An infinite regress is a series of appropriately related elements witha first member but no last member, where each element leads to orgenerates the next in some sense.^[1] An infinite regress argument is an argument that makes appeal to aninfinite regress. Usually such arguments take the form of objectionsto a theory, with the fact that the theory implies an infinite regressbeing taken to be objectionable.

There are two ways in which a theory’s resulting in an infiniteregress can form an objection to that theory. The regress might reveala bad feature of the theory—a feature that is not the regressitself, that we have independent reason to think is a reason to rejectthe theory. Or the fact that the theory results in the infiniteregress might itself be taken to be a reason to reject the theory. Theformer cases are the easier ones, since in those cases we do not haveto make a judgment as to whether the regress itself is objectionable,we only need to ask about the feature of the theory that the regressreveals. We will look at cases like this first, before turning tocases where the regress itself might be seen as a reason to reject atheory.

Many, many infinite regress arguments have been given throughout thehistory of philosophy, and we will not attempt here to survey even themost important of them. Rather, the aim will be to shed light on thekinds of regress argument that may be encountered, and the differentconsiderations that arise in different cases. As we proceed, however,we will see some particularly famous regress arguments as examples.(Rescher 2010 and Wieland 2014 survey some historical regresses.)^[2]

• 1. Regress and Theoretical Vices
  • 1.1 Regress and Contradiction
  • 1.2 Local Theoretical Vices
  • 1.3 Regress and Failure of Analysis
• 2. Foundations, Coherence, and Regress
• 3. Regress and Global and Local Explanation
• 4. Regress and Theoretical Virtues
• 5. Transmissive and Non-Transmissive Explanations
• 6. Coherence, Circularity, and Holism
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Regress and Theoretical Vices

Sometimes it is uncontroversial that a theory that generates aninfinite regress is objectionable, because the regress reveals thatthe theory suffers from some kind of theoretical vice that is a reasonto reject the theoryindependently of it yielding an infiniteregress. In these cases, an infinite regress argument can show us thatwe have reason to reject a theory, but it is not because the theoryyields a regressper se, but rather because it has this otherbad feature, and the regress has revealed that.

1.1 Regress and Contradiction

One such kind of case is when the very same principles of a theorythat generate the regress also lead to a contradiction. If this is sothen it does not matter what we think about infinite regress ingeneral, we will of course have reason to reject the theory, becauseit is contradictory. Two such examples are discussed by Daniel Nolan(2001), and we will recount one of them here (also cf. Clark 1988).The theory in question is Plato’s theory of Forms, and theregress objection is Parmenides’s Third Man objection, asreconstructed by Vlastos (1954).

Suppose some things, the \(X\)s, are alike in a certain way: theyshare some feature, \(F\). The theory of Forms says that if this is sothen there is some form, \(F\)-ness, in which the \(X\)s eachparticipate and in virtue of which they have that shared feature. Thetheory of Forms also says that Forms are self-predicated: the Form ofthe good is good, the Form of largeness is large, etc. So the Form of\(F\)-ness is \(F\). In that respect, then, it is like each of the\(X\)s. So not only are the \(X\)s all alike in a certain way, the\(X\)s and \(F\)-ness are all alike in a certain way. And so theremust be some Form in which each of the \(X\)s and \(F\)-nessparticipate, in virtue of which they have this shared feature.However, this Form cannot be \(F\)-ness, because the Form must bedistinct from the things that have that character that participate init, so the \(X\)s and \(F\)-ness must participate in a new form\(F_1\)-ness. But \(F_1\)-ness will, just like the \(X\)s and\(F\)-ness, be \(F_1\), given that Forms self-predicate, and so the\(X\)s, \(F\)-ness and \(F_1\)-ness are all alike in a certain way… and so on. We are off on an infinite regress.

Whatever one thinks about regresses in general, the principles thatgenerate this regress must be denied, for they lead to contradiction.The theory of Forms, as presented here at least, tells us (i) thatwhen some things are a certain way, they participate in a single Form,in virtue of which they are that way; (ii) that Formsself-predicate—Forms themselves are the way that things thatparticipate in them are; and (iii) that the Form is distinct from thethings that participate in it. Each of these three claims is essentialto generating the regress: without (i) we don’t get theexistence of a Form in the first place, without (ii) we don’tget the second list of things with a shared feature, so would stopsimply with the \(X\)s participating in some Form, and without (iii)we could conclude that just as the \(X\)s participate in \(F\)-ness,so do the \(X\)s and \(F\)-ness itself—there would be no needfor a new Form. So (i)–(iii) generate a regress, and they areeach needed to do so. But (i)–(iii) are inconsistent, and noregress argument is needed to show that. (i) and (ii) together entailthat Forms participate in themselves. (ii) tells us that \(F\)-ness is\(F\), and (i) tells us that if this is so it is in virtue of\(F\)-ness participating in \(F\)-ness, since that is howingeneral things get to be \(F\). So \(F\)-ness participates initself, and so by (iii) \(F\)-ness must be distinct from itself, sinceForms are distinct from that which participates in them. But nothingis distinct from itself: contradiction.

We don’t need an argument against infinite regresses to showthat this version of the theory of Forms is no good: that it iscontradictory is the best reason we could have to reject it. Now, thatthe theory is contradictory and that it leads to an intuitivelyworrying infinite regress are not unrelated. Beyond the mereontological profligacy involved in being committed to infinitely manyForms any time we notice that some things are some way (we will comeback to ontological profligacy and regress in section 4), what seemsintuitively problematic about the regress of Forms is that weshouldn’t get a new Form each time. The \(X\)sare \(F\), so we have the form of \(F\)-ness in which theyparticipate. We then get anew Form in which the \(X\)s and\(F\)-ness all participate. But why do we have a new Form? Wedon’t have a new shared feature, we have the very same sharedfeature we started with: just as the \(X\)s are all \(F\), so is\(F\)-ness itself. We alreadyhave the Form that makessomething \(F\): \(F\)-ness. The regress is troubling because weshouldn’t be invoking a new Form, but we have to because of theban on Forms participating in themselves. But this diagnosis of whythe regress is troubling is really just another way of stating thecontradiction at the heart of the theory: the Form \(F\)-ness issupposed to bein general the thing in virtue of which thingsget to be \(F\), but it also cannot be because it itself must be \(F\)and it cannot participate in itself. So the regress and thecontradiction are intimately related. As Nolan (2001, 528) puts it:“infinite regresses of this sort and the statement of formalcontradiction are different ways of bringing out [the same]unacceptable feature.”

This is an easy case, because we don’t have to adjudicate onwhether the fact that the theory leads to an infinite regress isitself objectionable. The principles that lead to regress also lead tocontradiction, and we know that a theory’s being contradictoryis a good reason to reject it, whether it leads to regress or not.

1.2 Local Theoretical Vices

More generally, if the features of a theory that result in an infiniteregress also result in some theoretical vice that we know to beobjectionableindependently of whether or not there is aregress, then we have a reason to reject that theory thatdoesn’t depend on the theory leading to regress. Sometimes, thetheoretical vice in question will be a global one: a feature that is areason to reject any theory that has it. Yielding a contradiction is,relatively uncontroversially^[3], such a vice. Other times, the feature in question might be a localvice: a feature that might be an unobjectionable feature of certaintheories, but a reason to reject a particular theory \(T\) because ofthe particular theoretical ambitions of \(T\) or as a result of otherthings we know about \(T\)’s subject matter.

For example, a theory might result in an infinite regress of entitiesand so entail that there are infinitely many things. This in itselfis, arguably, not objectionable. But it might be a local vice to atheory if we have independent reason to think that we are dealing witha finite domain. (See Nolan 2001, 531–532, and Nolan 2019.)(Some philosophers object to the very idea of reality containinginfinities. Aristotle, e.g., famously allowed that there could bepotential infinite series, but notcompletedinfinite series. (See Mendel 2017.)) But we will ignore such generalanti-infinitism in this entry, for it is infinity itself that suchtheorists take to be objectionable, not infiniteregresses perse.)

Peano’s axioms for arithmetic, e.g., yield an infinite regress.We are told that zero is a natural number, that every natural numberhas a natural number as a successor, that zero is not the successor ofany natural number, and that if \(x\) and \(y\) are natural numberswith the same successor, then \(x = y\). This yields an infiniteregress. Zero has a successor. It cannot be zero, since zero is notany natural number’s successor, so it must be a new naturalnumber: one. One must have a successor. It cannot be zero, as before,nor can it be one itself, since then zero and one would have the samesuccessor and hence be identical, and we have already said they mustbe distinct. So there must be a new natural number that is thesuccessor of one: two. Two must have a successor: three. And so on… And this infinite regress entails that there are infinitelymany things of a certain kind: natural numbers. But few have foundthis worrying. After all, there is no independent reason to think thatthe domain of natural numbers is finite—quite the opposite.

By contrast, consider the following two principles: (i) Every event ispreceded by another event that is its cause; (ii) The relation \(x\)precedes \(y\) is irreflexive (nothing precedes itself),asymmetric (if \(a\) precedes \(b\) then \(b\) does not precede \(a\))and transitive (if \(a\) precedes \(b\) and \(b\) precedes \(c\) then\(a\) precedes \(c\)).

This yields an infinite regress, at least from the assumption thatthere is at least one event. If there is an event, \(E_1\), then it ispreceded by its cause. That cause cannot be \(E_1\), as nothingprecedes itself and causes precede what they cause. So the cause of\(E_1\) must be a new event, \(E_2\). This event is preceded by itscause. This cannot be \(E_2\) for the same reasons as before, and itcannot be \(E_1\) because then each of \(E_1\) and \(E_2\) wouldprecede the other in violation of asymmetry. So the cause of \(E_2\)must be a new event, \(E_3\). \(E_3\) is preceded by its cause. Itcannot be \(E_3\) or \(E_2\) for reasons similar to before. And itcannot be \(E_1\), for then \(E_1\) would precede \(E_3\), but since\(E_3\) precedes \(E_2\) which precedes \(E_1\), transitivity entailsthat \(E_3\) precedes \(E_1\), and so \(E_1\) cannot precede \(E_3\)due to asymmetry. So the cause of \(E_3\) must be a new event,\(E_4\). And so on …

This regress of events is very similar to the regress of naturalnumbers. In each case we start from the claim that there is a thing ofa certain kind (a number or an event), and we have a principle thattells us that for each thing of that kind, there is another thing ofthat very kind that bears a certain relation to the previous one (itis its successor, or it is its preceding cause). We then havesupplementary principles that rule out the other thing of that kindbeing any of the things on our list so far, thus forcing us tointroduce a new thing of that kind, thus inviting the application ofthe principles to this new thing, and so onad infinitum. Butwhile the regress and resulting infinity of natural numbers isarguably unobjectionable, the regress of events seems problematic,because we have good empirical reasons to deny that there areinfinitely many events, each preceded by another. For either thatinfinite sequence of events takes place in a finite amount of time oran infinite one. We have good empirical reason to rule out the latteroption, since we have good empirical reason to think that there hasonly been a finite amount of past time: that time started a finitetime ago with the Big Bang. And we have good empirical reason to ruleout the former option, since the only way of fitting an infiniteseries of events, each preceded by another, into a finite stretch oftime is by having the time between them become arbitrarily small. Sofor example, \(A_2\) might be a minute before \(A_1\), and \(A_3\)half a minute before \(A_2\), and \(A_4\) a quarter of a minute before\(A_3\), etc. If the time between events \(A_n\) and \(A_{n + 1}\) isalways half the time between events \(A_{n - 1}\) and \(A_n\), we canfit infinitely many events into a two minute time-period. But thisrequires the time between events to become arbitrarily small, andthere is some reason to think that time is quantized, such that thereis a minimum length of time during which a change can occur, thusproviding a lower limit on the amount of time that can separate twotemporally distant events. (See Nolan 2008b for relevant discussionand further references.)

So while the numbers regress and the events regress are structurallyanalogous, we might find the principles that yield the regressobjectionable in one case but not the other, because while eachregress entails that there are infinitely many things of kind \(K\),whether that is a vice may depend on what kind \(K\) is, and whetherwe have independent reason to think that the domain of \(K\)s is afinite one. Yielding infinitely many things of kind \(K\) might be alocal vice when the kind in question is events separated in time, butnot when the kind in question is natural numbers structured by thesuccessor relation.

1.3 Regress and Failure of Analysis

In the previous section we saw two theories generating similarregresses, but where one is found unobjectionable whereas the other isfound objectionable due to the different things we think we know,independently of encountering these regress arguments, about thesubject matter of the theories. We could also have cases where asingle theory yields a regress that is objectionable by the lights ofone theorist and not another, as a result of their differingtheoretical commitments leading one but not the other to think that afeature revealed by the regress is a vice. (See Cameron 2022, section1.3.)

Consider Bradley’s regress. (Bradley 1893 [1968], (21–29).Note, however, that Bradley is very hard to interpret, and there ismuch debate concerning how to reconstruct his argument. See the entryonBradley’s regress for discussion.) We start with the demand to give an account ofpredication: what is it for \(A\) to be \(F\)? One answer is that itis for the particular \(A\) to be bound to the property of \(F\)-ness.But this answer yields a new predication: \(A\) is bound to\(F\)-ness. If the original monadic predication of \(A\) demands anaccount, this relational predication of \(A\) and \(F\)-ness alsodemands an account. Since before we posited a property correspondingto the monadic predicate and said that the property was bound to theobject that was the subject of predication, we should follow suit hereand posit a relation corresponding to the dyadic predicate and saythat it is bound to the two things that are the subjects of thatpredication. So we should posit a relation—let’s call itinstantiation—that binds together \(A\) and \(F\)-ness.But this yields another new predication: Instantiation binds \(A\) to\(F\)-ness. Now this triadic predication needs to be accounted for,and so we need a relation corresponding to the triadicpredicate—let’s call it the instantiation\(_2\)relation—that binds together the instantiation relation, \(A\),and \(F\)-ness. But this yields yet another predication, this time atetradic one: Instantiation\(_2\) binds Instantiation to \(A\) and\(F\)-ness… And so on. Each predicational fact tells us thatsome particulars, properties, and relations are bound together, whichforces us to posit a relation corresponding to that binding, whichgenerates the next predicational fact, and so onadinfinitum.

Is this regress objectionable? Arguably it depends on what we wantfrom an account of predication. If what you want is ananalysis of predication, then arguably this regressis objectionable. If you start off not understandingpredication—if you find it just utterly mysterious what‘\(A\) is \(F\)’ is meant to mean, given that this‘is’ is not identity—then this regress means thatthis account is not going to help you, for each answer simply invokesanother predication, which is exactly what you don’t understand.If you don’t understand ‘\(A\) is \(F\)’,you’re not going to understand ‘\(A\) is bound to\(F\)-ness’, or ‘Instantiation is bound to \(A\) and\(F\)-ness’, or etc., since the ‘is’ in those claimsis also not the ‘is’ of identity but the ‘is’of predication, which you find a mystery. However, if youunderstand predication perfectly fine and want simply anontological account of it—an account of what in theworldmakes true the true predications—then arguablythe regress isnot objectionable, because while there areinfinitely many true predications, these can all be made true by theone underlying state of the world: the state of affairs of theparticular \(A\) being bound to the property \(F\)-ness. Just as thisstate of affairs makes it true that \(A\) is \(F\), so does it make ittrue that \(A\) is bound to \(F\)-ness, and that this binding holdsbetween \(A\) and \(F\)-ness, and etc. We would have one ontologicalunderpinning for the infinitely many true predications. As Nolan(2008a, 182) puts it, we have “Ontology for each truth, and noinfinite regress of [states of affairs], but only of descriptions of[states of affairs]: and it should never be thought that an infiniteregress of descriptions is something to worry about per se”.(Cf. Armstrong 1974 and 1997 (157–8).) So whether or not we willfind this regress objectionable depends on what we demand of anaccount of predication. We can all agree that the regress shows thatthe account does not yield an analysis of predication. Is this atheoretical vice? It depends on whether or not predication requires ananalysis, and that depends on our theoretical goals.

Another regress that arguably fits this pattern is McTaggart’s(1908) regress argument against the reality of the A-series of time.The A-series of time is the sequence of times one of which is present,others past, and others future. (As opposed to the B-series, whichmerely says that some times are before others, some after others,etc., without singling out any as being past, present, or future.)But, says McTaggart, the times that are past,were present;the time that is presentwas future andwill bepast; the times that are futurewill be present andwillbe past. McTaggart concludes that we end up attributing eachA-property (is present, is past, andis future) toevery time (and therefore to every event in time). But this is absurd,because the A-properties are incompatible: to have one is to haveneither of the others. So we end up in contradiction: each time bothhas only one such property, and all such properties. McTaggartconcludes that the A-series cannot be real.

An obvious response to McTaggart’s argument is this: it’snot that some future timeis present and past as well,it’s merely that itwill be present andwillbe (later still) past. That is: no time and eventhasmore than one of the A-properties, it is merely the case that theyhave one A-property anddid andwill have another.But McTaggart thinks this response does not solve the problem, becauseit leads to regress. He says (ibid., 469):

If we avoid the incompatibility of the three characteristics byasserting that M is present, has been future, and will be past, we areconstructing a second A series, within which the first falls, in thesame way in which events fall within the first… the second Aseries will suffer from the same difficulty as the first, which canonly be removed by placing it inside a third A series. The sameprinciple will place the third inside a fourth, and so on without end.You can never get rid of the contradiction, for, by the act ofremoving it from what is to be explained, you produce it over again inthe explanation. And so the explanation is invalid.

McTaggart’s argument is difficult and philosophers disagree onhow to interpret it, but here is one interpretation. (See Dummett 1960and Mellor 1998 (72–74) for two among many presentations of theargument along similar lines. See Cameron 2015 (Ch. 2) and Skow 2015(Ch. 6) for recent discussion.)

The way things were and the way things will be seems to be part ofreality in a way that, for example, Bilbo’s finding the One Ringis not, it’s merely part of a fiction. History isnot afiction, it’s part of our world, so historical truths and futuretruths should, seemingly, be part of the overall account of how theworld is. But that is puzzling, given that thingschange and,hence, the way things were is incompatible with the way things arenow. How can they both contribute to the way realityis ifthey are incompatible? McTaggart’s argument focuses on aparticular instance of this concerning the A-properties.Caesar’s crossing the Rubicon is past. But itwaspresent, and so its presentness is a feature of our world’shistory. But our world’s history, as we just said, is part ofthe complete account of how our worldis, and so the completeaccount of how our world is includes both Caesar’s crossing theRubicon being past and it being present, and yet those areincompatible features. The defender of the A-series replies byinsisting that in giving the complete account of how reality is wehave to take seriously the fact that realitychanges and thatit is, therefore, different wayssuccessively, and there isno inconsistency in things being one way andthen another,incompatible, way. So it’s not that reality is such thatCaesar’s crossing the Rubicon is both past and present,it’s that reality is such that Caesar’s crossing theRubiconwas future, andwas present, and isnow past.

McTaggart responds by restating this response in terms of second-orderA-properties. To say that Caesar’s crossing the Rubiconwas future, andwas present, and isnowpast is to say that it has the properties of beingpastfuture (i.e. having been future),past present (i.e.having been present) andpresent past (i.e. now being past).And similar reasoning to the above suggests that every time and eventhas each of the nine possible second-order A-properties; and whileit’s not the case that any two of them are incompatible, certainpairs of them are. Nothing can be bothpast past andfuture future, for example: if \(E\) has been a mere pastevent, it can’t be true that \(E\) will be something that is yetto happen. And yet the complete account of reality seems to includeboth Caesar’s crossing the Rubicon being past past and its beingfuture future. After all, in 2000 BCE Caesar’s crossing theRubicon was future future, since 1000 BCE was the future andCaesar’s crossing the Rubicon would still be the future then.And Caesar’s crossing the Rubicon is past past just now, since1000 CE is past, and Caesar’s crossing the Rubicon was pastthen. But both 2000 BCE and the present are part of the overallhistory of the world, so the goings on at each time are part of whatthe world as a whole is like, and so Caesar’s crossing theRubicon is both past past and future future, and yet those areincompatible. And again, the defender of the A-series will respondthat it’s not that Caesar’s crossing the Rubiconis both past past and future future, it’s that it isnow past past andwas future future. And McTaggartwill respond that this is to invokethird-orderA-properties—beingpresent past past, beingpastfuture future, etc. And the same problem will arise, and invitethe same response, which will lead to the same problem concerningfourth-order A-properties, which will invite the same response again… and so on,ad infinitum.

Whether McTaggart’s regress is vicious has proven a subject ofmuch debate. Some philosophers see the regress as demonstrating thatany attempt to describe the world in A-theoretic terms is ultimatelyinconsistent, and see the A-theorist as merely invoking anotherinconsistent account of reality every time they attempt to explainaway this inconsistency. Mellor, for example, says (1998, 75)“[McTaggart’s] critics react by denying the viciousness ofthe regress. At every stage, they say, we can remove the apparentcontradiction by distinguishing the times at which the events haveincompatible [A-properties]. They ignore the fact that the way theydistinguish these times … only generates morecontradictions.” Others see McTaggart as simply making aconfused challenge at each stage—as mistakenly concluding fromthe fact that thingswere one way and arenowanother incompatible way that theyare both ways atonce—and continuing to make the same mistake in response to eachof the A-theorist’s correct explanations that in fact theincompatible properties are only ever had one after another, never atthe same time. Skow (2015, 87), e.g., says “At each stage[McTaggart] accuses objective becoming [i.e. things changing theirA-properties] of being inconsistent, and [the A-theorist] shows thatallegation to be false. And even an infinite sequence of falseallegations does not add up to a good argument.”

This is arguably another case where whether or not the regress isvicious by a philosopher’s lights will depend on theirbackground theoretical commitments. If one starts out happy with thenotion ofsuccession—i.e. of one thing being the caseandthen ceasing to be the case as something elsecomesto be the case—then one may be inclined to seeMcTaggart’s regress as entirely benign. We start out with a setof incompatible properties that are never had by anythingsimultaneously but are held by things successively; in stating thatthose properties are had successively we make salient a new set ofincompatible properties, but these are also never had by anythingsimultaneously, only successively; this makes salient yet another setof incompatible properties, and so onad infinitum. There isnever, at any stage, a contradiction, if the notion of succession isindeed in good standing, for we are never forced to say that a thinghas incompatible properties, only that a thing successively hasproperties that cannot be had simultaneously. If, by contrast, one issuspicious of the very notion ofsuccession—if one seesin it simply an attempt to paper over what is ultimately acontradiction inherent in the commitment to reality being each of twoincompatible ways, the way it supposedly is now and the way itsupposedly was or will be—then one will see in McTaggart’sregress an infinite sequence of contradictory accounts of how realityis, with each attempt at explaining away the contradiction simplyresulting in another such contradictory account. The regress, then,looks vicious or benign depending on whether one is content to grantthe legitimacy of the notion of temporal succession.

2. Foundations, Coherence, and Regress

In section 1 we looked at cases where an infinite regress is taken toreveal some feature that might, possibly depending on your othertheoretical commitments, be taken to reveal a feature of a theoryindependent of its leading to regress that is a reason toreject it. But sometimes the regress itself is taken to be anobjectionable feature of the theory that yields it.

Suppose that there is an \(X\) that is \(F\), and that to account forwhy \(X\) is \(F\) we need to appeal to another \(X\) that isalso \(F\). Now there is the question as to whythis \(X\) is\(F\), and so we need to appeal toanother \(X\) which isalso \(F\) …. If this proceedsad infinitum,with a new \(X\) invoked at each new stage in the process, there is aconcern that we end up without having accounted for the \(F\)-ness ofany of the \(X\)s. If this infinite regress argument is successfulthen our choices are either:

Foundationalism: To halt the regress by taking there to be afoundation—a set of \(X\)s whose \(F\)-ness is taken tobe basic, and by which we can account for the \(F\)-ness of all other\(X\)s.

or:

Coherentism: To resist an infinite regress by allowing a circular orholistic explanation of the \(F\)-ness of at least some \(X\)s.

Infinite regress arguments used to motivate Foundationalism orCoherentism appear in many different areas of philosophy. Here aresome highlights:

In metaphysics:

Metaphysicians have wanted to account for the very existence, ornature, of some things by appealing to things on which theyontologically depend: for example, a complex object exists and is theway it is because its parts exist and are the way they are; a setexists because its members exist; etc. (See Fine 1995 and Koslicki2013 for discussion.) But of course the things the dependent beingsdepend on must themselves exist as well. Some have been suspicious ofthe idea that this can go onad infinitum, with every thingbeing ontologically dependent on some new thing(s), and thus haveargued for Metaphysical Foundationalism: the view that there is acollection of absolutely fundamental^[4] entities upon which all else ultimately ontologically depends.Aquinas, e.g., holds that events are ontologically dependent on theircauses, and that an infinite regress of causes and effects would be aninfinite series of things each of which is ontologically dependent onthe next, and this is impossible.^[5] Thus he concludes that there must be a first cause of all else thatis itself uncaused—namely, God. We shall see more examples ofMetaphysical Foundationalists below. See also the supplementarydocument on

Metaphysical Foundationalism and the Well-Foundedness of Ontological Dependence.

(Metaphysical Coherentism—the view that ontological dependencecould be a holistic phenomenon—has received few defenders, butsee Barnes 2018, Bliss 2014, Cameron 2022 (Chs. 4 & 5), Nolan2018, Priest 2014 (Chs. 11 & 12), and Thompson 2016 and 2018 forsome discussion.)

In epistemology:

Epistemologists want to account for the justification of our beliefs.We do not want to believe at random, we want our beliefs to bejustified—that is, we want there to be areason to believe the propositions we believe. But thosereasons will be further propositions, and if our initial belief is tobe justified, so surely must the reasons for that belief be, and so wemust appeal to yet more propositions, and so on. Many—going backto Sextus Empiricus (Outlines of Pyrrhonism PH I,164–9)—have thought that this cannot proceedadinfinitum, and that the only serious options are EpistemicFoundationalism—the view that there is a class of propositionswhose justification does not come via some other justifiedpropositions, and that can provide a reason for everything else webelieve—or Epistemic Coherentism—the view that acollection of propositions can collectively be justified in virtue ofthe web of epistemic relations they stand in to one another. Thus Sosa(1980, 3) says “epistemology must choose between the solidsecurity of the ancient foundationalist pyramid and the riskyadventure of the new coherentist raft.” (See the entries onfoundationalist theories of epistemic justification andcoherentist theories of epistemic justification for surveys of Epistemic Foundationalism and Epistemic Coherentism,respectively).

In ethics:

As well as asking about the source of justification for our moralbeliefs (see e.g., Sinnott-Armstrong 1996), moral philosophers havebeen concerned with distinctively moral regresses, which arise notwhen we attempt to account for the justification of a moral claim, butrather when we attempt to account for the moral status of something byappealing to something else of the same moral status, and so on.Aristotle (Nicomachean Ethics, 1094a) thought that some things weregood because we desire them for the sake of something else that isgood. But ifeverything that is good is good simply becauseit aims at something else that is good, this would lead to a regressand “all desire would be futile and vain”. And soAristotle argues that there must be a Highest Good—somethingthat is desired for its own sake—that other things can be goodin virtue of aiming towards this highest good. Aristotle is a MoralFoundationalist: there is something whose goodness does not getexplained by reference to anything else, by means of which thegoodness of other things is accounted for.

(Explicit statements of anything other than Foundationalism in themoral case are hard to come by.^[6] But see Roberts 2017 for relevant discussion.)

However it is also always possible to simply embrace the regress andaccept:

Infinitism: The \(F\)-ness of each \(X\) is accounted for by factsinvolving a new \(X\) that is \(F\), and this proceedsadinfinitum.

Whether in metaphysics, epistemology, or ethics, Foundationalism hasoften been seen as the default, orthodox, view, with Coherentism beingseen as the radical alternative. Infinitism is often simply dismissed,or not even considered as a live option. Foundationalism andCoherentism are (often^[7]) motivated by the thought that if the \(F\)-ness of each \(X\) isaccounted for by appeal to a new \(X\) that is \(F\) then each ofthose infinitely many explanations fails. The thought is that eachexplanation of the \(F\)-ness of an \(X\) would be dependent on thesuccess of the next, a promissory note that is never paid if thatprocess does not end. Let’s examine this anti-Infinitistthought.

To focus our inquiry, consider the case of a complex object and itsproper parts. Some metaphysicians have considered the possibility thatthere aregunky objects: objects such that every part of themitself has proper parts. If \(A\) is gunky then it is composed of somethings, the \(X\)s, such that there is more than one of the \(X\)s,and \(A\) is not amongst the \(X\)s. Pick one of those \(X\)s,\(X_1\). \(X_1\) is composed of some things, the \(Y\)s, such thatthere is more than one of the \(Y\)s, and neither \(A\) nor \(X_1\) isamongst the \(Y\)s.^[8] Pick one of those \(Y\)s, \(Y_1\). \(Y_1\) is composed of somethings, the \(Z\)s, such that there is more than one of the \(Z\)s,and neither \(A\), nor \(X_1\), nor \(Y_1\) is amongst the \(Z\)s. Andso on. This process will never end: each item in the series is acollection of entities (a collection containing just one thing in thefirst case, and more than one thing in every subsequent case), andgiven the transitivity of parthood each thing in each collection willbe a part of \(A\) and hence—since \(A\) isexhypothesi gunky—will itself be composed of a collection ofproper parts, and so we can therefore pick any member from anycollection to generate the next item on the list. (Of course, a thingneed not—and if it is gunky,will not—have aunique decomposition: there can be two collections of things, the\(X\)s and the \(Y\)s, each of which compose \(A\), where none of the\(X\)s is amongst the \(Y\)s and vice versa. But all we need is thatthere aresome such collections, from which we can pickarbitrarily to get the next collection in the sequence.) Now take theinfinite series that consists of \(A\) as its first element, \(X_1\)as its second element, \(Y_1\) as its third, etc. So we form thisinfinite sequence by taking one item from each collection that formedthe previous infinite sequence: namely that item which was used togive us the collection of things that came next in the series.

Assuming that complex objects are ontologically dependent on theirproper parts, we now have an infinite regress of entities, each ofwhich is ontologically dependent on the next. Such an infinite regresshas been thought by some metaphysicians to be objectionable, leadingthem to reject the possibility of gunk. Leibniz, for example, arguesthat there cannot be only “beings by aggregation” (i.e.,composite objects), because this would lead to an infinite regress,with each being by aggregation being made up of further beings byaggregation, and so onad infinitum.

Leibniz’s idea seems to be that if each thing depends on someother, there could not be anything at all in the first place. Thethought is that ontologically dependent entities inherit theirexistence, or being, from that on which they depend; so if this chainof dependence does not terminate, the whole process couldn’t getoff the ground, and there would be nothing at all. Leibniz says(1686–87, 85):

Where there are only beings by aggregation [composite objects], thereare no real beings. For every being by aggregation presupposes beingsendowed with real unity [simples], because every being derives itsreality only from the reality of those beings of which it is composed,so that it will not have any reality at all if each being of which itis composed is itself a being by aggregation, a being for which wemust still seek further grounds for its reality, grounds which cannever be found in this way, if we must always continue to seek forthem.

By contrast, if the dependence runs in the other direction—if westart off with a fundamental entity whose being can then ground thebeing of each subsequent entity—there is arguably no problem ifthat process continuesad infinitum: thus the infinitesequence where you start with a thing, form its singleton set, thenformthat thing’s singleton set and so onadinfinitum has not been thought to be objectionable by those whoreject gunky objects, for it is the set that is ontologicallydependent on its members, notvice versa. (See Fine 1994 fordiscussion of the direction of ontological dependence. See Cameron2008 and Maurin 2007 for discussion of the difference between infiniteregresses where ontological dependence runs upwards from ones where itruns downwards. Cf. Clark 1988, and also Johansson 2009 and thediscussion in Maurin 2013.)^[9]

A contemporary sympathizer with Leibniz’s thought is JonathanSchaffer (2010). Unlike Leibniz, Schaffer grants the possibility ofgunky objects, but thinks that this possibility is precisely a reasontodeny that complex objects are (always) ontologicallydependent on their parts.^[10] Instead, Schaffer takes the possibility of there being no simplethings as a reason to hold that the dependence flows in the otherdirection: that parts are dependent on the wholes of which they areparts, and that every thing is thereby ontologically dependent on thebiggest thing that there is—the cosmos—that has everythingelse as a proper part.^[11] Since classical mereology guarantees that there is a biggestthing—the thing that has all else as proper parts—but itdoes not guarantee that there are any smallest things—thingsthat have no proper parts—it guarantees that if parts aredependent on the wholes of which they are parts then there will be afirst, ontologically fundamental, element, whereas if wholes aredependent on their parts then there is the possibility of an infiniteregress in which each thing is dependent on some further thing(s),with nothing being fundamental: a possibility in which, Schaffer(2010, 62) says (agreeing with Leibniz), “Being would beinfinitely deferred, never achieved”. Leibniz and SchafferadvocateMetaphysical Foundationalism: the view that therehave to be some things that are absolutely fundamental—dependenton nothing—on which all else ultimately depends.

But is it true that if the \(F\)-ness of each \(X\) is dependent onthe next \(X\) in the sequence being \(F\), and if this goes onwithout end, that we cannot explain why any \(X\) is \(F\)? Schafferclaims that in the case of an infinite regress of ontologicaldependence, with each entity depending on the next in the chain, andno independent entities, being would be “infinitely deferred,never achieved”. Why think this? The idea seems to be that adependent entity only has the being it has on condition of somethingelse having being. If \(A\) is ontologically dependent on \(B\) thenthe existence of \(A\) is a promissory note, only paid if \(B\) itselfexists. But if \(B\) is ontologically dependent on \(C\) then theexistence of \(B\) is a promissory note, only paid if \(C\) exists… and so on, so that if this process never stops, thepromissory note is never paid, in which case, allegedly, the existenceof all these things could never get off the ground in the firstplace.

An analogy may help. Suppose Anne has no sugar, and needs some. Shecan borrow a bag of sugar from Breanna. Now Anne has a bag of sugar.Where did it come from? Easy—it came from Breanna, who is now abag of sugar down. But suppose Breanna borrowed a bag of sugar fromCraig in order to then pass it on to Anne. Where did Anne’s bagof sugar come from then? Ultimately, from Craig, who ends up a bag ofsugar down. But suppose Craig borrowed a bag of sugar from Devi… and so on,ad infinitum. Then where did the bag ofsugar come from? At the end of the infinite sequence, Anne is one bagof sugar up, and nobody is a bag of sugar down, for everyone afterAnne simply borrowed a bag of sugar, and then passed it on to the nextperson in the chain. There’s an extra bag of sugar in the systemthat seems to have appeared as if by magic. If there is a finitesequence of borrowers, however long, then the last person in the chainends up a bag of sugar down, so that’s where the bag of sugarthat ends up with Anne ultimately comes from. But if the chain neverends, Anne ends up with a bag of sugar that doesn’t seem to havecome from anybody, as nobody has lost any bags of sugar—they alljust borrowed it to pass it on. The infinite regress seems to createsugar from nowhere: pleasant, perhaps, but metaphysically suspiciousall the same.

As with sugar, likewise with being—or justification, orgoodness, or whatever feature we aim to account for. If \(A\) dependson \(B\) and \(B\) is fundamental, where did \(A\)’s being comefrom? From \(B\). If \(A\) depends on \(B\) and \(B\) depends on\(C\), where did \(A\)’s being come from? Ultimately, from\(C\). And for any finite chain, no matter how long, we can say wherethe being of any dependent entity ultimately comes from: from thefundamental thing(s) at the bottom of the chain. But if the chain isinfinite, the being of any thing is, arguably, as mysterious asAnne’s new bag of sugar. The explanation of where it came fromis always postponed, and its presence in the system as a wholeunexplained. So, at least, goes the regress objection.

3. Regress and Global and Local Explanation

Distinguish between alocal explanation of the \(F\)-ness ofsome particular \(X\) and aglobal explanation of why thereare any things that are \(F\) at all. Some philosophers have arguedthat when we have an infinite regress, with the \(F\)-ness of each\(X\) being accounted for by appeal to another \(X\) that is \(F\),then we do indeed lack aglobal explanation of why there arethings that are \(F\), but we nevertheless have a local explanationfor each of the infinitely many \(X\)s as to whyit is \(F\).(This seems to be the position of Hume’s Cleanthes in Part IX ofHume 1779.)

Ricki Bliss, e.g., speaking of the infinite regress of ontologicallydependent entities, says (2013, 408): “In a reality thatadmitted of no foundations … although everything has itsreality accounted for in terms of that upon which it depends, we havefailed to explain how the whole lot of them—everything—hasany reality at all.” Compare to the infinite borrowers case: forany given instance of someone having received a bag of sugar, we canexplain where that bag of sugar came from: it came from the nextperson in the chain. But what we can’t explain is a global factabout the series as a whole: why is this bag of sugar in the series inthe first place?

But Bliss argues that it is not necessarily a mark against infinitelydescending chains of ontological dependence that it leaves this globalfact—why does anything have being in the firstplace?—unexplained. She says (ibid.):

[T]he regress is not designed to answer this question. All the regresscan tell us is how each individual member has the property underconsideration, namely, in dependence upon something else. Theappearance of an infinite regress should not lead us to conclude thatnothing within the regress has the property underconsideration—nor has its possession of that propertyunexplained—but rather that not everything about the possessionof the property that needs to be explained has been.

Contra Leibniz and Schaffer, then, Bliss rejects the idea that in aninfinitely descending chain of ontological dependence, being wouldnever be achieved. Having a propertydependent on somecondition is nevertheless to have that property, so there is nopressure, she argues, to conclude that nothing in the infinite serieswould exist. Rather, they all exist, and the existence of each isperfectly well accounted for: it exists because the next thing in thesequence does. Everything has its being merely on some condition, butthe condition is always met. Why there is an infinite chain ofexisting entities at all is not accounted for, but Bliss says it is amistake to think that the regress was ever supposed to account forthat.

Bliss concludes that whether or not an ontological infinite regress isvicious or benign depends on what we set out to give an account of. Ifall we want is an account of why each thing exists, then it is benign;but if we want an account of why there are things at all, it is vicious.^[12] She says (ibid., 414):

If \(x\) is grounded in \(y\) and \(y\) in \(z\), [and so onadinfinitum] and all that we are seeking for is an explanation ofhow or why \(x\) exists (as the thing that it is), an explanation ofhow or why \(y\) exists, and so onad infinitum, the regressis benign. Why? Because where \(z\) explains \(y\) and \(y\) explains\(x\) ourexplanans andexplanandum arenotof the same form. In order to explain facts about my existence, we canmake recourse to the existence of—or facts of the existenceof—my parents, my vital organs, etc. In order to explain thesefacts, we make recourse to further facts, and so on. At each stage, wehave a satisfactory explanation of that for which we are seeking one .. .

The regress is not benign, however, if what we are seeking anexplanation for is how anything exists, or has being, at all. For evenat infinity, what the regress shows is that we have not explainedwhere existence comes from. Even at infinity, we are still invokingthings that exist in order to explain how anything exists atall… We encounter, at each level, the explanatory failurecharacteristic of a vicious infinite regress: existents whoseexistence we seek an explanation for are explained in terms ofexistents. The existence of \(y\) may explain the existence of \(x\)but the existence of \(x\), \(y\), \(z\), and so onadinfinitum cannot help us explain how anything exists atall—where being comes from. Whether or not a regress of groundsis vicious, therefore, will depend upon the question for which we areseeking an answer.

Similar remarks are made by Graham Priest (2014, 186), who asks us toimagine an infinite sequence of objects, \(a_0\), \(a_{-1}\),\(a_{-2}\), … etc. Each of these can be in one of two states:active or passive. For each \(n \le 0\), if \(a_{n-1}\) is passive itdoes nothing, but if it is active it instantaneously makes \(a_n\)active as well; and the only way for \(a_n\) to become active is that\(a_{n-1}\) makes it so. Given this set-up there are only two possibleoptions: each object in the chain is active, or each is passive. Butboth are logically possible options: the fact that each object is onlyactive if the next object makes it active (and this sequence continueswithout end) gives us no reason, says Priest, to reject thepossibility of them being all active. The active status of each objectwould be accounted for, by the active status of the previous one.First we explain the active status of \(a_0\): it is explained by theactive status of \(a_{-1}\). Then we have a completely different thingto explain: the active status of \(a_{-1}\): it is explained by theactive status of \(a_{-2}\) … and so on. All such facts getexplained and since, Priest argues, it is a different fact beingaccounted for each time, this regress is not vicious. But echoingBliss, Priest admits that something is not explained. He says (ibid.,187) “[W]hat an infinite regress will not explain is why thewhole regress is as it is … the state of each \(a_n\) isdetermined by the state of \(a_{n-1}\), but there is nothing in thisstory to explain why the whole system is in theall activestate, as opposed to theall passive state. If there is suchan explanation, it must come from elsewhere.”

So if \(a\) can only exist if \(b\) exists for \(a\) to beontologically dependent on, and \(b\) can only exist if \(c\) existsfor \(b\) to be ontologically dependent on … and so onadinfinitum, either the whole infinite sequence of things exists,or none of them do. And if the whole infinite sequence exists, thereis no explanation (from within the sequence at least) as to whyanything exists at all. However, thereis an explanation foreach particular thing as to whyit exists: it exists becausethe next thing in the sequence does. If Bliss and Priest are correct,then whether or not an ontological infinite regress is vicious orbenign depends on our explanatory ambitions: are we attempting toexplain an (infinite) collection of particular existence facts: thatthis thing exists, thatthat thing exists, etc., orare we attempting to explain the global fact that things exist.Exactly the same infinitely regressing ontology can be vicious orbenign depending on one’s theoretical lights. Notice thesimilarity to the discussion in section 1.3 of Bradley’s andMcTaggart’s regresses: once again, whether or not there issomething inherently objectionable to an infinite regress may dependon our theoretical ambitions. Bliss (2019) argues that MetaphysicalFoundationalism appears to be motivated by an explanatory demand thateverything that is apt for explanation be explained in terms of factsthat do not require an explanation, but that this explanatory demandbegs the question in favor of Foundationalism.

Ross Cameron (2022) argues that when we have an infinite regress ofontological dependence or grounding, we might as a result of thatregress lack an explanation of the existence or nature of the entitieson the infinite chain, or of the facts on that chain. However, Cameronargues that this is not a reason to deny that such infinite regressesare impossible, and instead argues that we must pull apart the notionof metaphysical explanation from relations like ontological dependenceand grounding, which he calls metaphysical determination relations. If\(E_1\) is ontologically dependent on \(E_2\), which is ontologicallydependent on \(E_3\), which is ontologically dependent on \(E_4\)… and so onad infinitum, there might be nometaphysical explanation for certain facts concerning the existenceand nature of \(E_1\), \(E_2\), … etc., but, argues Cameron,this is compatible with each of the facts concerning those thingsbeing adequately grounded in reality, and hence with the reality ofsuch an infinite sequence. Infinite regresses of metaphysicaldetermination relations, argues Cameron, might require us to abandoncertain explanatory goals, but this does not render themimpossible.

4. Regress and Theoretical Virtues

Bliss and Priest, as we have seen, argue that while an ontologicalinfinite regress might leave some questions unanswered, there isnothing inherently objectionable, incoherent, or inconsistent in aninfinite regress of things each of which is ontologically dependent onthe next. However, even if such ontological infinite regresses arepossible, some metaphysicians argue that we may have good reason tothink that the actual world is not like that.

Nolan (2001) and Cameron (2008) argue that considerations oftheoretical parsimony can lead us to reject ontological infiniteregresses even if such regresses are not metaphysically impossible. Atheory that yields an ontological infinite regress of course therebyyields an infinite ontology. Even if we are not in the situation(discussed above in section 1.2) where we have independent knowledgethat we are dealing with a finite domain, this could still be a markagainst the theory, simply on the grounds that it is an unparsimoniousontology. So for example, we might object to the claim that materialobjects are gunky—with each part of them being divisible intofurther proper parts—not because there’s any inconsistencyin the hypothesis, or because it leads to an infinite chain ofontological dependence and thereby leaves the existence of all thingsunexplained, but simply because it means that whenever there is anobject somewhere, there are in fact infinitely many objects there. Agunky world is an ontologically extravagant world, and so we have thesame kind of reason to reject the hypothesis that things are gunky aswe have to reject needlessly complex hypotheses about how thingsbehave (e.g., the Ptolemaic theory of planetary motion with itsepicycle upon epicycle): other things being equal, we should prefersimpler theories and more economical ontologies over complex theoriesand more expansive ontologies.

To illustrate, Nolan considers the famous example of someonesuggesting that the Earth is held up by resting on the back of a giantturtle, which is in turn held up by resting on the back of anotherturtle, which is in turn … and so on, turtles all the way down.There does not appear to be an inconsistency hiding in this regress,nor does it thwart an attempt at analysis. For all we know, space isinfinite, so there’s no problem fitting all these turtles in, sowe’re not in a case where we know independently that we’redealing with a finite domain. This is not a regress that involvesontological dependence, so there are no concerns about the existenceof things going ungrounded. Of course, we have pretty good empiricalevidence that the infinite turtles hypothesis is false, since we havegone into space and can’t see any world turtles. But evenputting that aside—let’s suppose we’re consideringthe hypothesis prior to our going into space—there is somethingintuitively weird about the turtles hypothesis. Nolan suggests it isthe ontological extravagance of the view. He says (2011,534–5)

[T]he two turtle theory [the world rests on a turtle, which rests onanother turtle, which is unsupported] is stranger and more absurd thanthe one turtle theory, the three turtle theory worse than the two, atwenty-eight turtle theory worse even than the three, an exactly sevenmillion turtle theory loonier still, and so on. The infinite turtletheory, while perhaps more motivated than the finite turtle theories,seems to be in some respects the limit of an increasing sequence ofabsurdity.

Not everyone will agree that each additional turtle theory is moreobjectionable than the last, since the extra things being postulatedare of the same kind (world turtles) as the previous theory alreadycountenanced. David Lewis (1973, 87), e.g., held that while we shouldprefer theories that are morequalitativelyparsimonious—they postulate fewerkinds of thing thantheir rivals—there is no reason at all to prefer theories thatare merely morequantitatively parsimonious—theypostulate fewer thingsof the same kind as their rivals. Thisis controversial, however, and Nolan (1997) argues that quantitativeparsimony is a genuine reason to prefer a theory. If Nolan is correct,the four turtle theory is indeed worse than the three turtle theory,the ten turtle theory worse still, and the infinite turtle theoryworse (other things being equal) than any finite turtle theory.

Cameron applies considerations of theoretical parsimony to the case ofinfinite chains of ontological dependence. While allowing that thereis no impossibility in an infinite regress of things, eachontologically dependent on the next, Cameron argues that we can stillhave reason to reject such a theory on parsimony considerations.Cameron (2008, 12) says that what needs to be explained is theexistence of each dependent entity; and while he allows that in aninfinitely descending chain of ontologically dependent entities, thereis an explanation for why each dependent entity exists, there is nosingle explanation for whyall the dependent entities exist.Whereas if there is a collection of fundamental entities on which allthe dependent entities ultimately depend, these fundamental entitiesprovide a single unified explanation for why every dependent entityexists. Either way, everything that needs to be explained getsexplained, but Cameron says we have reason to prefer the unifiedexplanation over the infinitely many disparate explanations, since itis in general a theoretical virtue to provide a unified explanation.For example, a physical theory that postulates one unified force toexplain all phenomena would, other things being equal, be preferableto one that postulates four fundamental forces—gravity,electromagnetism, strong nuclear, and weak nuclear—even if thetwo theories explain exactly the same phenomena. (Orilia (2009)argues, contra Cameron, that there is no unified explanation providedby Metaphysical Foundationalism over theories with infiniteontological descent.)

If Nolan and Cameron are right it at most gives us aprotanto reason to reject a theory that leads to an ontologicalinfinite regress. Just as we can justifiably accept the more complexhypothesis over the simpler one because the more complex hypothesis ismore powerful (e.g.), so we might justifiably accept an ontologicalinfinite regress because there is some virtue afforded by the theorythat makes the cost worthwhile. Relatedly, Cameron (2008, 13–14)argues that this would not give us any reason to think thatontological infinite regresses are metaphysically impossible, at mostit gives us a reason to think they are not actual, which limits thepositions that such regress arguments can be used to argue for.

5. Transmissive and Non-Transmissive Explanations

In section 3 we considered the suggestion that if the explanation ofthe \(F\)-ness of each \(X\) appeals to another \(X\) that is \(F\),and so onad infinitum, then while the \(F\)-ness of eachindividual \(X\) can be accounted for, something is left unexplained:why there are things that are \(F\) at all. But arguably, not everyinfinite regress leaves even this global fact unexplained.

To say merely that the \(F\)-ness of each \(X\) is explained by appealto another \(X\) that is \(F\) leaves open a crucial question: doesthe \(F\)-ness of the new \(X\) play a role in the explanation of the\(F\)-ness of the initial \(X\)? Following Bob Hale (2002),let’s distinguish betweentransmissive explanations of\(F\)-ness, in which the \(F\)-ness of \(X_2\) plays a crucial role inexplaining the fact that \(X_1\) is \(F\), the \(F\)-ness of \(X_3\)plays a crucial role in explaining the fact that \(X_2\) is \(F\), andso on, fromnon-transmissive explanations of \(F\)-ness, inwhich the \(F\)-ness of \(X_1\) is explained by facts concerning\(X_2\), which is in fact \(F\), but where the \(F\)-ness of \(X_2\)is not crucial to explaining the \(F\)-ness of \(X_1\), and so on.Arguably an infinite regress oftransmissive explanations of\(F\)-ness leaves unexplained why anything is \(F\) in the firstplace, but an infinite regress of non-transmissive explanations neednot.

Simon Blackburn (1986) argued that any realist attempt to explain whythere is necessity in the world would fail because it faced a dilemma.Suppose we say that \(A\) is necessary because \(B\). \(B\) at leasthas to be true, but is \(B\) itself necessary? Blackburn thought wecould not answer no, because to explain the necessity of a necessarytruth \(A\) by appeal to a contingent truth would undermine thenecessity of \(A\). But to answer yes is to invite regress, for now weneed to explain the necessity of \(B\) by appeal to a necessary truth\(C\), and so onad infinitum. Now, whether the contingencyhorn is indeed vicious is debatable (see Hale 2002 and Cameron 2010for discussion), but focus on the regress involved in the necessityhorn. Hale (2002) argues that the realist about necessity can resistgetting into a vicious regress by distinguishing between transmissiveand non-transmissive explanations of the necessity of any givenproposition. Grant that the necessity of \(A\) can only be explainedby a proposition, \(B\), that is itself necessary: that explanationwill betransmissive if the necessity of \(B\) plays a rolein explaining the necessity of \(A\), otherwise it will benon-transmissive. Hale (ibid., 308–309) offers as anexample of a transmissive explanation of the necessity of \(A\) aproof of \(A\) with the necessary truth \(B\) as its sole premise. If\(A\) follows logically from a necessary truth, then \(A\) itself mustbe necessary. And in this case, the necessity of \(B\) plays a crucialrole in the explanation, for if I do not know whether \(B\) isnecessary, knowing that it entails \(A\) does not tell me whether\(A\) is necessary, for a contingent proposition can follow fromanother contingent proposition. In that case, \(A\)’s necessityseems to be hostage to \(B\)’s necessity, and so the ultimateexplanation of \(A\)’s necessity will seemingly involve whateverexplains \(B\)’s necessity, which is where Blackburn sensesregress. But Hale thinks there can also be non-transmissiveexplanations of necessity. He suggests: ‘Necessarily theconjunction of two propositions \(A\) and \(B\) is true only if \(A\)is true and \(B\) is true’ because ‘conjunction just isthat binary function of propositions which is true iff both itsarguments are true’ (ibid., 312). While he grants that theexplanans in this caseis necessary, Hale thinks that itsnecessity is no part of the explanation. All that is needed to explainwhy it is necessary that \(A \amp B\) is true only if \(A\) and \(B\)are both true is thetruth of the fact that conjunction justis that function which is true iff both its arguments are true. It maybe necessary the conjunction is that function, but that is not part ofthe explanation of the original necessity, and thus the necessity ofthe explanans is not appealed to in a way that makes the success ofthe explanation dependent now on explaining this further necessity.Perhaps there is a new question to be asked concerning why thisfurther claim is necessary, but our original explanation standsindependently of whether we can successfully answer this newquestion.

Another example. Epistemic Infinitists embrace the infinite regress ofreasons and argue that it is not vicious (see, e.g., Aikin 2005, 2011,Klein 1998, 2003, Peijnenburg 2007 and Atkinson & Peijnenburg2017). One response for the Infinitist to make to the regress argumentis the response from section 3: to hold that each belief is justifiedin virtue of the next one being justified, but to claim that this isnot a problem: that while the regress means we do not have anexplanation for why anything is justified in the first place, this isnot what the sequence wasmeant to explain—all that weneed is an explanation for each belief concerning whyit isjustified, and this we have. (See e.g., Aikin 2005, 197 and Klein2003, 727–729.)

However, the Infinitist may also simply deny that anything remainsunexplained in such a regress. Peter Klein (1998, 2003) holds not onlythat therecan be an infinite regress of justifications, butindeed that it isnecessary for \(S\) to be justified inbelieving that \(p\) that \(S\) have available to them an infinitenumber of propositions such that the first of these, \(r_1\), is areason for \(p\), the second, \(r_2\), is a reason for \(r_1\), thethird, \(r_3\), is a reason for \(r_2\), and so onadinfinitum. The regress objection seems to presuppose that \(r_1\)is a reason for \(p\)in virtue of (at least in part) \(r_2\)being a reason for \(r_1\), etc. (See e.g., Gillett 2003, 713.) Inwhich case, so the objection goes, the justificatory chain could notget off the ground, and nothing would be justified. But Klein (2003,720–723) denies that \(r_1\) is a reason for \(p\)in virtueof \(r_2\) being a reason for \(r_1\). There mustbe areason, \(r_2\), for \(r_1\), and there mustbe a distinctreason for \(r_2\), \(r_3\), and so on. Each of these infinitely manypropositions are justified, but each one’s being justified, saysKlein, does not hold in virtue of anyother being justified.The Infinitist demands that there is an infinite justificatorysequence, but that in itself is silent as to what justificationconsists in. The Infinitist can simply hold that there issome feature \(F\) such that for \(x\) to be a reason for \(y\),\(\langle x,y \rangle\) has to have \(F\), and that \(\langle r_1,p\rangle\) has \(F\), and \(\langle r_2,r_1 \rangle\) has \(F\), and soon. That feature could be the first element increasing the objectiveprobability of the second, or something else entirely (ibid., 722).The point is, it needn’t involve the second element itself beingjustified, and thus the Infinitist need not accept that thejustification of \(p\) from \(r_1\) isinherited from thejustification of \(r_1\) from \(r_2\), and so on. In Hale’sterminology, the explanation of the justification of a proposition byappeal to another justified proposition is a non-transmissive one:while one must appeal in the explanation to a proposition that is infact justified, the fact that it is justified plays no role in thatexplanation. Thus there is, arguably, no reason to think that theregress of epistemic justification is viciouseven if youdemand an explanation for why any of our beliefs are justified in thefirst place.

It’s worth reflecting on the difference between the epistemicregress and the ontological regress of dependent entities that makesKlein’s response here possible. Suppose that \(A\) isontologically dependent on \(B\) and \(B\) ontologically dependent on\(C\). It is very plausible that in this case, \(C\)’s existenceand/or nature is part of the explanation of \(A\)’s existenceand/or nature. In saying that \(A\) is ontologically dependent on\(B\) we are saying that \(A\) exists, or is the way it is, at leastpartly in virtue of \(B\)’s existence and/or nature. So \(B\)has to exist, or be the way it is, in order for \(A\) to exist, or bethe way it is. In explaining \(A\)’s existence/nature, we areappealing to \(B\)’s existence/nature, in which case it seemsthat anything that we need to explain \(B\)’s existence and/ornature—in this case \(C\)’s existence/nature—mustultimately be part of the explanation of \(A\)’s existenceand/or nature. That’s why when we have a chain of ontologicaldependence, the existence and/or nature of the first entity seems tobe ultimately dependent on not just the existence/nature of the secondentity in the chain, but on that of every subsequent entity in thechain: explanations of being appear to be transmissive. Which is why,if the chain is endless, we seem to lack an explanation as to whyanything exists at all.

But while it is overwhelmingly plausible that \(B\) can only serve asthe ontological ground of \(A\) because \(B\) itselfexists,or is the way it is, it isnot forced on us to hold that\(r_2\) can only be a reason for \(r_1\) because \(r_2\) is itselfjustified, and this is why Klein’s response to the epistemicregress is available. It need be no part of the explanation for why\(r_2\) is a reason for \(r_1\) that \(r_2\) itself be justified. AsKlein says, the entirety of the explanation for why \(r_2\) is areason for \(r_1\) might be simply that the objective probability of\(r_1\) given \(r_2\) is sufficiently high. That fact does not involve\(r_2\) being justified. Klein thinks that \(r_2\) mustbejustified if \(r_1\) is, but that need not be any part ofwhy\(r_2\) is a reason for \(r_1\), and thus there is no pressure to holdthat the justification of \(r_1\) by \(r_2\) is dependent on, orinherited by, the justification of \(r_2\) by \(r_3\), and so on. Sowhile there is indeed an infinite sequence of propositions, each ofwhich is a reason for the previous one on the list, at no stage is thefact that one proposition is a reason for another hostage to the factthat any other proposition is a reason for another. And so arguably,nothing remains unexplained: there can be a good explanation not onlyfor why each particular proposition is justified (\(r_5\) is justifiedby \(r_6\), etc.), but also for why there are any justifiedpropositions in the first place (there are propositions that raise theobjective probability of others, e.g.).^[13]

6. Coherence, Circularity, and Holism

The Coherentist resists regress by allowing a circular or holisticexplanation of the \(F\)-ness of at least some \(X\)s. This could beto simply allow straightforwardly circular explanations, such as that\(X_1\) is \(F\) in virtue of \(X_2\) being \(F\) and \(X_2\) is \(F\)in virtue of \(X_1\) being \(F\). But that is not the only option forthe Coherentist. Consider again the regress argument concerningjustification of belief: our belief \(p_1\) is justified by appeal to\(p_2\), which is in turn justified by appeal to \(p_3\), and so on.There is an assumption behind this regress: that what is to beexplained are the facts concerning the individual beliefs—why isthis one justified, then why isthat one justified,etc. An epistemic Coherentist such as Bonjour (1985) rejects thisassumption. It is not, primarily, individual beliefs that arejustified, it issystems of belief. A particular belief isjustified only in a derivative sense, by belonging to a justifiedsystem. A system of belief is justified because of the properties ofthe system as a whole, namely that the beliefs in it form a coherentsystem. Thus, justification is a holistic phenomenon: a collection ofbeliefs is justified because of what they, collectively, are like, notbecause of what each individual member of the system is like. Thisholistic explanation of where justification comes from is verydifferent from a circular explanation: a circular explanation tells usthat one individual’s being \(F\) explains another’s being\(F\) and also vice versa, but a holistic explanation tells us toabandon the idea of explaining an individual’s being \(F\) byappeal to another individual being \(F\), and instead hold that theexplanation forsome things being \(F\) can be the factsconcerning what that collection of things as a whole is like. (Ananalogous view concerning Metaphysical Coherentism and holisticexplanation is defended by Thompson (2018), and Cameron (2022, Ch.4).Cf. Barnes (2018).)

Sometimes a circular explanation might be warranted because we are nottrying to explain in virtue of what the \(X\)s are \(F\), but ratherwe are merely attempting to illuminate the \(X\)s being \(F\) byshowing how the \(X\)s relate. Consider the regress argument againstthe thesis that time passes given by J.J.C. Smart (1949, 484):

If time is a flowing river we must think of events taking time tofloat down this stream, and if we say ‘time has flowed fastertoday than yesterday’ we are saying that the stream flowed agreater distance today than it did in the same time yesterday. Thatis, we are postulating a second timescale with respect to which theflow of events along the first time dimension is measured …Furthermore, just as we thought of the first time dimension as astream, so will we want to think of the second time dimension as astream also; now the speed of flow of the second stream is a rate ofchange with respect to a third time dimension, and so we can go onindefinitely postulating fresh streams without being any bettersatisfied.

Here we start with our ordinary temporal dimension—what we mayhave supposed to be theonly temporal dimension—the onephilosophers have in mind when they say that time passes. Smartsupposes that if this dimension of time indeed passes then there mustbe a rate at which it passes.^[14] Whether that rate can slow down or speed up or if time always flowsat the same rate is not important, but there must be some rate atwhich it passes, thinks Smart. Now, just as we would measure the speedof a car, say, by measuring how much distance it covers in a givenamount of time, so, thinks Smart, we would have to measure the speedat which time itself passes by measuring how much time passes in agiven amount of time of some second temporal dimension. While the carcovers forty miles of road in the space of an hour, an hour of timepasses in the space of, say, two hours of this second temporaldimension. But then, how fast does this second dimension of time pass?We need a third temporal dimension to measure how long it takes for anhour of the second temporal dimension to pass. But how fast does thethird temporal dimension pass? And so on,ad infinitum. Smartconcludes that time does not pass.

A defender of the view that time passes could attempt to resistSmart’s regress by cutting off the regress at the second stageby claiming that there is a principled difference between the firsttemporal dimension and the second that results in the first temporaldimension passing at some rate, but not the second. Smart himselfconcludes that time does not pass, so can hardly object to thepostulation of temporal dimensions that do not pass. However,Smart’s regress can be resisted without abandoning the principlethat all temporal dimensions pass at some rate.

Ned Markosian (1993) points out that to give a rate is to compare twodifferent types of change. When we say that the car travels at fortymph, we are comparing one type of change—the car started off inone place and ended up forty miles distant—with another—itwas one time and it is now an hour later. In that case, we can alwaysget the rate of the second type of change by comparing back to thefirst. As Markosian says (ibid., 842): “If … I tell youthat Montana’s passing totals increased at the rate of 21 passesper game, then I have also told you that the games progressed at therate of one game per 21 completions by Montana.” So suppose wesay that the first temporal dimension passes at a rate of one hour forevery two hours of the second temporal dimension; there is no need toinvoke a third temporal dimension in order to state the rate at whichthe second passes, for we have already given that rate: an hour of thesecond temporal dimension passes for every half hour of the first.Indeed, as Markosian points out, we need not even invoke a secondtemporal dimension, for any time we give a rate of any ordinaryprocess with respect to time—such as that the Earth goes aroundthe sun once every year—we have thereby stated the rate at whichtime passes with respect to those ordinary processes: time passes atthe rate of one year for every orbit of the Earth around the sun.

Markosian’s maneuver is possible because in giving the rate ofone process of change by appeal to a second process of change we arenot saying whatmakes it the case that the first changeoccurs at the rate it does. What makes it the case—what are theontological grounds of the fact—that the car travels at 40 mph?Not that an hour of time passes while the car moves a distance of 40miles, for that is merely a re-description of the fact in question: away ofdescribing the rate. In the case of time itself,defenders of the view that time passes may plausibly claim that whatmakes it the case that time passes is simply the nature oftime: that it is in its nature to pass at the rate it does. Itdoesn’t pass at the rate it doesbecause of somerelation it stands in, either to ordinary processes of change or to asecond temporal dimension. Such relations allow us to informativelystate the rate of change, they do not provide thegrounds for it.

If we were providing the metaphysical grounds of rates of change,Smart might be right that this would lead to a vicious regress, sincearguably grounding is asymmetric (see e.g., Rosen 2010, 115).^[15] If the car travels at the speed it doesin virtue ofsomething to do with the passage of time then, arguably, time cannotpass at the rate it doesin virtue of anything to do with thespeed of the car, and so we need to appeal to the passage of a secondtemporal dimension to provide the ontological grounds of the rate ofpassage of the first. If the passage of this second temporal dimensiongrounds facts about the passage of the first temporal dimension, ititself cannot passin virtue of facts concerning the passageof the first, and so we need to appeal to a third temporal dimension,and so on. And even if we hold that it’spossible forthere to be infinitely descending chains of grounds, it seems absurdin this case to suppose that there are in fact infinitely manytemporal dimensions. But that is not what is going on. When we explainthe speed of the car by appeal to the passage of time, we’re notproviding the ontological grounds of its speed, we’re simplyshowing a connection between two things: the movement of the car andthe passage of time. Likewise for the rate of time’s passageitself: we are not seeking to provide the ontological grounds oftime’s passage in stating its rate, for the ontological groundsare plausibly just the nature of time itself. Rather, when we comparethe two changes, we are simply trying to illuminate one or both ofthose changes by pointing to the way they relate. That is the onlysense in which oneexplains the rate of one change whencomparing it to another kind of change: the mutual connection tells ussomething enlightening about each. It is not to give a metaphysicalexplanation in the sense of providing the metaphysicalgrounds of either rate of change. Compare: if I tell you thatthe value of a US dollar is 0.7 British pounds (andthereforethat the value of a British pound is 1.43 US dollars), this is not tosay that the US dollar has the worth that it hasin virtue ofstanding in this relation to the British pound. The US dollar relatesthus to the British poundbecause of what they are eachworth; they are not worth what they are in virtue of standing in thatrelationship. What makes it the case that the US dollar is worth whatit is is some incredibly complex set of facts concerning economics,monetary policy, etc. To give an exchange rate is not to give thegrounds of the value of the currencies, but merely to say somethingsubstantive about each value by stating their connections. Likewisewith rates of change, which is why Markosian is able to resistSmart’s regress in this manner. Coherentist explanations mightbe controversial when it comes to providing ontological grounds, butthey are less so when it comes to simply casting light on the natureof some phenomena by showing how they connect. So whether a regressargument even gets going will depend on the explanatory ambitions ofthe view being targeted.

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• Klein, Peter D., and Turri, John, “Infinitism in Epistemology”, inInternet Encyclopedia of Philosophy, James Fieser andBradley Dowden (editors).
• Dowden, Bradley, “The Infinite”, inInternet Encyclopedia of Philosophy, James Fieser andBradley Dowden (editors).

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Aristotle, Special Topics: mathematics |Bradley, Francis Herbert: Regress |cosmological argument |dependence, ontological |dialetheism |grounding, metaphysical |infinity |justification, epistemic: coherentist theories of |justification, epistemic: foundationalist theories of |McTaggart, John M. E. |space and time: supertasks |Zeno of Elea: Zeno’s paradoxes

Acknowledgments

Thanks to Elizabeth Barnes, Trenton Merricks, Daniel Nolan, JonathanSchaffer, Jason Turner, and Robbie Williams. Thanks to Aaron Cotnoirfor valuable comments on the material in the supplement.