## 1. Life



Donald Williams was born on 28 May 1899 in Crows Landing, California,
at that time a strongly rural district, and died in Fallbrook, also in
his beloved California, and also at that time far from cities, on 16
January 1983. His father was Joseph Cary Williams, who seems to have
been a jack of all countrymen's trades; his mother Lula Crow, a
local farmer's daughter. Donald was the first in his family to
pursue an academic education. After studies in English Literature at
Occidental College (BA 1923), he went to Harvard for his Masters, this
time in Philosophy (AM 1925). He then undertook further graduate study
in philosophy, first at the University of California at Berkeley
(1925-27), then at Harvard, where he took his PhD in 1928.



Also in 1928 he married Katherine Pressly Adams, from Lamar, Colorado,
whom he had met at Berkeley, where she was something of a
pioneer--a woman graduate student in psychology. In time, there
were two sons to the marriage. The couple spent a year in Europe in
1928-29 ("immersing himself in Husserl's
Phenomenology to the point of immunization", Firth, Nozick and
Quine 1983, p 246). Then Donald began his life's work as a
Professor of Philosophy, first spending ten years at the University of
California, Los Angeles, and then from 1939 until his retirement in
1967, at Harvard.



Throughout his long and distinguished career in philosophy he retained
a down-to-earth realism and naturalism in metaphysics, and a
conservative outlook on moral and political issues, characteristic of
his origins. A stocky, genial, and cheerful man, he found neither the
content nor the validation of ethics to be problematic. Although
originally a student of literature, whose first publication was a book
of poetry, he had not the slightest tincture of literary or academic
bohemianism. He was among the very least alienated academics of his
generation; which is not surprising, as his career was indeed one
version of The American Dream.


## 2. The Nature of Philosophy



The traditional ambition of philosophy, in epistemology and
metaphysics, is to provide a systematic account of the extent and
reliability of our knowledge, and on that basis, to provide a synoptic
and well-based account of the main features of Reality. When Williams
was in his prime, this ambition was largely repudiated as
inappropriate or unattainable, and a much more modest role for
philosophy was proposed. In setting forth his own position, in the
preface to the collection of his selected essays (1966, p.viii), he
lists some of these fashionable philosophies from the mid-twentieth
century:



> 
> 
> ...logical positivism, logical behaviorism, operationalism,
> instrumentalism, the miscellaneous American linguisticisms, the
> English varieties of Wittgensteinism, the Existentialisms, and Zen
> Buddhism...
> 
> 
> 



Each of these is, in its own way, a gospel of relaxation. They all
propose that, in place of the struggle to uncover how things are,
careful descriptions of how things appear will suffice. None is
ambitious enough to set about constructing a positive and systematic
epistemology and metaphysics.



Undeterred by this spirit of the age, Williams continued to insist
that philosophical issues, including those of traditional metaphysics,
are real questions with genuine answers (Fisher 2017). Conceptual
analysis, concentration on phenomenological description, or
exploration of the vagaries of language, may have their (subordinate)
place, but to elevate them to a central position is an evasion of
philosophy's main task.



Still worse was the suggestion that philosophical questions are mere
surface expressions of a philosopher's underlying
psycho-pathology. The claim of Morris Lazerowitz to that effect,
suggesting that Bradley's Absolute Idealism was no more than an
intellectual's poorly expressed death wish, or that
McTaggart's argument against the reality of time a panic fear of
change, he met with a stern rebuke (1959, pp 133-56).



He set forth a Realist philosophy on traditional empiricist
principles: "He thought that practically everything was right
out there where it belonged" (Firth, Nozick and Quine 1983, p
246). He maintained that while all knowledge of fact rests on
perceptual experience, it is not limited to the perceptually given,
but can be extended beyond that by legitimate inference (1934a). In
this way his Realism can develop the breadth and depth required to do
justice to all the scientific techniques which so far surpass mere
perception.



Williams's empiricism extended to philosophy itself. He
challenged the prevailing orthodoxy that philosophy is a purely *a
priori* discipline. He emphasized the provisional character of
much philosophizing, and the striking absence of knock-down arguments
in philosophic controversy ("Having Ideas In The Head"
1966, 189-211).


## 3. Metaphysics--Cosmology



Following his own prescription for an affirmative and constructive
philosophy, Williams worked steadily towards the development of his
own distinctive position in metaphysics. He introduced the useful
division of the subject into Speculative Cosmology, which deals with
the basic elements making up the world we live in, such as matter,
mind, and force, together with the relations between them, and
Analytic Ontology, which explores the fundamental categories, such as
Substance and Property, and how they relate to one another.
Speculative Cosmology, in particular, needs to be open to developments
in the fundamental sciences, and so needs to be seen as always
provisional and *a posteriori.* Analytic Ontology, the
exploration of the categories of being, is a more purely reflective
and a priori discipline aiming to elucidate the range of different
elements in any universe.



To begin with Speculative Cosmology, Williams's position has
three main features.



It is Naturalistic. The natural world of space, time, and matter, with
all its constituents, is a Reality in its own right. Contrary to all
Idealist metaphysics, this world, except for the finite minds that are
to found within it, is independent of any knowing mind. Moreover, it
is the only world. There are no divinities or supernatural powers
beyond the realm of Nature ("Naturalism and the Nature of
Things" 1966, pp 212-238). Even mathematical realities
belong in the natural world: numbers--at least natural
ones--as abstractions from clusters, and geometrical objects as
abstractions from space. The philosophy of mathematics was an aspect
of his position that was never fully worked through.



Second, his position is not only Naturalist but also Materialist. This
Materialism is not of the rather crude kind that supposes that every
reality is composed of a solid, crunchy substance, the stuff that
makes up the atoms of Greek speculation. Any spatio-temporal reality,
whether an 'insubstantial' property, such as a color, or
something as abstract as a relation, such as farther-away-than, or
faster-than, so long as it takes its place as a spatio-temporal
element at home in the world of physics, is accepted as part of this
one great spatio-temporal world. Williams's Materialism is thus
one which can accept whatever physical theory posits as the most
plausible foundation for the natural sciences, provided that it
specifies a world which develops according to natural law, without any
teleological (final) causes.



The Mental is accommodated in the same way. Although Mind is
unquestionably real, mental facts are as spatio-temporally located as
any others ("The Existence of Consciousness", "Mind
as a Matter of Fact" 1966, pp 23-40, 239-261). The
Mental is not an independent realm parallel to and equal to the
Material, but rather a tiny, rather insignificant fragment of Being,
dependent upon, even if not reducible to, the physical or biological
nature of living beings. Williams has a capacious conception of the
Material--his position could perhaps be better described as
Spatio-Temporal Naturalism.



Thirdly, Williams's metaphysics is 4-dimensional. Or, more
precisely, given the development of multi-dimensional string theories
since his time, it takes Time to be a dimension in the same way that
Space has dimensions. The first step is to insist, against Aristotle
and his followers, that statements about the future, no less than
those concerning the present and the past, are timelessly true or
false. They need not await the event they refer to, in order to gain a
truth value ("The Sea Fight Tomorrow" 1966, pp
262-288.) This encourages the further view that the facts that
underpin truths about the future are (timelessly) Real. All points in
time are (timelessly) Real, as are all points on any dimension of our
familiar Space. Whatever account is to be given of Change, it does not
consist in items gaining or losing Reality. This stance receives
powerful support from physical theory. Williams embraced and argued
for the conception of Time as a fourth dimension introduced by
Minkowski's 'Block Universe' interpretation of
Einstein's Theory of Special Relativity. A consequence of this
is that the experience of the flow or passage of Time must be some
sort of illusion. Williams embraced that consequence, and argued for
it in a celebrated paper ('The Myth of Passage',
1951).


## 4. Metaphysics--Ontology



Apart from his Materialistic or Spatio-Temporal Naturalism,
Williams's major contributions to metaphysics lie in the realm
of ontology, and concern the fundamental constituents of Being. His
key proposal is that properties are indeed real--in fact Reality
consists in nothing but properties--but that these properties are
not Universals, as commonly supposed, but particulars with unique
spatio-temporal locations. The structure of Reality comprises a single
fundamental category, Abstract Particulars, or "tropes".
Tropes are particular cases of general characteristics. A general
characteristic, or Universal, such as redness or roundness, can occur
in any one of indefinitely many instances. Williams' focus was
on the particular case of red which occurs as the color, for example,
of a particular rose at a specific location in space and time, or the
particular case of circularity presented by some particular coin in my
hand on a single, particular occasion. These tropes are as particular,
and as grounded in place and time, as the more familiar objects, the
rose and the coin, to which they belong.



These tropes are the building blocks of the world. In his analogy,
they provide 'the Alphabet of Being' from which the
entities belonging to more complex categories--objects,
properties, relations, events--can be constructed, just as words
and sentences can be built using the letters of the alphabet. Familiar
objects such as shoes and ships and lumps of sealing wax, and their
parts as revealed by empirical scientific investigation, such as
crystals, molecules and atoms, are concrete particulars or things. In
Williams's scheme, each of these consists in a
*compresent* *cluster* of tropes--the particular
thing's particular shape, size, temperature, and consistency,
its translucency, or acidity, or positive charge, and so on. All the
multitude of different tropes that comprise some single complex
particular do so by virtue of their sharing one and the same place, or
sequence of places, in Space-Time. That is what
'compresent' means. There is no inner substratum or
individuator to hold all the tropes together. The tropes are
individuals in their own right, and do not inhere in any thing-like
particular. So Williams's view is a No-Substance theory, or,
otherwise described, a theory in which each trope is itself a simple
Humean substance, capable of independent existence.



Universal properties and quantities such as acidity and velocity,
which are common to many objects, are not beings in their own right,
but *resemblance classes* of individual tropes. If two objects
match in color, both being red, for example, the tropes of color
belonging to each are separate tropes, both being members of the class
of similar color tropes which constitutes Redness.



Relations are treated along the same lines. If London is Larger-than
Edinburgh, and Dublin Larger-than Belfast, we have two instances of
the Larger-than relation, two relational tropes. And they, along with
countless other cases, all belong to the resemblance class whose
members are all and only the cases of Larger-than. This account denies
that, literally speaking, there is any single entity which is
simultaneously fully present in two different cases of the same color,
or temperature, or whatever. So it is a No-Universals view, and often
described as a version of Nominalism. This is understandable, but it
is better to confine the term 'Nominalism' to the denial
of the reality of properties at all. So far from denying properties,
on Williams's theory the entire world consists in nothing but
tropes, which are properties construed as particulars. So his position
is better described not as Nominalist but as Particularist.



Tropes provide an elegant and economical base for an ontology. Unlike
almost all competing ontologies, that of Williams rests on just one
basic category, which can be used in the construction not just of
things and their properties and relations, but of further categories,
such as events and processes. Events are replacements of the tropes
that are to be found in a given location. Processes are sequences of
such changes. Trope theory is well placed to furnish an attractive
analysis of causality, as involving power tropes that govern and drive
the transformations to be found in events and processes.



It can also be of use in other areas of philosophy, for example in
valuation theory, where the existence of many tropes, rather than one
single unitary reality, can explain our sometimes divided attitudes
toward what, on a Substance ontology, we would regard as the same
thing. Something can be good in some respects (tropes), but not in
others. To view the manifest world as comprising, for the most part,
clusters of compresent tropes makes explicit the complexity of the
realities with which we are ordinarily in contact.


## 5. Objections to Williams's Trope Ontology



Many philosophers have admitted tropes into their scheme of things:
Aristotle, Locke, Spinoza and Leibniz, for example. What is
distinctive in Williams is not that tropes are admitted as a category,
but as the *only* fundamental category, a trope-based form of
what Schaffer calls "property primitivism." (Schaffer
2003, 125). All else is constructed out of tropes, including concrete
particulars and general properties, whereas tropes themselves are not
constructed out of anything else, for example, out of a substance, a
universal and the relation of exemplification. Not surprisingly,
various aspects of Williams' trope primitivism have been
subjected to serious philosophical challenges either directly or
indirectly.



Some philosophers reject all forms of property primitivism, including
that of Williams, on the grounds that properties cannot serve as the
only independent elements of being. Properties must be had by objects
and a property cannot be instantiated in isolation from wholly
distinct properties so they lack the requisite independence, the
capacity of existing in any combination with "wholly distinct
existences." (Some philosophers go so far as to claim that if a
property *P* is a trope, then *P* is dependent on the
specific thing that has it and so could not have existed without that
thing existing; see Mulligan, Simons, & Smith 1984, Heil 2003.)
There are two ways for this objection to go. (1) Armstrong takes the
ostensive fact that properties must be had by objects to establish the
dependence of properties on non-property particulars, substrata, and,
thus, the falsity of property primitivism (Armstrong 1989, 115). (2)
Alternately, one can take the apparent fact that there can be no
properties that are not clustered with other properties to show that
properties cannot play this role.



In response to (1), one can challenge the assumption that if
properties must be had by objects, then they are not capable of
independent existence. Ross Cameron, for example, suggests that
properties might both be capable of independent
existence--including existing without substrata--and not be
capable of existing without being the property of something. (2006,
104). The dependence of properties on objects is compatible with
property primitivism if one adopts a bundle theory of concrete
particulars. In that case, properties are always had by objects since
they are always found in bundles, even if only a bundle of one, but
exist without substrata.



As for (2), some philosophers reject the requirement that if
properties are the only ultimate constituents of reality, then they
must be capable of existing in isolation from all other wholly
distinct properties (Simons 1994; Denkel 1997). Properties can be both
the only ultimate constituents of reality and inter-dependent. This
response requires the rejection of the Humean principle that the basic
independent units of being can exist in any combination, including
unaccompanied (Schaffer 2003, 126). A very different response to (2)
rejects the necessity of trope clustering altogether (Williams 1966,
97; Campbell 1981, 479; Schaffer 2003).



> 
> 
> Plausible though it be, however, that a color or a shape cannot exist
> by itself, I think we have to reject the notion of a standard of
> concreteness. ... (Williams 1966, 97)
> 
> 
> 



At best, it is a contingent matter that there are no tropes that are
not compresent with any other tropes--for example, a mass trope
on its own. (Campbell goes so far as to suggest that there is reason
to think that there are actual cases of free-floating tropes (1981,
479)).



Even if the trope primitivist can get around these quite general
objections, there remains the most serious objection to
Williams' brand of trope primitivism, an objection that is
specific to Williams' conception of tropes, depending on more
than the assumption that tropes are properties. The charge is that a
Williamsonian trope is not genuinely simple, but complex embracing, at
least, an element that furnishes the nature or content of the trope,
and an element providing its particularity (Hochberg 1988; Armstrong
2005; Moreland 1985; Ehring 2011). In short, Williamsonian tropes are
constructed out of something else, making them incompatible with trope
primitivism.



This objection is based on two assumptions. First, under
Williams' conception, the nature of a trope is a non-reducible,
intrinsic matter that is not determined by relations to anything else,
including resemblance relations to other tropes or memberships in
various natural classes of tropes. And, second, anything that stands
in more than one arbitrarily different relation, each of which is
grounded intrinsically in that entity, must be complex since that
entity will have intrinsic "aspects" that are not
identical to each other. The two relevant relations are numerical
difference from other tropes and resemblance to other tropes, each of
which is grounded intrinsically in the trope relata under the Williams
conception. Hence, there are "intrinsic aspects" of each
trope that are not identical to each other, a particularity-generating
component and a nature-generating component.



In response to this "complexity" objection, Campbell
claims that the distinction between a trope's nature and its
particularity is merely a "formal" distinction, a product
of different levels of abstraction, and not a real distinction between
different components of a trope. One should no more distinguish a
particularity-component from a nature-component of a trope than
distinguish components of warmth and orangeness in an orange
trope:



> 
> 
> To recognize the case of orange as warm is not to find a new feature
> in it, but to treat it more abstractly, less specifically, than in
> recognising it as a case of orange. (Campbell 1990, 56-7;
> further in Maurin 2005; Fisher 2020)
> 
> 
> 



Alternately, Ehring suggests that we grant this objection, but
preserve trope simplicity by switching to a non-Williamsonian
conception of tropes, according to which a trope's nature is
determined by its memberships in various natural classes of tropes
rather than intrinsically, thereby sidestepping one of the assumptions
operative in the objection (2011; discussion in Hakkarainen and
Keinanen 2017).



Coming under criticism as well is Williams' resemblance-based
account of general characteristics and property agreement. According
to Williams, a fully determinate general color
characteristic--say, the shade of red that characterizes this
shirt and this chair---is just the set of all tropes that exactly
resemble the red trope of this shirt. Different objects of that
"same" shade of red each possess a different trope from
this set of exactly similar red tropes. But this analysis seems to
generate an infinite regress. If trope *t*1 is
related to trope *t*2 by resemblance trope
*r*1, *t*2 is related to trope
*t*3 by resemblance trope *r*2,
and *t*3 is related to *t*1 by
resemblance trope *r*3, then these resemblance
tropes will also resemble each other, giving rise to further
resemblance tropes, and so on. To stop this vicious regress, the
objection continues, resemblance must be taken to be a universal (Daly
1997, 150).



One response to this objection tries to stop the regress before it
starts by denying that there are any resemblance trope-relations
holding between tropes. In particular, the trope theorist might follow
Oliver's advise



> 
> 
> to avoid saying that when two tropes are exactly similar ...,
> there exists a relation-trope of exact similarity ... holding
> between the two tropes. (1996, 37)
> 
> 
> 



There are no resemblance-tropes corresponding to these resemblance
predicates. Another response denies that resemblance relations mark an
addition to our ontology and, hence, there can be no regress of
resemblance relations. Campbell, for example, argues that the
successive resemblance relations in the regress are nothing over and
above the non-relational tropes that ultimately ground these
relations. Since resemblance is an internal relation, it supervenes on
these ground-level non-resemblance tropes, but supervenient
"additions" are not real additions to one's ontology
(Campbell 1990, 37).



There is also a whole host of objections in the literature to
Williams' trope bundle theory. According to Williams, concrete
particulars are not substrata instantiating various properties. They
are wholly constructed out of tropes, forming bundles of tropes, the
trope constituents of which are pairwise tied together by a
compresence relation. Compresence, in turn, is collocation for
Williams, "the unique congress in the same volume." (In
order to allow for non-spatial objects, Williams grants the
possibility of "locations" in systems analogous to space
(1966, 79).) One immediate worry, raised by Campbell, is the
possibility that there may be cases of overlapping objects
demonstrating that collocation is not sufficient for compresence
(1990, footnote 5, 175). In response, the trope bundle theorist can
opt for the view that compresence is non-reducible. However, even with
this revision there remain significant objections concerning the
possibility of accidental properties, the possibility of change, and
an apparent vicious regress of compresence relations.



Bundle theory has been charged with ruling out the possibility of
accidental properties in concrete particulars. Bundles of properties
have all of their constituent properties essentially. Objects
generally do not. This chair could have been blue instead of red, but
the bundle of properties that characterize the chair could not have
failed to include that red property. One way around this objection is
proposed by O'Leary-Hawthorne and Cover: combine bundle theory
(although for them properties are universals) with a specific account
of modality, a counterpart semantics for statements about ordinary
particulars (1998). What makes it true that a particular object
*o* could have had different properties is that a non-identical
counterpart to *o*, *n*, in another possible world has
different properties than does *o*. As long as there is a
possible world in which there is a object-bundle, *n*, that is
non-identical counterpart to *o* and differs from *o*
with respect to its properties, then *o* could have differed in
just that way.



Simons suggests a very different approach. He proposes to replace
unstructured bundles with "nucleus theory." An object
consists of an inner core of essential tropes and an outer band of
accidental tropes, but no non-property substratum (1994).



In like manner, bundle theory has been charged with ruling out the
possibility of change in objects. The same bundle complex cannot be
composed of one set of property at one time, but a different set at a
different time. Concrete objects, on the other hand, can and do
change. In response, it has been suggested that this objection loses
its force if bundle theory is combined with a four-dimensionalist
account of object persistence (Casullo 1988; see also Ehring 2011).
For example, if ordinary objects are spacetime worms made up of
appropriately related instantaneous temporal parts that are themselves
complete bundles of compresent tropes, then change can be read as a
matter of having different temporal parts that differ in their
constitutive properties. Another response to the change-is-impossible
objection is to adopt Simons' "nucleus theory" in
place of traditional bundle theory. Nucleus theory seems to allow for
change in the outer band of accidental properties (Simons 1994).



Trope bundle theory has also been accused of giving rise to
Bradley-style vicious regress. According to trope bundle theory, for
an object o to exist, its tropes must be mutually compresent. However,
it would seem, for trope *t*1 to be compresent with
trope *t*2, they must be linked by a compresence
trope, say, *c*1. But, the existence of
*t*1, *t*2, and
*c*1 is insufficient to make it the case that
*t*1 and *t*2 are compresent since
these tropes could each be parts of different, non-overlapping
bundles. So for *t*1 and *t*2 to
be linked by compresence trope *c*1,
*c*1 must be compresent with *t*1
by way of a further compresence relation, say *c*2
(and with *t*2 by, say, *c*3) and
so on, giving rise to either a vicious, or at least, uneconomical
regress (Maurin 2010, 315). In response, one can try to break the link
between the relevant predicates and tropes. Oliver suggests that the
trope theorist should reject the assumption that there are any
relation-tropes corresponding to the predicate "....is
compresent with..." even though that predicate has some
true applications (1996, 37). This response might be indirectly
supported by reference to Lewis's claim that it is an impossible
task to give an analysis of all predications since any analysis will
bring into play a new predication, itself requiring analysis (Lewis
1983, 353).



A second, quite different response is modeled on Armstrong's
view that "instantiation" is a "tie," not a
relation (since "the thisness and nature are incapable of
existing apart"), and, hence, it is not subject to a relation
regress. (1978, 109). The idea is that the union of compresent tropes
is too intimate to speak of a relation between them since the tropes
of the same object could not have existed apart from each other. This
response, however, requires more than generic dependencies--for
example, that this specific mass trope requires the existence of some
solidity trope or other--since generic dependencies would not
guarantee that the specific mass and solidity tropes, say, in this
particular bundle could not have existed without being compresent.



Maurin provides an alternative response that grants the existence of
compresence relation-tropes, but denies the regress on the grounds
that relation-tropes, including compresence relations, necessitate the
existence of that which they relate. There is no need for further
compresence relations holding between a compresence-relation and its
terms (2002, 164). A fourth response, from Ehring, suggests that this
regress can be stopped once it is recognized that compresence is a
self-relating relation, a relation that can take itself as a relatum.
The supposed infinite regress for the bundle theorist involves an
unending series of compresence tropes, *c*1,
*c*2, ..., and *c**n*,
but the series, *c*1, *c*2,
..., and *c**n* is taken to be infinite
because it is assumed that each "additional" compresence
trope is *not* identical to the immediate preceding compresence
trope in the series. However, if compresence is a
"self-relating" relation, this assumption may be false
(2011).



Finally, there is an objection to the very notion of a trope and,
hence, to the foundations of trope primitivism (and, perhaps, to any
ontology that includes tropes). The idea is that if properties are
tropes, then exactly similar properties can be swapped across objects,
but there is no such possibility.



> 
> 
> If the redness of this rose is exactly similar to but numerically
> distinct from the redness of that rose, then the redness of this rose
> could have been the redness of that rose and vice versa. But this is
> not really a possibility and, thus, properties are not tropes.
> (Armstrong 1989, 131-132)
> 
> 
> 



A similar argument is based on the possibility of tropes swapping
positions in space (Campbell 1990, 71). "Property
swapping," it is claimed, is an unreal possibility since
property swapping would make no difference to the world. (Note that
the cross-object version of this objection cannot get off the ground
if tropes are not transferable between objects, a view that is found
in (Martin 1980), although Martin rejects trope primitivism since he
posits substrata in addition to tropes).



One response to the no-swapping objection, given by Campbell and
Labossiere, rejects the assumption that trope swaps would make no
difference to the world. For example, although the effects of
"swapped" situations would be exactly similar in nature,
those effects would differ in their causes (Campbell 1990, 72;
Labossiere 1993, 262). Schaffer, on the other hand, denies that the
trope theory is automatically committed to the possibility of trope
swapping (2001). If tropes are individuated by times/locations and a
counterpart theory of modality is right, then trope swapping is ruled
out:



> 
> 
> The redness which would be here has exactly the same inter- and
> intraworld resemblance relations as the redness which actually is
> here, and the same distance relations, and hence it is a better
> counterpart than the redness which would be *there*. (Schaffer
> 2001, 253).
> 
> 
> 



What is clear is that Williams' trope ontology remains at the
center of a vibrant and ongoing debate, counting as a serious option
among a small field of contenders. His brand of trope primitivism is
certainly of more than merely historical interest.



Williams has also left a wider, if less plainly manifest legacy as a
metaphysician. For a discussion of his influence on David Lewis, and
through him on later thinkers, see Fisher 2015.


## 6. The Ground of Induction



The Problem of Induction is the problem of vindicating as rational our
unavoidable need to generalize beyond our current evidence to
comparable cases that we have not yet observed, or that never will be
observed. Without inductive inferences of this kind, not only all
science but all meaningful conduct of everyday life is paralyzed.
Indeed, Williams prefaced his philosophical treatment with a
declaration of the evils of inductive skepticism in eroding rational
standards in general, even in politics:



> 
> 
> In the political sphere, the haphazard echoes of inductive skepticism
> which reach the liberal's ear deprive him of any rational right
> to champion liberalism, and account already as much as anything for
> the flabbiness of liberal resistance to dogmatic encroachments from
> the left or the right. (Williams 1947, pp 15-20)
> 
> 
> 



David Hume, in the eighteenth century, had shown that all such
inferences must involve risk: no matter how certain our premises, no
inductively reached conclusion can have the same degree of certainty.
Hume went further, and held that inductive reasoning provides no
rational support whatever for its conclusions. This is Hume's
famous inductive scepticism.



Williams was almost alone in his time in holding not only that the
problem does admit of a solution, but in presenting a novel solution
of his own. To do this, he needed to argue against Hume, and
Hume's twentieth century successors Bertrand Russell and Karl
Popper, who had declared the problem insoluble, and also against
contemporaries such as P. F. Strawson and Paul Edwards, who had
claimed that there was no real problem at all (Russell 1912, Chapter
6; Popper 1959; Edwards 1949, pp 141-163; Strawson 1952, pp
248-263). Williams tackled the problem head-on. In *The
Ground of Induction* (1947) he makes original use of results
already established in probability theory, whose significance for the
problem of induction he was the first to appreciate. He treats
inductive inference as a special case of the problem of validating
sampling techniques. Among any population, that is, any class of
similar items, there will be a definite proportion having any possible
characteristic. For example, among the population of penguins, 100%
will be birds, about 50% will be female, some 10%, perhaps, will be
Emperor penguins, some 35%, perhaps, will be more than seven years
old, and so on. This is the *complexion* of the population,
with regard to femaleness, or whatever.



Now in the pure mathematics of the relations between samples and
populations, Jacob Bernoulli had in the 18th century
established the remarkable fact that, for populations of any large
size whatever (say above 2500), the vast majority of samples of 2500
or more closely match in complexion the population from which they are
drawn. In the case of our penguins, for example, if the population
contains 35% aged 7 years or more, well over nine tenths of samples of
2500 penguins will contain between 33% and 37% aged 7 years or more.
This is a necessary, purely mathematical, fact. In the language of
statistics, the vast majority of reasonably sized samples are
*representative* of their population, that is, closely resemble
it in complexion.



Bernoulli's result enables us to infer from the complexion of a
population to the complexion of most reasonably-sized samples taken
from it, since the complexion of most samples closely resembles the
complexion of the population. Williams's originality was this:
he noticed that resemblance is symmetrical. If we can prove, as
Bernoulli did, that most samples resemble the population from which
they are drawn, then conversely the population's complexion
resembles the complexion of most of the samples.



This brings us to the problem of induction. What our observations of
the natural world provide us with can be regarded as samples from
larger populations. For example the penguins we have observed up to
this point provide us with a sample of the wider population of all
penguins, at all times, whether observed or not. What can we infer
about this wider population from the sample we have? That, in all
probability, the population's complexion is close to that of the
sample. We *may* of course, have an atypical sample before us.
But with samples of more than 2500, well over 90% represent the
population fairly closely, so the odds are against it.



Thus Williams assimilates the problem of induction to an application
of the statistical syllogism (also called direct inference or the
proportional syllogism). A standard syllogism concerns complexions of
100%, and has a determinate conclusion: if *all S are P*, and
this present item is an *S*, then it must be *P*. A
statistical syllogism deals with complexions of less than 100%, and
its conclusion is not definite but only probable: for example, if
*95% of S are P,* this present *S* is *probably
P*. Some logicians claim that the probability in question is
exactly 0.95, but Williams does not need to rely on that additional
claim. It is enough that the probability be high. Applying the
statistical syllogism to Williams' reversal of the Bernoulli
result, we have: 95% of reasonably-sized samples are closely
representative of their population, so the sample we have, provided
there are no grounds to think otherwise, is probably one of them. On
that basis, we are rationally entitled to infer that the population
probably closely matches the sample, whose complexion is known to us.
The inference is only probable. Induction cannot deliver certainty. In
any given case, it is abstractly possible that our sample may be a
misleading, unrepresentative one. But to expect an inference from the
observed to the unobserved to yield certainty is to expect the
impossible.


## 7. Objections to Williams' proposed solution to the problem of induction



Williams' treatment of induction created quite a stir when it
appeared, but attracted criticisms of varying power from commentators
wedded to more defeatist attitudes, and was eclipsed by the Popperian
strategy of replacing an epistemology of confirmation with one that
focused on refutation. It thus exercised less influence than it
deserves. It is a closely reasoned, deductively argued defense of the
rationality of inductive inference, well meriting continued
attention.



Its reliance on *a priori* reasoning (as opposed to any
contingent principles such as the "uniformity of nature"
or the action of laws of nature) means that it should hold in all
possible worlds. Williams' argument would thus be easily
defeated by the exhibition of a possible world in which inductive
reasoning did not work (in the sense of mostly yielding false
conclusions). However, critics of Williams have not offered such a
possible non-inductive world. Chaotic worlds are not non-inductive (as
the induction from chaos to more chaos is correct in them), while a
world with an anti-inductive demon, who falsifies, say, most of the
inductive inferences I make, is not clearly non-inductive either
(since although most of *my* inductions have false conclusions,
it does not follow that most inductions in general have false
conclusions).



Marc Lange (2011) does however propose a counterexample, arising from
the "purely formal" nature of Williams' argument.
Should it not apply equally to "grue" as to
"green"? Objects are grue if they are green up to some
future point in time, and blue thereafter. The problem is to show that
our sample, to date, of green things is not a sample of things
actually grue. (Williams's own views on the grue problem are
given in his "How Reality is Reasonable" in (2018)).



Stove (1986, 131-144) argued that his more specific version of
Williams's argument (see below) was not subject to the
objection, and that the failure of induction in the case of
'grue' showed that inductive logic was not purely
formal--but then neither was deductive logic.



Other criticisms arise from the suggestion that the proportional
syllogism in general is not a justified form of inference without some
assumption of randomness. Any proportional syllogism (with exact
numbers) is of the form


* The proportion of *F*s that are *G* is
*r*.
* *a* is *F*.
* So, the probability that a is *G* is *r*.



(Or, if we let *B* be the proposition that the proportion of
*F*s that are *G* is *r*, and we let
*p*(*h* | *e*) be the conditional
probability of hypothesis *h* on evidence *e*, then we
can express the above proportional syllogism in the language of
probability as follows: *p*(*Ga* | *Fa*
& *B*) = *r*.)



Do we not need to assume that *a* is chosen
"randomly", in the sense that all *F*s have an
equal chance of being chosen? Otherwise, how do we know that
*a* is not chosen with some bias, which would make its
probability of being *G* different from *r*?



Defenders of the proportional syllogism (McGrew 2003; Campbell and
Franklin 2004) argue that no assumption of randomness is needed. Any
information about bias would indeed change the probability, but that
is a trivial fact about any argument. An argument infers from given
premises to a given conclusion; a different argument, with a different
force, moves from some other (additional) premises to that conclusion.
Given just that the vast majority of airline flights land safely, I
can have rational confidence that my flight will land safely, even
though there are any number of other possible premises (such as that I
have just seen the wheels fall off) that would change the probability
if I added them to the argument. The fact that the probability of the
conclusion on *other* evidence would be different is no reason
to change the probability assessment on the given evidence.



Similar reasoning applies to any other property that *a* may
have (or, in the case of the Williams argument, that the sample may
have), such as having been observed or being in the past. If there is
some positive reason to think that property relevant to the conclusion
(that *a* is *G*), that reason needs to be explained; if
not, there is no reason to believe it affects the argument and the
original probability given by the proportional syllogism stands.



Serious criticisms specific to Williams's argument have been
based on claims that the proportional syllogism, though correct in
general, has been misapplied by Williams. Any proportional
syllogism,


* The proportion of *F*s that are *G* is
*r*.
* *a* is *F*.
* So, the probability that *a* is *G* is
*r*,



is subject to the objection that, in the case at hand, there is
actually further information about the *F*s that is relevant to
the conclusion *Ga*. For example, in


* The proportion of candidates who will be appointed to the board of
Albert Smith Corp is 10%.
* Albert Smith Jr is a candidate.
* So, the probability that Albert Smith Jr will be appointed is
10%,



it is arguable that information is hidden in the proper names that is
favorably relevant to the younger Smith's chance of success. The
question then is whether the same could happen with the proportional
syllogism in Williams' argument for induction. Maher (1996)
argues that there is such a problem. In this version of
Williams' argument:


* The proportion of large samples whose complexion approximately
matches the population is over 95%.
* *S* is a large sample.
* So, the probability that the complexion of *S* matches the
population is over 95%,



is there, typically, any further information about the sample
*S* that is relevant to whether it matches the population? The
potentially relevant information one has is the proportion in
*S* of the attribute to be predicted (blue, or whatever it
might be). Can that be relevant to matching?



It certainly can be relevant. For example, if the proportion of blue
items in the sample is 100%, it suggests that the population
proportion is close to 100%; that is positively relevant to matching
since samples of near-homogeneous populations are more likely to
match. (For example, if the population proportion is 100%,
*all* samples match the population.) Conversely, fewer samples
match the population when the population proportion is near one
half.



David Stove (1986), in defending Williams' argument, proposed to
avoid this problem by taking a more particular case of the argument
which would not be subject to the objection. Stove proposed:



> 
> 
> If *F* is the class of ravens, *G* the class of black
> things, *S* a sample of 3020 ravens, *r* = 0.95, and
> 'match' means having proportion within 3% of the
> population proportion, then it is evident that 'The proportion
> of *F*s that are *G*s is *x*' is not
> substantially unfavorable to matching (for all *x*).
> 
> 
> 



For even in the worst case, when the proportion of black things is one
half, it is still true that the vast majority of samples match the
population (Stove 1986, 66-75).



Stove's reply emphasizes how strong the mathematical truth about
matching of samples is: it is not merely that, for any population size
and proportion, one can find a sample size and degree of match such
that samples of that size mostly match; it is the much stronger result
that one can fix the sample size and degree of match beforehand,
without needing knowledge of the population size and proportion. For
example, the vast majority of 3020-size samples match to within
3%--irrespective of the population size (provided of course that
it is larger than 3020) and the proportion of objects with the
characteristic under investigation..



Maher also objects that the attribute itself, such as
"blue", might be *a priori* relevant to its
proportion in the sample and hence to matching. For example, if
"blue" is one of a large range of possible colors, then
*a priori* it is unlikely that an individual is blue and
unlikely that a sample will have many blue items. As with any Bayesian
reasoning, a prior probability close to zero (or one) requires a lot
of evidence to overcome; in this case, we would conclude that a sample
with a high proportion of blue items was most likely a coincidence,
and the posterior probability of the sample matching the population
would still not be high.



Scott Campbell (2001), in reply to Maher, argues that priors do not
dominate observations in the way Maher suggests. By analogy, suppose
that, while blindfolded, I throw a dart at a dartboard. I am told that
99 of the 100 spots on the board are the same color, and that there
are 145 choices of color for the spots. After throwing, I observe just
the spot I have hit and find it is blue. Then (in the absence of
further information), the chance is very high that almost all the
other spots are blue. The prior improbability of blue does not prevent
that. In the same way, the fact that the vast majority of samples
match the population gives good reason to suppose that the observed
sample does too, irrespective of any prior information of the kind
Maher advances.



Williams' defense of induction thus has resources to supply
answers to the criticisms that have been made of it. It remains the
most objectivist and ambitious justification of induction.