give the nod to the more parsimonious of the two.  After all, the stricture is not against 'multiplying entities' tout court, but against mutiplyng entities beyond necessity, i.e., in excess of what is needed for purposes of adequate explanation.   But in the situation before us, Jeff's theory is not explanatorily adequate.  It completely fails as an explantion of why there is a universe rather no universe or some other universe.

If the universe has an explanation then it must be in terms of a noncontingent explainer.  As you appreciate, if such an entity exists, then it is necessary, and if it does not, then it is impossible.  But the rest of your reasoning is dubious which is why your friend is not buying it. The point you need to insist on is that Jeff is not offering an adequate alternative explanation.  He falsely assumes that the collection of contingent beings is a necessary being.  It is not.  It is as contingent as its members.

That aside, it doesn’t seem to me that an infinite regress of instances seeking [needing?] explanation really is conceivable EVEN IF actual infinities per se are conceivable. A necessary being may be temporally eternal. That’s one thing. But an infinite regress of contingent beings, each created by the previous? I don’t see how such a regress is conceivable, or how it embodies the necessity Jeff agrees has to be posited in order to explain the existence of the world. Surely if every member in an infinite regress is contingent, then the regress is contingent and the whole thing in need of the same explanation any particular member needs, no? We can’t reify the regress per se and attribute necessity to IT while positing the contingency of every member.

Right.  That's exactly the point I made above.  But surely such a regress is conceivable in the manner I explained above.  Just don't use the world 'create' because that muddies the waters.

Lastly, wouldn’t it be the case in such a regress that every member god would HAVE to create something, so that no one of them could be free to not create at all? That seems to follow. If any member in the regress is free to not create at all, and every member is created, then any member might not have been created at all (which is just to say each is contingent). But that is to posit the contingency of the regress and thus abandon its explanatory value. No? Yes?

I agree.  Jeff's suggestion is much stronger if he thinks of the regress as one of ordinary empirical causes in tandem with the assumption that causation is not probabilistic but deterministic.  But if he is talking about a regress of free gods, then an added dimension of contingency comes in via the libertarian free will of these gods.

Am I nuts? Personally I think an infinite regress of created/contingent beings is impossible.

You are not 'nuts.'  You are basically right.  But it is not clear that an infinite regress of contingent beings is impossible.  Why should it be impossible?  There are benign infinite regresses.  What you want to say is that an infinite regress of contingent beings cannot do any explanatory work re: the question, Why does the universe exist?

So far, then, Tom 1, Jeff 0.

Regress? What Regress? Truth-Making Revisited

Ed continues to repeat his regress argument against truth-makers, despite my hurling invective at it.  I think I called it "breathtakingly rotten" or something equally offensive, all in good fun of course:

I have argued (e.g. here and here that the notion of a ‘truthmaker’ leads to an infinite regress. If there is such a truthmaker, an entity that makes a proposition like ‘Socrates sits’ true - let it be A - then it comes into existence when Socrates sits down, and ceases to exist when he stands up. But then there would have to be a further truthmaker for A existing. I.e. the sentence “A exists” can be true or false, and so requires a further truthmaker B, that makes it true when B exists. But then “B exists” requires yet another truthmaker, and so on ad infinitum.

Now what is the regress supposed to be?  There is an entity  A and it makes-true sentence s.  A is not a sentence, or any other type of representation.  Since we can talk about A, we can say 'A exists.'  'A exists' is contingently true, so it too needs a truth-maker.  So far, so good.

Ed assumes that the truth-maker for "A exists' must be distinct from the truth-maker for s.  Without this assumption, the regress can't get started.  Therefore, to show that his regress argument is bogus, it suffices to show that one and the same entity A can serve as the truth-maker for both s and 'A exists.'

Suppose the truth-maker of 'Tom is tired' is the fact, Tom's being tired.  Now consider the sentence 'Tom's being tired exists.'  I claim that the truth-maker of both sentences is Tom's being tired.   I conclude that there is no regress.

To appreciate this you must note that while 'Tom is tired' is a predication, 'Tom's being tired exists' is not.  It is an existential sentence like 'Tom exists.'  So while the predication requires a fact for its truth-maker, the existential sentence does not.  It does not need a fact as a truth-maker any more than 'Tom exists' does.  The truth-maker of the latter is just Tom.  The truth-maker of  'Tom's being tired' is not the fact, Tom'sbeing tired's existence, but just  Tom's being tired.

There is a second reason why the regress cannot arise.  Ed is a nominalist. He eschews propositions and believes only in sentences.  Well, there is no need for there to be the sentence 'A exists'!  If no one says that A exists, then there is no sentence 'A exists.'  And of course nonexistent sentences do not need truth-makers.   And if someone does say that A exists, there is no need that he, or anyone else, say that the truth-maker of 'A exists' exists.  So for this reason too the regress can't get started.

Ed ends his post on this strange note: "If we buy the idea of a ‘truthbearer’ (a proposition, a thought, whatever), the idea of a ‘truthmaker’ comes with it."  That's plainly false.  That there are truth-bearers is self-evident; that there are truth-makers is not.  Must I dilate further on this self-evident point?  Second, if the quoted sentence is true, and Ed's regress argument is sound, the upshot is that there are no truth-bearers, which is absurd.  In effect, Ed has provided a reductio ad absurdum of his own claim that there are no truth-makers!

What Ed says about representation and the representation of the faithfulness of a representation would require a separate post to discuss.  But I sense the conflation of epistemological questions with ontological ones.

An Infinite Regress Argument Against Truth-Makers? Round Two

The truth-maker of 'Tom sits' cannot be Tom.  Otherwise it would also be the truth-maker of 'Tom stands' which is the logical contrary of the first sentence.  And that won't do, as London Ed appreciates.  But now what about 'Tom exists'?  This too is a contingent sentence, and so it too needs a truth-maker.  I say the truth-maker is Tom.  The truth-maker of 'Tom sits' is a fact, the fact of Tom's being seated. This fact is a complex having Tom himself and the property of being seated as constitutents.  (Let's not worry about what holds these constituents together!)  The truth-maker of 'Tom exists,' however, is not a fact having Tom and the property of existence as constituents.

Why the asymmetry?  Because existence is not a property in the same sense of 'property' in which being-seated is a property.  I won't repeat the many arguments I have given on this blog and in my articles and book.

But suppose you, like Ed, see symmetry where I see asymmetry.  You think that the truth-maker of 'Tom exists'  is the fact of Tom's existence, or the fact of Tom's existing.   Call this truth-making fact T.  Since T exists, and exists contingently, 'T exists' needs a truth-maker.  I am willing to concede that a vicious infinite regress then arises, though the matter is not entirely clear.

But what does this show?  I say it shows that the assumption that existence is a property is mistaken. 

The dialectical situation is this.  There are plenty of arguments why existence cannot be a property.  And we have good reason to admit truth-makers for contingent truths.  So in the case of contingent existential truths like 'Tom exists' we should say that it is the referent of the subject term itself that is the truth-maker.

Frege's Regress

Some of us of a realist persuasion hold that at least  some truths have need of worldly correlates that 'make them true.' This notion that (some) truths need truthmakers  is a variation on the ancient theme that truth implies a correspondence of what-is said or what-is-thought with what-is.  You all know the passages in Aristotle where this theme is sounded.

Example. Having just finished my drink, the thought expressed by an assertive utterance of 'My glass is empty' is true. But the thought is not just true; it is true because of the way things are 'outside' my mind. The glass (in reality) is (in reality) empty. So the realist says something like this: the thought (proposition, judgmental content, etc.) is true in virtue of the obtaining of a truthmaking state of affairs or fact. The thought is true because the fact obtains or exists, where 'because' does not have a causal sense but expresses the asymmetrical relation of truthmaking. The fact is the ontological ground (not the cause) of the thought's being true.

One might wonder whether this realist theory of truth leads to an infinite regress, and if it does, whether the regress is vicious. Some cryptic remarks in Gottlob Frege's seminal article, "The Thought: A Logical Inquiry," suggest a regress argument against the correspondence theory of truth.

For Frege, a thought (Gedanke) or proposition is the sense (Sinn) of a context-free declarative sentence. 'Snow is white' and its German translation Schnee ist weiss are examples of context-free declarative sentences. 'Context-free' means that all indexical elements have been extruded including verb tenses.  When we say that a sentence such as 'Snow is white' is true, what we are really saying is that the sense of this sentence is true.  The primary truth-vehicles are propositions, sentences being truth-bearers only insofar as they express true propositions.

Now could the being-true of a sentential sense consist in its correspondence to something else? Frege rejects this notion: "In any case, being true does not consist in the correspondence of this sense with something else, for otherwise the question of truth would reiterate itself to infinity." (Philosophical Logic, ed. Strawson, p. 19) A little earlier, Frege writes,

For what would we then have to do to decide whether something were true? We should have to enquire whether it were true that an idea and a reality, perhaps, corresponded in the laid-down respect. And then we should be confronted by a question of the same kind and the game could begin again. So the attempt to explain truth as correspondence collapses. And every other attempt to define truth collapses too. (Ibid.)

What exactly is Frege's argument here? We begin by noting that

1. Necessarily, for any proposition p, it is true that p iff p.

This equivalence, which I hope nobody will deny, gives rise to an infinite regress, call it the truth regress. For from (1) we can infer that if snow is white, then it is true that snow is white, and
iterating the operation, if it is true that snow is white, then it is true that it is true that snow is white, and so on without end. This is an infinite regress all right, but it is obviously benign. For if
we establish the base proposition, Snow is white, then we ipso facto establish all the iterations. Our establishing that snow is white does not depend on a prior establishing that it is true that snow is white. In general, our establishing of any proposition in the infinite series does not depend on having first established the next proposition in the series. The truth regress, though infinite, is benign.

Note that if the truth-regress were vicious, then the notion of truth itself would have been shown to be incoherent. For the truth-regress is a logical consequence of the equivalence principle (1) above, a principle that simply unpacks our understanding of 'true.' So if the truth-regress were vicious, then (1) would not be unproblematic, as it surely is.

It follows that if Frege's Regress is to amount to a valid objection to the definition of truth as correspondence, "and [to] every other attempt to define truth," then Frege's Regress must be different from the truth regress. In particular, it must be a vicious regress. Only vicious infinite regresses have the force of philosophical refutations.  But then what is Frege's Regress? Consider

2. Necessarily, for any p, it is true that p iff *p* corresponds to reality.

One can think up counterexamples to (2), but the precise question before us is whether (2) issues in a vicious infinite regress. Now what would this regress (progress?) look like? Let 'T(p)' abbreviate
'it is true that p.' And let 'C*p*' abbreviate '*p* corresponds to reality.' (The asterisks function like Quine's corners.) The regress, then, looks like this:

3. p iff T(p) iff C*p* iff T(C*p*) iff C(T(C*p*)) iff T(C(T(C*p*) iff C(T(C(T(C*p*)) . . .
   
Is (3) a vicious regress? It would be vicious if one could establish T(p) only by first establishing C*p* and so on. But if these two terms have the same sense, in the way that the first and second terms have the same sense, then (3) will be as benign as the truth regress.  Suppose that 'It is true that p' and '*p* corresponds to reality' have  the same sense. Suppose in other words that the correspondence theory of truth is the theory that the sense or meaning of these distinct sentences is the same. It would then follow that to establish that it is true that p and to establish that *p* corresponds to reality would come to the same thing, whence it would follow that the regress is benign.

For the regress to be vicious, the second and third terms must differ in sense. For again, if the second and third terms do not differ in sense, then to establish one is to establish the other, and it would not be case that to establish that it is true that p one would first have to establish that *p* corresponds to reality or to some chunk of reality. But if the second and third terms do not differ in sense, then it appears that the regress doesn't get started at all. For the move from the second term to the third to be valid, the entailment must be grounded in the sense of the second term: the third term must merely unpack the sense of the second term. If, however, the two terms are not sense-connected, then no infinite regress is ignited.

My interim conclusion is that it is not at all clear that Frege's Regress is either benign, or not a regress at all, and therefore not at all clear that it constitutes a valid objection to theories of truth, in particular to the theory that truth resides in correspondence.

REFERENCE: Peter Carruthers, "Frege's Regress," Proc. Arist. Soc., vol. LXXXII, 1981/1982, pp. 17-32.
 

An Infinite Regress Argument Against Truth-Makers?

Edward, the proprietor of Beyond Necessity,  presents an infinite regress argument against truth-makers.  Here it is:

. . . I reject the idea of a truthmaker altogether. If there is such a truthmaker, let it be A, it comes into existence when Socrates sits down, and ceases to exist when he stands up. If it were something real – let’s say a candle flame, which comes into existence when we light the candle, and ceases to exist when we blow it out – then there would have to be a further truthmaker for A existing. I.e. the sentence “A exists” can be true or false, and so requires a further truthmaker B, that makes it true when B exists. But then “B exists” requires yet another truthmaker, and so on ad infinitum. That is absurd. Therefore, there are no truthmakers.

I am not sure Ed understands what a truth-maker is.  Here is a Philosophy 101 explanation.  Suppose we have some true contingent declarative sentence such as 'Tom is tired.' The truth-maker theorist maintains that for contingent true sentences, there is more to the sentence than its being true.  There  must be something external to the sentence, something that is not a sentence, that 'makes it true.'  If you deny this, then you are saying that the sentence is just true and that there is no explanation of its being true in terms of anything  extralinguistic.  And surely that is absurd, assuming you are not some sort of linguistic idealist.  'Tom is tired' cannot just be true; it is true because there exists a man to whom 'Tom' refers and this man is in a certain state.

Could Tom by himself be the truth-maker of 'Tom is tired'?  No.  For if he were, then he would also be the truth-maker of 'Tom is manic' -- which is absurd.  This is why truth-maker theorists (not all but most) introduce facts or states of affairs as truth-makers.  David Armstrong is a prominent contemporary example.

Now what are we to make of Edward's argument?  The argument seems to be that if sentence s has a truthmaker t, then the sentence 't exists' must also have a truth-maker, call it t*.  But then the sentence 't* exists' must itself have a truth-maker, t**, and so on ad infinitum.

Now this is a terrible, a thoroughly and breath-takingly rotten, argument which is why no one in the literature (to the best of my knowledge) has ever made it.  Suppose that 'Tom is tired' is made-true by the fact of Tom's being tired.  Call this fact F.  If  'Tom is tired' is true, then F exists, whence it follows that 'F exists' is true.  (This of course assumes that there is the sentence 'F exists,' an assumption I will grant  arguendo.)  Since 'F exists' is contingent, we can apply the truth-maker principle and ask for its truth-maker.  But surely its truth-maker is just F.  So there is no regress at all, let alone an infinite regress, let alone a vicious infinite regress.  (Please note that only vicious infinite regresses have the force of refutations.)  'Tom is tired' has F as its truth-maker, and 'F exists' has the very same F as its truth-maker.  Tom's being tired makes true both 'Tom is tired' and 'Tom's being tired exists.'  No regress.

So Ed's argument is a complete non-starter.  There are, however, plausible arguments against facts as truth-makers.  See my Facts category. 

Another Example of a Vicious Infinite Regress: Philosophical Investigations, Sec. 239

I am collecting examples of infinite regress arguments in philosophy. See the category Infinite Regress Arguments.  Here is one that is suggested by section 239 of Wittgenstein's Philosophical Investigations. When I hear the word 'red,' how do I know which color is being referred to?  The following answer might be given:  'Red' refers to the color of the mental image that hearing the word elicits.  But then the question arises once again:  How do I know that the color of the mental image is the color to which 'red' refers?  Do I need a criterion for that as well?  If I do, then I am embarked upon an infinite regress, one that is vicious.

Why is it vicious?  Most of us know which color 'red' refers to.  But how do we know it?  To ask how we know this is to request an epistemological (and therefore a philosophical) explanation.  But if the explanation is that 'red' refers to the color of the mental image that hearing the word elicits, then, although we have answered the initial question, we have  answered it in a way that allows the posing of a second question of the same form as the first.  And so on.

Original and Derived Intentionality, Circles, and Regresses

1. Original/Derived Intentionality. All will agree that there is some sort of distinction to be made here. A map is not about a chunk of terrain just in virtue of the map's physical and geometrical properties. Consider the contour lines on a topographical map. The closer together, the steeper the terrain. But that closer together should mean steeper is a meaning assigned by the community of map-makers and map-users. This meaning is not intrinsic to the map qua physical object. Closer together might have meant anything, e.g., that the likelihood of falling into an abandoned mine shaft is greater.

So some things derive their referential and semantic properties from other things. What about these other things? I draw you a map so that you can find my camp. I use the Greek phi to mark my camp and the Greek psi to mark the camp of a heavily-armed crazy man that you are well advised to avoid. I intend that phi designate my camp. That intending (narrow sense) is a case of intentionality (broad sense). This is not in dispute. What is in dispute is whether my intending is a case of original or of derived intentionality.

If the latter, then a regress ensues which appears to be both infinite and vicious. But before discussing this further, I need to bring in another point.

Infinite Regresses: Vicious, Benign, Virtuous?

I haven't yet said anything particularly illuminating about the criteria of viciousness, the criteria that would allow us to sort infinite regresses into the vicious and the non-vicious. I should address the problem of criteria.  But in this installment I want to suggest that we may need to make a tripartite distinction among vicious, benign, and virtuous regresses.  If a regress is not vicious, then it is non-vicious. But it may be that non-vicious regresses come in two kinds, the benign and the virtuous.  Here is a crude analogy.  Fearing cancer, I have a certain growth checked out by a medico.  It is determined that the growth is not malignant, but benign. But no one will say that the growth serves a useful purpose.  If something is harmless, it does not follow that it is helpful.  And if something is not vitiating, it does not follow that it is 'empowering.'

Thus I am suggesting that we not refer to non-vicious regresses as virtuous, as some writers do.  For if a regress is merely benign or harmless or innocuous, it does not follow that it is explanatorily useful or helpful.  And it may be that there are some infinite regresses that serve an explanatory purpose.  These would deserve to be called virtuous.  But we need some examples.

I don't endorse the following example, but it is worth thinking about.  Suppose we want to explain why the universe exists, and we want to do so without recourse to anything transcendent of the universe: we seek a satisfactory immanent explanation.  Suppose further, contrary to current cosmology, that the universe always existed.  Let's also assume that to explain the parts of a whole is to explain the whole.  To adapt an example of Paul Edwards, suppose the Three Stooges are hanging out at the corner of Hollywood and Vine on a certain afternoon.  To explain why the boys are there at that time it suffices to explain why Larry is there, why Moe is there, and why Curly Joe is there.  Having explained why each is there, one has explained why the trio is there.  It would be senseless to demand an explanation of why the trio is there after one has been given satisfactory explanations of why each member of the trio is there.  The trio is not something over and above its members.

Applying this Hume-Edwards principle -- the principle that to explain the parts of a whole is to explain the whole -- to the universe, one could say that to explain why the universe exists it suffices to explain why each phase of the universe exists, so that, if each phase of the universe has an explanation, then eo ipso the universe has an explanation.  Now if the universe is temporally infinite in the past direction, and each phase of the universe is caused by an earlier phase, then every phase of the universe has a causal explanation in terms of an earlier phase.  Since no phase, no temporal part, of the universe lacks an explanation, and since the universe as process just is the whole of these temporal parts, and since to explain the parts of a whole is to explain the parts, it seems to follow that the universe is self-explanatory, that its existence can be accounted for in wholly immanent terms. It looks as if a beginningless universe could be causa sui. Let us assume arguendo that this very bad argument I have just inflicted on you is not bad.

In this argument it appears that the infinite regress of causes does positive explanatory work.  For if there were a temporally first event, or a temporally first phase of the universe, then one could demand an explanation of it, and this demand could not be immanently satisfied.  But if every event or phase has an explanation in terms of an earlier event or phase, then this demand cannot be made.  What we have then is a putative example of an actually infinite regress that is not merely harmless, but positively helpful unto explanation.

Hence my suggestion:  we ought to make a three-fold distinction among vicious, benign (harmless), and virtuous (helpful) infinite regresses.  And thus we ought not conflate benign regresses with virtuous regresses.  Virtuous regresses are a proper subset of benign regresses (since every explanatorily hepful regress is explanatorily harmless), which implies that there are benign regresses that are not virtuous.

Now the example I gave of a virtuous infinite regress is not a very convincing one, or at least it is not convincing to me.  Are there better examples of virtuous infinite regresses, infinite regresses that do positive explanatory work?

What say you, Jan?  Francesco?

Vicious and Benign Regresses Again

What is the difference between a vicious and a benign infinite regress?  We ought to look at a number of examples.  Here is one.  An entailment of a proposition p is any proposition that is a logical consequence of p.  Now consider

1. Every proposition has entailments.

2. To know a proposition one must know its entailments.

(1) gives rise to infinite series.  The entailments of a proposition are themselves propositions, so that if every proposition has entailments, then for every proposition there is an infinite series of propositions.  For example, p entails ~~P, which entails ~~~~P, and so on ad infinitum. There is nothing problematic here.

(2), however, engenders a vicious infinite regress.  For if to know a given proposition I must know its entailments, then to know a given proposition I must know infinitely many propositions.  But I cannot know infinitely many propositions.  So (2) implies that I cannot know any proposition. 

What makes the regress vicious in the second case?  What does viciousness consist in? It has to do with (2)'s being explanatory.  (2) proposes a philosophical explanation: one knows a proposition by knowing its entailments.  (2) proposes a theory as to what knowing a proposition consists in.  But the explanation is faulty.  Suppose p entails q which entails r which entails s, and so on.  The theory proposes that in order to know p, I must know q. But to know q I must know r, and so on.  This implies the impossibility of my knowing p.  Viciousness, then, is the property of being explanatorily unsuccessful.

Perhaps we can hazard the following general formulation. A vicious infinite regress is an infinite regress that arises in the context of an attempted philosophical explanation when the explanation given permits the question that was to be answered to arise at successively higher levels ad infinitum.  In the above example, to know that p one must know p's entailments, but to know them, one must know their entailments, and so on endlessly.

Now consider this pair:

3. Every event has a cause.

4. To explain an event one must explain its causes.

(3) engenders an infinite series: if every event has a cause, and causes are events, then there is an infinite regress of events. But the regress is benign.  (4), however, is the answer to a philosophical question about the nature of explanation: What is it to explain an event?  (4) proposes a philosophical explanation of explanation, namely, that to explain an event one must explain its causes.  But this theory leads to a vicious infinite regress.  Suppose z has y as a cause.  The theory implies that to explain z one must explain y.  But y is an event, so to explain it one must explain its cause x, and so on infinitely.  The regress is vicious because it sets an impossible standard of explanation: if to explain an event one must explain every event in its causal ancestry, then no event can be explained. So (4) is false.

 

Infinite Regresses: Vicious and Benign

The peripatetic (not Peripatetic) Kevin Kim once asked me:

Are all infinite regresses (regressions?) vicious? Why the pejorative label? Of the many things I don't understand, this must be near the top of my list, and it's an ignorance that dates back to my undergrad Intro to Philosophy days. When I first read the Thomistic cosmological proofs, I found myself wondering why Aquinas had such trouble countenancing the possibility that, as the lady says, "it's turtles all the way down."

Without a first, there can't be a second... so what? It doesn't follow that there must be a first element to a series. What makes a temporally infinite series (of moments, causes/effects, etc.) impossible?

Here is the answer I gave him, considerably expanded and updated:

1. No, not all infinite regresses are vicious. Some are, if not 'virtuous,' at least benign. The term 'benign' is standardly used. The truth regress is an example of a benign infinite regress. Let p be any proposition. And let 'T' stand for the operator 'It is true that ( ).' Clearly, p entails T(p). The operation is iterable. So T(p) entails T(T(p)). And so on, ad infinitum or ad indefinitum if you prefer. The resulting infinite series is wholly unproblematic. Whether you call this a progression or a regression, it doesn't cause any conceptual trouble.