## 1. Regress and Theoretical Vices



Sometimes it is uncontroversial that a theory that generates an
infinite regress is objectionable, because the regress reveals that
the theory suffers from some kind of theoretical vice that is a reason
to reject the theory *independently* of it yielding an infinite
regress. In these cases, an infinite regress argument can show us that
we have reason to reject a theory, but it is not because the theory
yields a regress *per se*, but rather because it has this other
bad 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 theory
that generate the regress also lead to a contradiction. If this is so
then it does not matter what we think about infinite regress in
general, we will of course have reason to reject the theory, because
it 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 the
regress objection is Parmenides's Third Man objection, as
reconstructed by Vlastos (1954).



Suppose some things, the \(X\)s, are alike in a certain way: they
share some feature, \(F\). The theory of Forms says that if this is so
then there is some form, \(F\)-ness, in which the \(X\)s each
participate and in virtue of which they have that shared feature. The
theory of Forms also says that Forms are self-predicated: the Form of
the 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 there
must be some Form in which each of the \(X\)s and \(F\)-ness
participate, in virtue of which they have this shared feature.
However, this Form cannot be \(F\)-ness, because the Form must be
distinct from the things that have that character that participate in
it, 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 that
generate this regress must be denied, for they lead to contradiction.
The theory of Forms, as presented here at least, tells us (i) that
when some things are a certain way, they participate in a single Form,
in virtue of which they are that way; (ii) that Forms
self-predicate--Forms themselves are the way that things that
participate in them are; and (iii) that the Form is distinct from the
things that participate in it. Each of these three claims is essential
to generating the regress: without (i) we don't get the
existence of a Form in the first place, without (ii) we don't
get the second list of things with a shared feature, so would stop
simply 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 need
for a new Form. So (i)-(iii) generate a regress, and they are
each needed to do so. But (i)-(iii) are inconsistent, and no
regress argument is needed to show that. (i) and (ii) together entail
that 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 how *in
general* things get to be \(F\). So \(F\)-ness participates in
itself, and so by (iii) \(F\)-ness must be distinct from itself, since
Forms are distinct from that which participates in them. But nothing
is distinct from itself: contradiction.



We don't need an argument against infinite regresses to show
that this version of the theory of Forms is no good: that it is
contradictory is the best reason we could have to reject it. Now, that
the theory is contradictory and that it leads to an intuitively
worrying infinite regress are not unrelated. Beyond the mere
ontological profligacy involved in being committed to infinitely many
Forms any time we notice that some things are some way (we will come
back to ontological profligacy and regress in section 4), what seems
intuitively problematic about the regress of Forms is that we
*shouldn**'t* get a new Form each time. The \(X\)s
are \(F\), so we have the form of \(F\)-ness in which they
participate. We then get a *new* Form in which the \(X\)s and
\(F\)-ness all participate. But why do we have a new Form? We
don't have a new shared feature, we have the very same shared
feature we started with: just as the \(X\)s are all \(F\), so is
\(F\)-ness itself. We already *have* the Form that makes
something \(F\): \(F\)-ness. The regress is troubling because we
shouldn't be invoking a new Form, but we have to because of the
ban on Forms participating in themselves. But this diagnosis of why
the regress is troubling is really just another way of stating the
contradiction at the heart of the theory: the Form \(F\)-ness is
supposed to be *in general* the thing in virtue of which things
get to be \(F\), but it also cannot be because it itself must be \(F\)
and it cannot participate in itself. So the regress and the
contradiction are intimately related. As Nolan (2001, 528) puts it:
"infinite regresses of this sort and the statement of formal
contradiction are different ways of bringing out [the same]
unacceptable feature."



This is an easy case, because we don't have to adjudicate on
whether the fact that the theory leads to an infinite regress is
itself objectionable. The principles that lead to regress also lead to
contradiction, and we know that a theory's being contradictory
is 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 infinite
regress also result in some theoretical vice that we know to be
objectionable *independently* of whether or not there is a
regress, then we have a reason to reject that theory that
doesn't depend on the theory leading to regress. Sometimes, the
theoretical vice in question will be a global one: a feature that is a
reason 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 local
vice: a feature that might be an unobjectionable feature of certain
theories, but a reason to reject a particular theory \(T\) because of
the particular theoretical ambitions of \(T\) or as a result of other
things we know about \(T\)'s subject matter.



For example, a theory might result in an infinite regress of entities
and so entail that there are infinitely many things. This in itself
is, arguably, not objectionable. But it might be a local vice to a
theory if we have independent reason to think that we are dealing with
a finite domain. (See Nolan 2001, 531-532, and Nolan 2019.)
(Some philosophers object to the very idea of reality containing
infinities. Aristotle, e.g., famously allowed that there could be
*potential* infinite series, but not *completed*
infinite series. (See Mendel 2017.)) But we will ignore such general
anti-infinitism in this entry, for it is infinity itself that such
theorists take to be objectionable, not infinite *regresses per
se.*)



Peano's axioms for arithmetic, e.g., yield an infinite regress.
We are told that zero is a natural number, that every natural number
has a natural number as a successor, that zero is not the successor of
any natural number, and that if \(x\) and \(y\) are natural numbers
with the same successor, then \(x = y\). This yields an infinite
regress. Zero has a successor. It cannot be zero, since zero is not
any natural number's successor, so it must be a new natural
number: 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 same
successor and hence be identical, and we have already said they must
be distinct. So there must be a new natural number that is the
successor of one: two. Two must have a successor: three. And so on
... And this infinite regress entails that there are infinitely
many things of a certain kind: natural numbers. But few have found
this worrying. After all, there is no independent reason to think that
the domain of natural numbers is finite--quite the opposite.



By contrast, consider the following two principles: (i) Every event is
preceded 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 that
there is at least one event. If there is an event, \(E\_1\), then it is
preceded by its cause. That cause cannot be \(E\_1\), as nothing
precedes 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 its
cause. This cannot be \(E\_2\) for the same reasons as before, and it
cannot be \(E\_1\) because then each of \(E\_1\) and \(E\_2\) would
precede 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. It
cannot be \(E\_3\) or \(E\_2\) for reasons similar to before. And it
cannot be \(E\_1\), for then \(E\_1\) would precede \(E\_3\), but since
\(E\_3\) precedes \(E\_2\) which precedes \(E\_1\), transitivity entails
that \(E\_3\) 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 natural
numbers. In each case we start from the claim that there is a thing of
a certain kind (a number or an event), and we have a principle that
tells us that for each thing of that kind, there is another thing of
that very kind that bears a certain relation to the previous one (it
is its successor, or it is its preceding cause). We then have
supplementary principles that rule out the other thing of that kind
being any of the things on our list so far, thus forcing us to
introduce a new thing of that kind, thus inviting the application of
the principles to this new thing, and so on *ad infinitum*. But
while the regress and resulting infinity of natural numbers is
arguably unobjectionable, the regress of events seems problematic,
because we have good empirical reasons to deny that there are
infinitely many events, each preceded by another. For either that
infinite sequence of events takes place in a finite amount of time or
an infinite one. We have good empirical reason to rule out the latter
option, since we have good empirical reason to think that there has
only been a finite amount of past time: that time started a finite
time ago with the Big Bang. And we have good empirical reason to rule
out the former option, since the only way of fitting an infinite
series of events, each preceded by another, into a finite stretch of
time is by having the time between them become arbitrarily small. So
for 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 \(A\_{n + 1}\) is
always half the time between events \(A\_{n - 1}\) and \(A\_n\), we can
fit infinitely many events into a two minute time-period. But this
requires the time between events to become arbitrarily small, and
there is some reason to think that time is quantized, such that there
is a minimum length of time during which a change can occur, thus
providing a lower limit on the amount of time that can separate two
temporally distant events. (See Nolan 2008b for relevant discussion
and further references.)



So while the numbers regress and the events regress are structurally
analogous, we might find the principles that yield the regress
objectionable in one case but not the other, because while each
regress entails that there are infinitely many things of kind \(K\),
whether that is a vice may depend on what kind \(K\) is, and whether
we have independent reason to think that the domain of \(K\)s is a
finite one. Yielding infinitely many things of kind \(K\) might be a
local vice when the kind in question is events separated in time, but
not when the kind in question is natural numbers structured by the
successor relation.


### 1.3 Regress and Failure of Analysis



In the previous section we saw two theories generating similar
regresses, but where one is found unobjectionable whereas the other is
found objectionable due to the different things we think we know,
independently of encountering these regress arguments, about the
subject matter of the theories. We could also have cases where a
single theory yields a regress that is objectionable by the lights of
one theorist and not another, as a result of their differing
theoretical commitments leading one but not the other to think that a
feature revealed by the regress is a vice. (See Cameron 2022, section
1.3.)



Consider Bradley's regress. (Bradley 1893 [1968], (21-29).
Note, however, that Bradley is very hard to interpret, and there is
much debate concerning how to reconstruct his argument. See the entry
on
 Bradley's regress
 for discussion.) We start with the demand to give an account of
predication: what is it for \(A\) to be \(F\)? One answer is that it
is 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 an
account, this relational predication of \(A\) and \(F\)-ness also
demands an account. Since before we posited a property corresponding
to the monadic predicate and said that the property was bound to the
object that was the subject of predication, we should follow suit here
and posit a relation corresponding to the dyadic predicate and say
that it is bound to the two things that are the subjects of that
predication. So we should posit a relation--let's call it
*instantiation*--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 triadic
predicate--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 a
tetradic one: Instantiation\(\_2\) binds Instantiation to \(A\) and
\(F\)-ness... And so on. Each predicational fact tells us that
some particulars, properties, and relations are bound together, which
forces us to posit a relation corresponding to that binding, which
generates the next predicational fact, and so on *ad
infinitum*.



Is this regress objectionable? Arguably it depends on what we want
from an account of predication. If what you want is an
*analysis* of predication, then arguably this regress
*is* objectionable. If you start off not understanding
predication--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 that
this account is not going to help you, for each answer simply invokes
another 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 claims
is also not the 'is' of identity but the 'is'
of predication, which you find a mystery. However, if you
*understand* predication perfectly fine and want simply an
*ontological account* of it--an account of what in the
world *makes true* the true predications--then arguably
the regress is *not* objectionable, because while there are
infinitely many true predications, these can all be made true by the
one underlying state of the world: the state of affairs of the
particular \(A\) being bound to the property \(F\)-ness. Just as this
state of affairs makes it true that \(A\) is \(F\), so does it make it
true that \(A\) is bound to \(F\)-ness, and that this binding holds
between \(A\) and \(F\)-ness, and etc. We would have one ontological
underpinning for the infinitely many true predications. As Nolan
(2008a, 182) puts it, we have "Ontology for each truth, and no
infinite regress of [states of affairs], but only of descriptions of
[states of affairs]: and it should never be thought that an infinite
regress of descriptions is something to worry about per se".
(Cf. Armstrong 1974 and 1997 (157-8).) So whether or not we will
find this regress objectionable depends on what we demand of an
account of predication. We can all agree that the regress shows that
the account does not yield an analysis of predication. Is this a
theoretical vice? It depends on whether or not predication requires an
analysis, 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, which
merely 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 present *was* future and *will be*
past; the times that are future *will be* present and *will
be* past. McTaggart concludes that we end up attributing each
A-property (*is present, is past,* and *is future*) to
every time (and therefore to every event in time). But this is absurd,
because the A-properties are incompatible: to have one is to have
neither of the others. So we end up in contradiction: each time both
has only one such property, and all such properties. McTaggart
concludes that the A-series cannot be real.



An obvious response to McTaggart's argument is this: it's
not that some future time *is* present and past as well,
it's merely that it *will be* present and *will
be* (later still) past. That is: no time and event *has*
more than one of the A-properties, it is merely the case that they
have one A-property and *did* and *will* have another.
But McTaggart thinks this response does not solve the problem, because
it leads to regress. He says (ibid., 469):



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



McTaggart's argument is difficult and philosophers disagree on
how to interpret it, but here is one interpretation. (See Dummett 1960
and Mellor 1998 (72-74) for two among many presentations of the
argument 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 of
reality in a way that, for example, Bilbo's finding the One Ring
is not, it's merely part of a fiction. History is *not* a
fiction, it's part of our world, so historical truths and future
truths should, seemingly, be part of the overall account of how the
world is. But that is puzzling, given that things *change* and,
hence, the way things were is incompatible with the way things are
now. How can they both contribute to the way reality *is* if
they are incompatible? McTaggart's argument focuses on a
particular instance of this concerning the A-properties.
Caesar's crossing the Rubicon is past. But it *was*
present, and so its presentness is a feature of our world's
history. But our world's history, as we just said, is part of
the complete account of how our world *is*, and so the complete
account of how our world is includes both Caesar's crossing the
Rubicon being past and it being present, and yet those are
incompatible features. The defender of the A-series replies by
insisting that in giving the complete account of how reality is we
have to take seriously the fact that reality *changes* and that
it is, therefore, different ways *successively*, and there is
no inconsistency in things being one way and *then* another,
incompatible, way. So it's not that reality is such that
Caesar's crossing the Rubicon is both past and present,
it's that reality is such that Caesar's crossing the
Rubicon *was* future, and *was* present, and is
*now* past.



McTaggart responds by restating this response in terms of second-order
A-properties. To say that Caesar's crossing the Rubicon
*was* future, and *was* present, and is *now*
past is to say that it has the properties of being *past
future* (i.e. having been future), *past present* (i.e.
having been present) and *present past* (i.e. now being past).
And similar reasoning to the above suggests that every time and event
has each of the nine possible second-order A-properties; and while
it's not the case that any two of them are incompatible, certain
pairs of them are. Nothing can be both *past past* and
*future future*, for example: if \(E\) has been a mere past
event, it can't be true that \(E\) will be something that is yet
to happen. And yet the complete account of reality seems to include
both Caesar's crossing the Rubicon being past past and its being
future future. After all, in 2000 BCE Caesar's crossing the
Rubicon was future future, since 1000 BCE was the future and
Caesar's crossing the Rubicon would still be the future then.
And Caesar's crossing the Rubicon is past past just now, since
1000 CE is past, and Caesar's crossing the Rubicon was past
then. But both 2000 BCE and the present are part of the overall
history of the world, so the goings on at each time are part of what
the world as a whole is like, and so Caesar's crossing the
Rubicon is both past past and future future, and yet those are
incompatible. And again, the defender of the A-series will respond
that it's not that Caesar's crossing the Rubicon
*is* both past past and future future, it's that it is
*now* past past and *was* future future. And McTaggart
will respond that this is to invoke *third-order*
A-properties--being *present past past*, being *past
future future*, etc. And the same problem will arise, and invite
the same response, which will lead to the same problem concerning
fourth-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 of
much debate. Some philosophers see the regress as demonstrating that
any attempt to describe the world in A-theoretic terms is ultimately
inconsistent, and see the A-theorist as merely invoking another
inconsistent account of reality every time they attempt to explain
away this inconsistency. Mellor, for example, says (1998, 75)
"[McTaggart's] critics react by denying the viciousness of
the regress. At every stage, they say, we can remove the apparent
contradiction by distinguishing the times at which the events have
incompatible [A-properties]. They ignore the fact that the way they
distinguish these times ... only generates more
contradictions." Others see McTaggart as simply making a
confused challenge at each stage--as mistakenly concluding from
the fact that things *were* one way and are *now*
another incompatible way that they *are* both ways at
once--and continuing to make the same mistake in response to each
of the A-theorist's correct explanations that in fact the
incompatible properties are only ever had one after another, never at
the same time. Skow (2015, 87), e.g., says "At each stage
[McTaggart] accuses objective becoming [i.e. things changing their
A-properties] of being inconsistent, and [the A-theorist] shows that
allegation to be false. And even an infinite sequence of false
allegations does not add up to a good argument."



This is arguably another case where whether or not the regress is
vicious by a philosopher's lights will depend on their
background theoretical commitments. If one starts out happy with the
notion of *succession*--i.e. of one thing being the case
and *then* ceasing to be the case as something else *comes
to be* the case--then one may be inclined to see
McTaggart's regress as entirely benign. We start out with a set
of incompatible properties that are never had by anything
simultaneously but are held by things successively; in stating that
those properties are had successively we make salient a new set of
incompatible properties, but these are also never had by anything
simultaneously, only successively; this makes salient yet another set
of incompatible properties, and so on *ad infinitum*. There is
never, at any stage, a contradiction, if the notion of succession is
indeed in good standing, for we are never forced to say that a thing
has incompatible properties, only that a thing successively has
properties that cannot be had simultaneously. If, by contrast, one is
suspicious of the very notion of *succession*--if one sees
in it simply an attempt to paper over what is ultimately a
contradiction inherent in the commitment to reality being each of two
incompatible ways, the way it supposedly is now and the way it
supposedly was or will be--then one will see in McTaggart's
regress an infinite sequence of contradictory accounts of how reality
is, with each attempt at explaining away the contradiction simply
resulting in another such contradictory account. The regress, then,
looks vicious or benign depending on whether one is content to grant
the 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 to
reveal some feature that might, possibly depending on your other
theoretical commitments, be taken to reveal a feature of a theory
*independent* of its leading to regress that is a reason to
reject it. But sometimes the regress itself is taken to be an
objectionable feature of the theory that yields it.



Suppose that there is an \(X\) that is \(F\), and that to account for
*why* \(X\) is \(F\) we need to appeal to another \(X\) that is
also \(F\). Now there is the question as to why *this* \(X\) is
\(F\), and so we need to appeal to *another* \(X\) which is
*also* \(F\) .... If this proceeds *ad infinitum*,
with a new \(X\) invoked at each new stage in the process, there is a
concern that we end up without having accounted for the \(F\)-ness of
any of the \(X\)s. If this infinite regress argument is successful
then our choices are either:



Foundationalism: To halt the regress by taking there to be a
*foundation*--a set of \(X\)s whose \(F\)-ness is taken to
be 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 or
holistic explanation of the \(F\)-ness of at least some \(X\)s.



Infinite regress arguments used to motivate Foundationalism or
Coherentism appear in many different areas of philosophy. Here are
some highlights:



*In metaphysics:*



Metaphysicians have wanted to account for the very existence, or
nature, of some things by appealing to things on which they
ontologically depend: for example, a complex object exists and is the
way it is because its parts exist and are the way they are; a set
exists because its members exist; etc. (See Fine 1995 and Koslicki
2013 for discussion.) But of course the things the dependent beings
depend on must themselves exist as well. Some have been suspicious of
the idea that this can go on *ad infinitum*, with every thing
being ontologically dependent on some new thing(s), and thus have
argued for Metaphysical Foundationalism: the view that there is a
collection of absolutely
 fundamental[4]
 entities upon which all else ultimately ontologically depends.
Aquinas, e.g., holds that events are ontologically dependent on their
causes, and that an infinite regress of causes and effects would be an
infinite series of things each of which is ontologically dependent on
the next, and this is
 impossible.[5]
 Thus he concludes that there must be a first cause of all else that
is itself uncaused--namely, God. We shall see more examples of
Metaphysical Foundationalists below. See also the supplementary
document on



Metaphysical Foundationalism and the Well-Foundedness of Ontological Dependence.



(Metaphysical Coherentism--the view that ontological dependence
could be a holistic phenomenon--has received few defenders, but
see Barnes 2018, Bliss 2014, Cameron 2022 (Chs. 4 & 5), Nolan
2018, Priest 2014 (Chs. 11 & 12), and Thompson 2016 and 2018 for
some 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 be
*justified*--that is, we want there to be a
*reason* to believe the propositions we believe. But those
reasons will be further propositions, and if our initial belief is to
be justified, so surely must the reasons for that belief be, and so we
must appeal to yet more propositions, and so on. Many--going back
to Sextus Empiricus (Outlines of Pyrrhonism PH I,
164-9)--have thought that this cannot proceed *ad
infinitum*, and that the only serious options are Epistemic
Foundationalism--the view that there is a class of propositions
whose justification does not come via some other justified
propositions, and that can provide a reason for everything else we
believe--or Epistemic Coherentism--the view that a
collection of propositions can collectively be justified in virtue of
the web of epistemic relations they stand in to one another. Thus Sosa
(1980, 3) says "epistemology must choose between the solid
security of the ancient foundationalist pyramid and the risky
adventure of the new coherentist raft." (See the entries on
 foundationalist theories of epistemic justification
 and
 coherentist 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 moral
beliefs (see e.g., Sinnott-Armstrong 1996), moral philosophers have
been concerned with distinctively moral regresses, which arise not
when we attempt to account for the justification of a moral claim, but
rather when we attempt to account for the moral status of something by
appealing to something else of the same moral status, and so on.
Aristotle (Nicomachean Ethics, 1094a) thought that some things were
good because we desire them for the sake of something else that is
good. But if *everything* that is good is good simply because
it aims at something else that is good, this would lead to a regress
and "all desire would be futile and vain". And so
Aristotle argues that there must be a Highest Good--something
that is desired for its own sake--that other things can be good
in virtue of aiming towards this highest good. Aristotle is a Moral
Foundationalist: there is something whose goodness does not get
explained by reference to anything else, by means of which the
goodness of other things is accounted for.



(Explicit statements of anything other than Foundationalism in the
moral 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 and
accept:



Infinitism: The \(F\)-ness of each \(X\) is accounted for by facts
involving a new \(X\) that is \(F\), and this proceeds *ad
infinitum*.



Whether in metaphysics, epistemology, or ethics, Foundationalism has
often been seen as the default, orthodox, view, with Coherentism being
seen as the radical alternative. Infinitism is often simply dismissed,
or not even considered as a live option. Foundationalism and
Coherentism are
 (often[7])
 motivated by the thought that if the \(F\)-ness of each \(X\) is
accounted for by appeal to a new \(X\) that is \(F\) then each of
those infinitely many explanations fails. The thought is that each
explanation of the \(F\)-ness of an \(X\) would be dependent on the
success of the next, a promissory note that is never paid if that
process does not end. Let's examine this anti-Infinitist
thought.



To focus our inquiry, consider the case of a complex object and its
proper parts. Some metaphysicians have considered the possibility that
there are *gunky* objects: objects such that every part of them
itself has proper parts. If \(A\) is gunky then it is composed of some
things, 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 that
there is more than one of the \(Y\)s, and neither \(A\) nor \(X\_1\) is
amongst the
 \(Y\)s.[8]
 Pick one of those \(Y\)s, \(Y\_1\). \(Y\_1\) is composed of some
things, 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. And
so on. This process will never end: each item in the series is a
collection of entities (a collection containing just one thing in the
first case, and more than one thing in every subsequent case), and
given the transitivity of parthood each thing in each collection will
be a part of \(A\) and hence--since \(A\) is *ex
hypothesi* gunky--will itself be composed of a collection of
proper parts, and so we can therefore pick any member from any
collection to generate the next item on the list. (Of course, a thing
need not--and if it is gunky, *will* not--have a
unique 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 that
there are *some* such collections, from which we can pick
arbitrarily to get the next collection in the sequence.) Now take the
infinite series that consists of \(A\) as its first element, \(X\_1\)
as its second element, \(Y\_1\) as its third, etc. So we form this
infinite sequence by taking one item from each collection that formed
the previous infinite sequence: namely that item which was used to
give us the collection of things that came next in the series.



Assuming that complex objects are ontologically dependent on their
proper parts, we now have an infinite regress of entities, each of
which is ontologically dependent on the next. Such an infinite regress
has been thought by some metaphysicians to be objectionable, leading
them to reject the possibility of gunk. Leibniz, for example, argues
that 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 by
aggregation, and so on *ad infinitum*.



Leibniz's idea seems to be that if each thing depends on some
other, there could not be anything at all in the first place. The
thought is that ontologically dependent entities inherit their
existence, or being, from that on which they depend; so if this chain
of dependence does not terminate, the whole process couldn't get
off the ground, and there would be nothing at all. Leibniz says
(1686-87, 85):



> 
> 
> Where there are only beings by aggregation [composite objects], there
> are no real beings. For every being by aggregation presupposes beings
> endowed with real unity [simples], because every being derives its
> reality 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 it
> is composed is itself a being by aggregation, a being for which we
> must still seek further grounds for its reality, grounds which can
> never be found in this way, if we must always continue to seek for
> them.
> 
> 
> 



By contrast, if the dependence runs in the other direction--if we
start off with a fundamental entity whose being can then ground the
being of each subsequent entity--there is arguably no problem if
that process continues *ad infinitum:* thus the infinite
sequence where you start with a thing, form its singleton set, then
form *that* thing's singleton set and so on *ad
infinitum* has not been thought to be objectionable by those who
reject gunky objects, for it is the set that is ontologically
dependent on its members, not *vice versa*. (See Fine 1994 for
discussion of the direction of ontological dependence. See Cameron
2008 and Maurin 2007 for discussion of the difference between infinite
regresses where ontological dependence runs upwards from ones where it
runs downwards. Cf. Clark 1988, and also Johansson 2009 and the
discussion in Maurin
 2013.)[9]



A contemporary sympathizer with Leibniz's thought is Jonathan
Schaffer (2010). Unlike Leibniz, Schaffer grants the possibility of
gunky objects, but thinks that this possibility is precisely a reason
to *deny* that complex objects are (always) ontologically
dependent on their
 parts.[10]
 Instead, Schaffer takes the possibility of there being no simple
things as a reason to hold that the dependence flows in the other
direction: that parts are dependent on the wholes of which they are
parts, and that every thing is thereby ontologically dependent on the
biggest thing that there is--the cosmos--that has everything
else as a proper
 part.[11]
 Since classical mereology guarantees that there is a biggest
thing--the thing that has all else as proper parts--but it
does not guarantee that there are any smallest things--things
that have no proper parts--it guarantees that if parts are
dependent on the wholes of which they are parts then there will be a
first, ontologically fundamental, element, whereas if wholes are
dependent on their parts then there is the possibility of an infinite
regress 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 be
infinitely deferred, never achieved". Leibniz and Schaffer
advocate *Metaphysical Foundationalism*: the view that there
have to be some things that are absolutely fundamental--dependent
on nothing--on which all else ultimately depends.



But is it true that if the \(F\)-ness of each \(X\) is dependent on
the next \(X\) in the sequence being \(F\), and if this goes on
without end, that we cannot explain why any \(X\) is \(F\)? Schaffer
claims that in the case of an infinite regress of ontological
dependence, with each entity depending on the next in the chain, and
no independent entities, being would be "infinitely deferred,
never achieved". Why think this? The idea seems to be that a
dependent entity only has the being it has on condition of something
else having being. If \(A\) is ontologically dependent on \(B\) then
the existence of \(A\) is a promissory note, only paid if \(B\) itself
exists. But if \(B\) is ontologically dependent on \(C\) then the
existence of \(B\) is a promissory note, only paid if \(C\) exists
... and so on, so that if this process never stops, the
promissory note is never paid, in which case, allegedly, the existence
of all these things could never get off the ground in the first
place.



An analogy may help. Suppose Anne has no sugar, and needs some. She
can 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 a
bag of sugar down. But suppose Breanna borrowed a bag of sugar from
Craig in order to then pass it on to Anne. Where did Anne's bag
of sugar come from then? Ultimately, from Craig, who ends up a bag of
sugar down. But suppose Craig borrowed a bag of sugar from Devi
... and so on, *ad infinitum*. Then where did the bag of
sugar come from? At the end of the infinite sequence, Anne is one bag
of sugar up, and nobody is a bag of sugar down, for everyone after
Anne simply borrowed a bag of sugar, and then passed it on to the next
person in the chain. There's an extra bag of sugar in the system
that seems to have appeared as if by magic. If there is a finite
sequence of borrowers, however long, then the last person in the chain
ends up a bag of sugar down, so that's where the bag of sugar
that ends up with Anne ultimately comes from. But if the chain never
ends, Anne ends up with a bag of sugar that doesn't seem to have
come from anybody, as nobody has lost any bags of sugar--they all
just borrowed it to pass it on. The infinite regress seems to create
sugar from nowhere: pleasant, perhaps, but metaphysically suspicious
all the same.



As with sugar, likewise with being--or justification, or
goodness, or whatever feature we aim to account for. If \(A\) depends
on \(B\) and \(B\) is fundamental, where did \(A\)'s being come
from? 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 where
the being of any dependent entity ultimately comes from: from the
fundamental thing(s) at the bottom of the chain. But if the chain is
infinite, the being of any thing is, arguably, as mysterious as
Anne's new bag of sugar. The explanation of where it came from
is always postponed, and its presence in the system as a whole
unexplained. So, at least, goes the regress objection.


## 3. Regress and Global and Local Explanation



Distinguish between a *local* explanation of the \(F\)-ness of
some particular \(X\) and a *global* explanation of why there
are any things that are \(F\) at all. Some philosophers have argued
that 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 a *global* explanation of why there are
things that are \(F\), but we nevertheless have a local explanation
for each of the infinitely many \(X\)s as to why *it* is \(F\).
(This seems to be the position of Hume's Cleanthes in Part IX of
Hume 1779.)



Ricki Bliss, e.g., speaking of the infinite regress of ontologically
dependent entities, says (2013, 408): "In a reality that
admitted of no foundations ... although everything has its
reality accounted for in terms of that upon which it depends, we have
failed to explain how the whole lot of them--everything--has
any reality at all." Compare to the infinite borrowers case: for
any given instance of someone having received a bag of sugar, we can
explain where that bag of sugar came from: it came from the next
person in the chain. But what we can't explain is a global fact
about the series as a whole: why is this bag of sugar in the series in
the first place?



But Bliss argues that it is not necessarily a mark against infinitely
descending chains of ontological dependence that it leaves this global
fact--why does anything have being in the first
place?--unexplained. She says (ibid.):



> 
> 
> [T]he regress is not designed to answer this question. All the regress
> can tell us is how each individual member has the property under
> consideration, namely, in dependence upon something else. The
> appearance of an infinite regress should not lead us to conclude that
> nothing within the regress has the property under
> consideration--nor has its possession of that property
> unexplained--but rather that not everything about the possession
> of the property that needs to be explained has been.
> 
> 
> 



Contra Leibniz and Schaffer, then, Bliss rejects the idea that in an
infinitely descending chain of ontological dependence, being would
never be achieved. Having a property *dependent* on some
condition is nevertheless to have that property, so there is no
pressure, she argues, to conclude that nothing in the infinite series
would exist. Rather, they all exist, and the existence of each is
perfectly well accounted for: it exists because the next thing in the
sequence does. Everything has its being merely on some condition, but
the condition is always met. Why there is an infinite chain of
existing entities at all is not accounted for, but Bliss says it is a
mistake to think that the regress was ever supposed to account for
that.



Bliss concludes that whether or not an ontological infinite regress is
vicious or benign depends on what we set out to give an account of. If
all 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 on *ad
> infinitum*] and all that we are seeking for is an explanation of
> how or why \(x\) exists (as the thing that it is), an explanation of
> how or why \(y\) exists, and so on *ad infinitum*, the regress
> is benign. Why? Because where \(z\) explains \(y\) and \(y\) explains
> \(x\) our *explanans* and *explanandum* are *not*
> of the same form. In order to explain facts about my existence, we can
> make recourse to the existence of--or facts of the existence
> of--my parents, my vital organs, etc. In order to explain these
> facts, we make recourse to further facts, and so on. At each stage, we
> have a satisfactory explanation of that for which we are seeking one .
> . .
> 
> 
> 
> The regress is not benign, however, if what we are seeking an
> explanation for is how anything exists, or has being, at all. For even
> at infinity, what the regress shows is that we have not explained
> where existence comes from. Even at infinity, we are still invoking
> things that exist in order to explain how anything exists at
> all... We encounter, at each level, the explanatory failure
> characteristic of a vicious infinite regress: existents whose
> existence we seek an explanation for are explained in terms of
> existents. The existence of \(y\) may explain the existence of \(x\)
> but the existence of \(x\), \(y\), \(z\), and so on *ad
> infinitum* cannot help us explain how anything exists at
> all--where being comes from. Whether or not a regress of grounds
> is vicious, therefore, will depend upon the question for which we are
> seeking an answer.
> 
> 
> 



Similar remarks are made by Graham Priest (2014, 186), who asks us to
imagine 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 it
does 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 possible
options: each object in the chain is active, or each is passive. But
both are logically possible options: the fact that each object is only
active if the next object makes it active (and this sequence continues
without end) gives us no reason, says Priest, to reject the
possibility of them being all active. The active status of each object
would be accounted for, by the active status of the previous one.
First we explain the active status of \(a\_0\): it is explained by the
active status of \(a\_{-1}\). Then we have a completely different thing
to explain: the active status of \(a\_{-1}\): it is explained by the
active status of \(a\_{-2}\) ... and so on. All such facts get
explained and since, Priest argues, it is a different fact being
accounted for each time, this regress is not vicious. But echoing
Bliss, Priest admits that something is not explained. He says (ibid.,
187) "[W]hat an infinite regress will not explain is why the
whole regress is as it is ... the state of each \(a\_n\) is
determined by the state of \(a\_{n-1}\), but there is nothing in this
story to explain why the whole system is in the *all active*
state, as opposed to the *all passive* state. If there is such
an explanation, it must come from elsewhere."



So if \(a\) can only exist if \(b\) exists for \(a\) to be
ontologically dependent on, and \(b\) can only exist if \(c\) exists
for \(b\) to be ontologically dependent on ... and so on *ad
infinitum*, either the whole infinite sequence of things exists,
or none of them do. And if the whole infinite sequence exists, there
is no explanation (from within the sequence at least) as to why
anything exists at all. However, there *is* an explanation for
each particular thing as to why *it* exists: it exists because
the next thing in the sequence does. If Bliss and Priest are correct,
then whether or not an ontological infinite regress is vicious or
benign depends on our explanatory ambitions: are we attempting to
explain an (infinite) collection of particular existence facts: that
*this* thing exists, that *that* thing exists, etc., or
are we attempting to explain the global fact that things exist.
Exactly the same infinitely regressing ontology can be vicious or
benign depending on one's theoretical lights. Notice the
similarity to the discussion in section 1.3 of Bradley's and
McTaggart's regresses: once again, whether or not there is
something inherently objectionable to an infinite regress may depend
on our theoretical ambitions. Bliss (2019) argues that Metaphysical
Foundationalism appears to be motivated by an explanatory demand that
everything that is apt for explanation be explained in terms of facts
that do not require an explanation, but that this explanatory demand
begs the question in favor of Foundationalism.



Ross Cameron (2022) argues that when we have an infinite regress of
ontological dependence or grounding, we might as a result of that
regress lack an explanation of the existence or nature of the entities
on the infinite chain, or of the facts on that chain. However, Cameron
argues that this is not a reason to deny that such infinite regresses
are impossible, and instead argues that we must pull apart the notion
of metaphysical explanation from relations like ontological dependence
and grounding, which he calls metaphysical determination relations. If
\(E\_1\) is ontologically dependent on \(E\_2\), which is ontologically
dependent on \(E\_3\), which is ontologically dependent on \(E\_4\)
... and so on *ad infinitum*, there might be no
metaphysical explanation for certain facts concerning the existence
and nature of \(E\_1\), \(E\_2\), ... etc., but, argues Cameron,
this is compatible with each of the facts concerning those things
being adequately grounded in reality, and hence with the reality of
such an infinite sequence. Infinite regresses of metaphysical
determination relations, argues Cameron, might require us to abandon
certain explanatory goals, but this does not render them
impossible.


## 4. Regress and Theoretical Virtues



Bliss and Priest, as we have seen, argue that while an ontological
infinite regress might leave some questions unanswered, there is
nothing inherently objectionable, incoherent, or inconsistent in an
infinite regress of things each of which is ontologically dependent on
the next. However, even if such ontological infinite regresses are
possible, some metaphysicians argue that we may have good reason to
think that the actual world is not like that.



Nolan (2001) and Cameron (2008) argue that considerations of
theoretical parsimony can lead us to reject ontological infinite
regresses even if such regresses are not metaphysically impossible. A
theory that yields an ontological infinite regress of course thereby
yields an infinite ontology. Even if we are not in the situation
(discussed above in section 1.2) where we have independent knowledge
that we are dealing with a finite domain, this could still be a mark
against the theory, simply on the grounds that it is an unparsimonious
ontology. So for example, we might object to the claim that material
objects are gunky--with each part of them being divisible into
further proper parts--not because there's any inconsistency
in the hypothesis, or because it leads to an infinite chain of
ontological dependence and thereby leaves the existence of all things
unexplained, but simply because it means that whenever there is an
object somewhere, there are in fact infinitely many objects there. A
gunky world is an ontologically extravagant world, and so we have the
same kind of reason to reject the hypothesis that things are gunky as
we have to reject needlessly complex hypotheses about how things
behave (e.g., the Ptolemaic theory of planetary motion with its
epicycle upon epicycle): other things being equal, we should prefer
simpler theories and more economical ontologies over complex theories
and more expansive ontologies.



To illustrate, Nolan considers the famous example of someone
suggesting that the Earth is held up by resting on the back of a giant
turtle, which is in turn held up by resting on the back of another
turtle, 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 is
infinite, so there's no problem fitting all these turtles in, so
we're not in a case where we know independently that we're
dealing with a finite domain. This is not a regress that involves
ontological dependence, so there are no concerns about the existence
of things going ungrounded. Of course, we have pretty good empirical
evidence that the infinite turtles hypothesis is false, since we have
gone into space and can't see any world turtles. But even
putting that aside--let's suppose we're considering
the hypothesis prior to our going into space--there is something
intuitively weird about the turtles hypothesis. Nolan suggests it is
the ontological extravagance of the view. He says (2011,
534-5)



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



Not everyone will agree that each additional turtle theory is more
objectionable than the last, since the extra things being postulated
are of the same kind (world turtles) as the previous theory already
countenanced. David Lewis (1973, 87), e.g., held that while we should
prefer theories that are more *qualitatively*
parsimonious--they postulate fewer *kinds* of thing than
their rivals--there is no reason at all to prefer theories that
are merely more *quantitatively* parsimonious--they
postulate fewer things *of the same kind* as their rivals. This
is controversial, however, and Nolan (1997) argues that quantitative
parsimony 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 theory
worse (other things being equal) than any finite turtle theory.



Cameron applies considerations of theoretical parsimony to the case of
infinite chains of ontological dependence. While allowing that there
is no impossibility in an infinite regress of things, each
ontologically dependent on the next, Cameron argues that we can still
have reason to reject such a theory on parsimony considerations.
Cameron (2008, 12) says that what needs to be explained is the
existence of each dependent entity; and while he allows that in an
infinitely descending chain of ontologically dependent entities, there
is an explanation for why each dependent entity exists, there is no
single explanation for why *all* the dependent entities exist.
Whereas if there is a collection of fundamental entities on which all
the dependent entities ultimately depend, these fundamental entities
provide a single unified explanation for why every dependent entity
exists. Either way, everything that needs to be explained gets
explained, but Cameron says we have reason to prefer the unified
explanation over the infinitely many disparate explanations, since it
is in general a theoretical virtue to provide a unified explanation.
For example, a physical theory that postulates one unified force to
explain all phenomena would, other things being equal, be preferable
to one that postulates four fundamental forces--gravity,
electromagnetism, strong nuclear, and weak nuclear--even if the
two theories explain exactly the same phenomena. (Orilia (2009)
argues, contra Cameron, that there is no unified explanation provided
by Metaphysical Foundationalism over theories with infinite
ontological descent.)



If Nolan and Cameron are right it at most gives us a *pro
tanto* reason to reject a theory that leads to an ontological
infinite regress. Just as we can justifiably accept the more complex
hypothesis over the simpler one because the more complex hypothesis is
more powerful (e.g.), so we might justifiably accept an ontological
infinite regress because there is some virtue afforded by the theory
that makes the cost worthwhile. Relatedly, Cameron (2008, 13-14)
argues that this would not give us any reason to think that
ontological infinite regresses are metaphysically impossible, at most
it gives us a reason to think they are not actual, which limits the
positions 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 of
the \(F\)-ness of each \(X\) appeals to another \(X\) that is \(F\),
and so on *ad infinitum*, then while the \(F\)-ness of each
individual \(X\) can be accounted for, something is left unexplained:
why there are things that are \(F\) at all. But arguably, not every
infinite regress leaves even this global fact unexplained.



To say merely that the \(F\)-ness of each \(X\) is explained by appeal
to another \(X\) that is \(F\) leaves open a crucial question: does
the \(F\)-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 between *transmissive* explanations of
\(F\)-ness, in which the \(F\)-ness of \(X\_2\) plays a crucial role in
explaining 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 \(F\), and
so on, from *non-transmissive* explanations of \(F\)-ness, in
which 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 of *transmissive* explanations of
\(F\)-ness leaves unexplained why anything is \(F\) in the first
place, but an infinite regress of non-transmissive explanations need
not.



Simon Blackburn (1986) argued that any realist attempt to explain why
there is necessity in the world would fail because it faced a dilemma.
Suppose we say that \(A\) is necessary because \(B\). \(B\) at least
has to be true, but is \(B\) itself necessary? Blackburn thought we
could not answer no, because to explain the necessity of a necessary
truth \(A\) by appeal to a contingent truth would undermine the
necessity of \(A\). But to answer yes is to invite regress, for now we
need to explain the necessity of \(B\) by appeal to a necessary truth
\(C\), and so on *ad infinitum*. Now, whether the contingency
horn is indeed vicious is debatable (see Hale 2002 and Cameron 2010
for discussion), but focus on the regress involved in the necessity
horn. Hale (2002) argues that the realist about necessity can resist
getting into a vicious regress by distinguishing between transmissive
and non-transmissive explanations of the necessity of any given
proposition. Grant that the necessity of \(A\) can only be explained
by a proposition, \(B\), that is itself necessary: that explanation
will be *transmissive* if the necessity of \(B\) plays a role
in explaining the necessity of \(A\), otherwise it will be
*non-transmissive*. Hale (ibid., 308-309) offers as an
example of a transmissive explanation of the necessity of \(A\) a
proof of \(A\) with the necessary truth \(B\) as its sole premise. If
\(A\) follows logically from a necessary truth, then \(A\) itself must
be necessary. And in this case, the necessity of \(B\) plays a crucial
role in the explanation, for if I do not know whether \(B\) is
necessary, knowing that it entails \(A\) does not tell me whether
\(A\) is necessary, for a contingent proposition can follow from
another contingent proposition. In that case, \(A\)'s necessity
seems to be hostage to \(B\)'s necessity, and so the ultimate
explanation of \(A\)'s necessity will seemingly involve whatever
explains \(B\)'s necessity, which is where Blackburn senses
regress. But Hale thinks there can also be non-transmissive
explanations of necessity. He suggests: 'Necessarily the
conjunction of two propositions \(A\) and \(B\) is true only if \(A\)
is true and \(B\) is true' because 'conjunction just is
that binary function of propositions which is true iff both its
arguments are true' (ibid., 312). While he grants that the
explanans in this case *is* necessary, Hale thinks that its
necessity is no part of the explanation. All that is needed to explain
why it is necessary that \(A \amp B\) is true only if \(A\) and \(B\)
are both true is the *truth* of the fact that conjunction just
is that function which is true iff both its arguments are true. It may
be necessary the conjunction is that function, but that is not part of
the explanation of the original necessity, and thus the necessity of
the explanans is not appealed to in a way that makes the success of
the explanation dependent now on explaining this further necessity.
Perhaps there is a new question to be asked concerning why this
further claim is necessary, but our original explanation stands
independently of whether we can successfully answer this new
question.



Another example. Epistemic Infinitists embrace the infinite regress of
reasons and argue that it is not vicious (see, e.g., Aikin 2005, 2011,
Klein 1998, 2003, Peijnenburg 2007 and Atkinson & Peijnenburg
2017). One response for the Infinitist to make to the regress argument
is the response from section 3: to hold that each belief is justified
in virtue of the next one being justified, but to claim that this is
not a problem: that while the regress means we do not have an
explanation for why anything is justified in the first place, this is
not what the sequence was *meant* to explain--all that we
need is an explanation for each belief concerning why *it* is
justified, and this we have. (See e.g., Aikin 2005, 197 and Klein
2003, 727-729.)



However, the Infinitist may also simply deny that anything remains
unexplained in such a regress. Peter Klein (1998, 2003) holds not only
that there *can* be an infinite regress of justifications, but
indeed that it is *necessary* for \(S\) to be justified in
believing that \(p\) that \(S\) have available to them an infinite
number of propositions such that the first of these, \(r\_1\), is a
reason for \(p\), the second, \(r\_2\), is a reason for \(r\_1\), the
third, \(r\_3\), is a reason for \(r\_2\), and so on *ad
infinitum*. 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.) In
which case, so the objection goes, the justificatory chain could not
get off the ground, and nothing would be justified. But Klein (2003,
720-723) denies that \(r\_1\) is a reason for \(p\) *in virtue
of* \(r\_2\) being a reason for \(r\_1\). There must *be* a
reason, \(r\_2\), for \(r\_1\), and there must *be* a distinct
reason for \(r\_2\), \(r\_3\), and so on. Each of these infinitely many
propositions are justified, but each one's being justified, says
Klein, does not hold in virtue of any *other* being justified.
The Infinitist demands that there is an infinite justificatory
sequence, but that in itself is silent as to what justification
*consists* in. The Infinitist can simply hold that there is
some 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 so
on. That feature could be the first element increasing the objective
probability of the second, or something else entirely (ibid., 722).
The point is, it needn't involve the second element itself being
justified, and thus the Infinitist need not accept that the
justification of \(p\) from \(r\_1\) is *inherited* from the
justification of \(r\_1\) from \(r\_2\), and so on. In Hale's
terminology, the explanation of the justification of a proposition by
appeal to another justified proposition is a non-transmissive one:
while one must appeal in the explanation to a proposition that is in
fact justified, the fact that it is justified plays no role in that
explanation. Thus there is, arguably, no reason to think that the
regress of epistemic justification is vicious *even if* you
demand an explanation for why any of our beliefs are justified in the
first place.



It's worth reflecting on the difference between the epistemic
regress and the ontological regress of dependent entities that makes
Klein's response here possible. Suppose that \(A\) is
ontologically dependent on \(B\) and \(B\) ontologically dependent on
\(C\). It is very plausible that in this case, \(C\)'s existence
and/or nature is part of the explanation of \(A\)'s existence
and/or nature. In saying that \(A\) is ontologically dependent on
\(B\) we are saying that \(A\) exists, or is the way it is, at least
partly 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 be
the way it is. In explaining \(A\)'s existence/nature, we are
appealing to \(B\)'s existence/nature, in which case it seems
that anything that we need to explain \(B\)'s existence and/or
nature--in this case \(C\)'s existence/nature--must
ultimately be part of the explanation of \(A\)'s existence
and/or nature. That's why when we have a chain of ontological
dependence, the existence and/or nature of the first entity seems to
be ultimately dependent on not just the existence/nature of the second
entity in the chain, but on that of every subsequent entity in the
chain: explanations of being appear to be transmissive. Which is why,
if the chain is endless, we seem to lack an explanation as to why
anything exists at all.



But while it is overwhelmingly plausible that \(B\) can only serve as
the ontological ground of \(A\) because \(B\) itself *exists*,
or is the way it is, it is *not* forced on us to hold that
\(r\_2\) can only be a reason for \(r\_1\) because \(r\_2\) is itself
justified, and this is why Klein's response to the epistemic
regress 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. As
Klein says, the entirety of the explanation for why \(r\_2\) is a
reason 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 \(r\_2\) must *be*
justified if \(r\_1\) is, but that need not be any part of *why*
\(r\_2\) is a reason for \(r\_1\), and thus there is no pressure to hold
that the justification of \(r\_1\) by \(r\_2\) is dependent on, or
inherited by, the justification of \(r\_2\) by \(r\_3\), and so on. So
while there is indeed an infinite sequence of propositions, each of
which is a reason for the previous one on the list, at no stage is the
fact that one proposition is a reason for another hostage to the fact
that any other proposition is a reason for another. And so arguably,
nothing remains unexplained: there can be a good explanation not only
for why each particular proposition is justified (\(r\_5\) is justified
by \(r\_6\), etc.), but also for why there are any justified
propositions in the first place (there are propositions that raise the
objective probability of others,
 e.g.).[13]


## 6. Coherence, Circularity, and Holism



The Coherentist resists regress by allowing a circular or holistic
explanation of the \(F\)-ness of at least some \(X\)s. This could be
to 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 for
the Coherentist. Consider again the regress argument concerning
justification 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 be
explained are the facts concerning the individual beliefs--why is
*this one* justified, then why is *that one* justified,
etc. An epistemic Coherentist such as Bonjour (1985) rejects this
assumption. It is not, primarily, individual beliefs that are
justified, it is *systems* of belief. A particular belief is
justified only in a derivative sense, by belonging to a justified
system. A system of belief is justified because of the properties of
the system as a whole, namely that the beliefs in it form a coherent
system. Thus, justification is a holistic phenomenon: a collection of
beliefs is justified because of what they, collectively, are like, not
because of what each individual member of the system is like. This
holistic explanation of where justification comes from is very
different from a circular explanation: a circular explanation tells us
that one individual's being \(F\) explains another's being
\(F\) and also vice versa, but a holistic explanation tells us to
abandon the idea of explaining an individual's being \(F\) by
appeal to another individual being \(F\), and instead hold that the
explanation for *some things* being \(F\) can be the facts
concerning what that collection of things as a whole is like. (An
analogous view concerning Metaphysical Coherentism and holistic
explanation is defended by Thompson (2018), and Cameron (2022, Ch.4).
Cf. Barnes (2018).)



Sometimes a circular explanation might be warranted because we are not
trying to explain in virtue of what the \(X\)s are \(F\), but rather
we are merely attempting to illuminate the \(X\)s being \(F\) by
showing how the \(X\)s relate. Consider the regress argument against
the 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 to
> float down this stream, and if we say 'time has flowed faster
> today than yesterday' we are saying that the stream flowed a
> greater distance today than it did in the same time yesterday. That
> is, we are postulating a second timescale with respect to which the
> flow of events along the first time dimension is measured ...
> Furthermore, just as we thought of the first time dimension as a
> stream, so will we want to think of the second time dimension as a
> stream also; now the speed of flow of the second stream is a rate of
> change with respect to a third time dimension, and so we can go on
> indefinitely postulating fresh streams without being any better
> satisfied.
> 
> 
> 



Here we start with our ordinary temporal dimension--what we may
have supposed to be the *only* temporal dimension--the one
philosophers have in mind when they say that time passes. Smart
supposes that if this dimension of time indeed passes then there must
be a rate at which it
 passes.[14]
 Whether that rate can slow down or speed up or if time always flows
at the same rate is not important, but there must be some rate at
which it passes, thinks Smart. Now, just as we would measure the speed
of a car, say, by measuring how much distance it covers in a given
amount of time, so, thinks Smart, we would have to measure the speed
at which time itself passes by measuring how much time passes in a
given amount of time of some second temporal dimension. While the car
covers forty miles of road in the space of an hour, an hour of time
passes in the space of, say, two hours of this second temporal
dimension. But then, how fast does this second dimension of time pass?
We need a third temporal dimension to measure how long it takes for an
hour of the second temporal dimension to pass. But how fast does the
third temporal dimension pass? And so on, *ad infinitum*. Smart
concludes that time does not pass.



A defender of the view that time passes could attempt to resist
Smart's regress by cutting off the regress at the second stage
by claiming that there is a principled difference between the first
temporal dimension and the second that results in the first temporal
dimension passing at some rate, but not the second. Smart himself
concludes that time does not pass, so can hardly object to the
postulation of temporal dimensions that do not pass. However,
Smart's regress can be resisted without abandoning the principle
that all temporal dimensions pass at some rate.



Ned Markosian (1993) points out that to give a rate is to compare two
different types of change. When we say that the car travels at forty
mph, we are comparing one type of change--the car started off in
one place and ended up forty miles distant--with another--it
was one time and it is now an hour later. In that case, we can always
get the rate of the second type of change by comparing back to the
first. As Markosian says (ibid., 842): "If ... I tell you
that Montana's passing totals increased at the rate of 21 passes
per game, then I have also told you that the games progressed at the
rate of one game per 21 completions by Montana." So suppose we
say that the first temporal dimension passes at a rate of one hour for
every two hours of the second temporal dimension; there is no need to
invoke a third temporal dimension in order to state the rate at which
the second passes, for we have already given that rate: an hour of the
second temporal dimension passes for every half hour of the first.
Indeed, as Markosian points out, we need not even invoke a second
temporal dimension, for any time we give a rate of any ordinary
process with respect to time--such as that the Earth goes around
the sun once every year--we have thereby stated the rate at which
time passes with respect to those ordinary processes: time passes at
the rate of one year for every orbit of the Earth around the sun.



Markosian's maneuver is possible because in giving the rate of
one process of change by appeal to a second process of change we are
not saying what *makes it the case* that the first change
occurs at the rate it does. What makes it the case--what are the
ontological 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 40
miles, for that is merely a re-description of the fact in question: a
way of *describing* the rate. In the case of time itself,
defenders of the view that time passes may plausibly claim that what
*makes it the case* that time passes is simply the nature of
time: that it is in its nature to pass at the rate it does. It
doesn't pass at the rate it does *because* of some
relation it stands in, either to ordinary processes of change or to a
second temporal dimension. Such relations allow us to informatively
*state* the rate of change, they do not provide the
*grounds* 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, since
arguably grounding is asymmetric (see e.g., Rosen 2010,
 115).[15]
 If the car travels at the speed it does *in virtue of*
something to do with the passage of time then, arguably, time cannot
pass at the rate it does *in virtue of* anything to do with the
speed of the car, and so we need to appeal to the passage of a second
temporal dimension to provide the ontological grounds of the rate of
passage of the first. If the passage of this second temporal dimension
grounds facts about the passage of the first temporal dimension, it
itself cannot pass *in virtue of* facts concerning the passage
of the first, and so we need to appeal to a third temporal dimension,
and so on. And even if we hold that it's *possible* for
there to be infinitely descending chains of grounds, it seems absurd
in this case to suppose that there are in fact infinitely many
temporal dimensions. But that is not what is going on. When we explain
the speed of the car by appeal to the passage of time, we're not
providing the ontological grounds of its speed, we're simply
showing a connection between two things: the movement of the car and
the passage of time. Likewise for the rate of time's passage
itself: we are not seeking to provide the ontological grounds of
time's passage in stating its rate, for the ontological grounds
are plausibly just the nature of time itself. Rather, when we compare
the two changes, we are simply trying to illuminate one or both of
those changes by pointing to the way they relate. That is the only
sense in which one *explains* the rate of one change when
comparing it to another kind of change: the mutual connection tells us
something enlightening about each. It is not to give a metaphysical
explanation in the sense of providing the metaphysical
*grounds* of either rate of change. Compare: if I tell you that
the value of a US dollar is 0.7 British pounds (and *therefore*
that the value of a British pound is 1.43 US dollars), this is not to
say that the US dollar has the worth that it has *in virtue of*
standing in this relation to the British pound. The US dollar relates
thus to the British pound *because* of what they are each
worth; they are not worth what they are in virtue of standing in that
relationship. What makes it the case that the US dollar is worth what
it is is some incredibly complex set of facts concerning economics,
monetary policy, etc. To give an exchange rate is not to give the
grounds of the value of the currencies, but merely to say something
substantive about each value by stating their connections. Likewise
with rates of change, which is why Markosian is able to resist
Smart's regress in this manner. Coherentist explanations might
be controversial when it comes to providing ontological grounds, but
they are less so when it comes to simply casting light on the nature
of some phenomena by showing how they connect. So whether a regress
argument even gets going will depend on the explanatory ambitions of
the view being targeted.