1
TROPES, UNIVERSALS, AND MATERIAL
OBJECTS
Of many leaves, we can say that they are green, of many diverse living beings that
they are human, and of
many actions that they are just, even though they are very different from each
other. How can two very different things
share one only
property? What allows us to apply predicates like ‘…is
green’, ‘…is a human being’, and ‘…is just’ to many different things, in order
to generalize, to discover unity in the multiplicity – a capacity that seems so
fundamental to human understanding?
There are at least three kinds of answers to this
question: realism,
nominalism, and trope theory. In what follows I will very shortly
explain the gist of the first two ontological approaches, saying something more
extended in favor of the third approach.
Realisms
The realism of universals is the doctrine according to which we can
distinguish the same property in many different particulars (individual
things) simply because this property is a universal, that is, something
able to be in some way instantiated (exemplified) in those many
different particulars. This would be, for instance, the abstract property of redness.
All particulars that are recognized by us as red are red because they
instantiate the universal property of redness. The same with more complex
properties, like the property of being human, instantiated by any human
being,
or relational properties, like the property of being father,
instantiated by all fathers and their children. For realist philosophers, it is
by the multiple instantiations of these universals that we can say the same of
many, in sum: to grasp generality.
There are two basic kinds of
realism: platonic and aristotelian. The difference between them is that in
platonic realism universals, properties (ideas or forms) exist before
things, or, more precisely, in complete independence of the empirical
particulars, while in Aristotelian realism the properties (forms) exist in some
way in the things.
In platonic realism universals
are abstract entities understood as non-empirical (transcendent), unchangeable
and indestructible, existing outside space and time. The modern concept of instantiation
(exemplification) is explanatorily insufficient: how can empirical things like
green leaves exemplify something that is hovering outside space and time.
It is not pre-defined how many empirical particulars can instantiate one only
abstract universal property: maybe none, maybe only one, maybe an indefinite
number of them. Important is to note that since platonic universals are
abstract, they are independent of the existence of the empirical world: if our
world ceases to exist, the world of property-ideas will continue eternally existing;
Platonic universals are world-independent. The epistemological consequence of
this is that our experience with the world serves only to help us to remember
universals innately present in our minds: Plato is a rationalist philosopher.
There are many problems with platonic realism. The biggest problem, in my view,
is simply that of intelligibility. It seems impossible for our intellect
to grasp non-empirical entities outside space and time, like the red-in-itself,
the humanity-in-itself, the justice-in-itself, the paternity itself. We simply cannot conceive them. It
seems rather that we are fooled by the external clothes of our language in the
belief that we are conceiving something more than just the spelling of some
meaningful words.
In its Aristotelian version,
universals are not abstract non-spatiotemporal entities. They are empirical
entities existing in the material objects as if they a chocolate layer on
pieces of cake. This means that, differently from Platonism, the universals
only exist because the empirical world exists; they are world-dependent. The
epistemological consequence of this is that the universals must be abstracted
(separated) from the particulars by means of experience, which means that
Aristotle, in opposition to Plato, was a kind of empiricist. There are,
however, also serious problem with the Aristotelian solution: that one
empirical universal property simultaneously instantiates itself in many
different material objects without dividing itself sounds still more incoherent
than Plato’s abstract universal idea doing the same. How is it possible that
the redness of this red tomato is at the same time present in the redness of a
fire extinguisher and in a red dwarf star situated millions light years from
the earth that long ago does not exist anymore? How can the humanity be
actualized in Gandhi and in Socrates remaining one and the same humanity? So
understood the Aristotelian doctrine seems still more hopeless than that of
Plato.
Strangely enough, realism has had
a huge influence on the history of philosophy and still today has talented
defenders, especially among people with an aptitude for formal sciences and,
perhaps, with some mystical disposition.
The most revealing positive
arguments in favor of realism are perhaps those of a linguistic nature. One of
them is the so-called nominalization or abstract reference. This
is usually achieved when predicates are placed in the position of the subject
of a sentence, which seems to make them to actually refer to the universal.
Here are some examples of sentences containing nominalized predicates, which I
adapted from M. J. Loux:
A
1. Courage is a moral virtue.
2. Wisdom is the object of philosophical
life.
3. Suzi prefers red to blue.
The nominalizations ‘courage’, ‘wisdom’, ‘the red’, ‘the blue’ seem here to
name the respective universal ideas.
The usual reaction against
examples such as the ones mentioned above is that the grammatical form of our
sentences hides what should be really considered, namely, their logical
structure, which shows what really should be thought in the fully analyzed
sentences.
From this perspective, here is what the three sentences of Ain fact mean:
A*
1.
All
that is courageous has at least one moral virtue. Or: “For all x, if x
is courageous, then x possesses at least one moral virtue”. Or still: “(x)
(Cx) → Ex (Vx)”.
2.
If
one studies philosophy, one should seek to be wise. Or: “For any x, if x
studies philosophy, x should seek to be wise”. Or still: “(x) (Px
→ Wx)”
3.
If
all other properties of things considered make no difference in choice, Suzi
prefers red things to blue things. Or: “For all x and for all y,
if for s x and y are indifferent in other respects, if x
is red and y is blue, s prefers x to y”. Or still:
“(x) (y) (Isxy & (Rx & By → Psxy)”.
Logically analyzed the sentences are evidenced
as referring as lacking any reference to universals, and the terms that seemed
to refer as names of universals – courage, wisdom, red, blue – were all shifted
to a predicate position.
It was
much easier for Plato and Aristotle to be realists. Neither of them knew
anything of mathematical logic. Because of this it was easy for them to confuse
the grammatical clothes of our language with its logical structure. For them, a
sentence like “Socrates is wise” would have the same logical structure as
“Wisdom is valuable”. Consequently, it was easy for them to believe that the
subject ‘Wisdom’ should refer to an object in the same way as the subject
‘Socrates’ refers to a real object. As they could not find any object like
wisdom in the visible world, their solution was to invent an entity called
universal, which for Plato were an eternal abstract idea outside the visible world
and for Aristotle should be a form that would be shared under the many objects
belonging to the visible empirical world but remaining (according to some
interpretations) in some magical way one and the same.
The
realist can insist on his solution, giving further examples of sentences that
although do not incorporate nominalizations of predicates, seem to affirm
something about universal properties or attributes. Some examples:
B
1.
This
tomato and that fire extinguisher have the same red color.
2.
This
color has been exemplified many times,
3.
John
has the same character traits as Mary.
Nonetheless, we can always invent paraphrases
that eschew the supposed appeal to universals, for instance:
B*
1.
This
tomato and that fire extinguisher have colors that are identical to each
other.
2.
Colors
similar to the color of this object here (say, a tomato) were found in
many other objects.
3.
All
of John’s character traits and all of Mary’s character traits are similar.
In all these paraphrases, supposed statements about universals are
eliminated, remaining only references to objects of the empirical world: the
tomato, the extinguisher, material objects, John, Mary... Moreover, as we will see, the
ontology of tropes gives a nice justification to these paraphrases.
2. Nominalisms
It seems that Middle-Age philosophers were
sufficiently bored with the unsurmountable problems of realism to invent
something opposite of it, a doctrine called nominalism. The most fundamental contention
of nominalism is that universals do not exist; only individuals exist.
For nominalism there are no universal
properties because there is nothing that can be instantiated in more than one
particular. Only the individuals named by the subject term exist. A vivid
proposal of nominalism was given by the first known nominalist, the medieval
philosopher Roscelin (1050-1125 D.C.). According to him, only individuals
exist, and universals are only flatus vocis, that is, emissions of
sound. If I say, “This tomato is red”, the subject refers to an object, the
tomato, but the predicate ‘…is red’ is only a sound without correspondence with
anything in the world!
Later forms of nominalism were
more sophisticated and plausible. I will consider here only three forms of
nominalism: predicative, resemblance, and class-nominalism.
According to predicate nominalism,
if I say, “This tomato is red” and “this fire extinguisher is red”, it is a
property of the predicative term ‘red’ to apply to both objects. This seems
plainly insufficient because it does not explain why a predicate applies
to some objects but does not apply to another.
A less unsatisfactory form of
nominalism is the so-called resemblance nominalism. The fire extinguisher is
red because it resembles the red tomato, which can be used as a paradigm.
All objects that resemble the model or that resemble one another by being red
form the class of red particulars. There is, against this proposal, an obvious
objection: if a fire extinctor resembles a tomato, it is because there is
something by means of which they resemble one another, and this thing cannot be
other than the fact that both are red; but in this case, it seems that we are
using red again as designating a universal. Moreover, it seems that the only
way to explain the similarity is by appealing to the aspect by which
different particulars are similar, and this aspect, one can argue, either is a
universal or, as we will see later, a trope.
Still, a form of nominalism is
class-nominalism. According to class-nominalism, if I say, “this tomato is red”
and “that flag is red”, I can apply the predicate red to both objects because
they are both members of the same class of red objects. This is already
problematic, since it seems that for the class-nominalist red should be
identified with the class of red objects, which is counterintuitive enough.
Another problem with
class-nominalism is that of co-extensive properties. Consider the class of chordates
(animals with hearth) and the class of renates (animals with kidneys).
Since all animals with heart also have kidneys and vice-versa, we should say
that the property of having a heart is the same as the property of having a
kidney. But this is absurd.
David Lewis tried to circumvent
this problem embracing modal realism. He suggests that all possible worlds are
real, differing our world from the others by being not only real but also
actual. Consequently, the class to be considered by the class-nominalist must
be not only that belonging to our own world, but of any possible worlds. Since
there must be possible worlds where there are animals with hearth but not with
kidneys and vice-versa, the two classes do not need to be the same. In any
case, class-nominalism does not explain why we decide that one object, for
instance, a tomato, should belongs to the class of red particulars. If it is
because of it is redness, we fall back into realism; if it is because it has
the property of being red, we fall back into trope ontology.
One could object that I am not
considering the arguments in sufficient details. My answer is that the above
presented difficulties should be more than sufficient to convince philosophers
they are digging in the wrong place. More than thousand years of discussion
without any expectative of a concrete result should be sufficient to convince
reasonable persons that traditional (and much of the contemporary) ontology is
out of track. For this reason, I will pass to the next account, which seems to
me to be the only view able to save ontology from death by consumption.
3. Trope theory
According to the trope theory, first proposed by Donald Williams,
our whole world and even any possible world – as a world that we can conceive –
is totally constituted by its tropes. But what are tropes? For Williams, a
trope is an abstract particular in a very special sense, that is, as
a property that remains after we abstract (exclude) all other properties of a
singular object. For instance: I have a tomato: after I
abstract its weight, its softness, its size, its shape, its flavor… the only
thing that remains is its red color. This particularized red color is a trope!
It is no universal of redness, but a particularized property of
being red, present in space and time. I could do the same with the weight, the
softness, the size, etc. They are all tropes. Another way to explain tropes is
to say that they are nothing but spatiotemporally localizable properties,
which can be identified with our pre-philosophical use of the word ‘property’.
Thus, the red of a certain tomato, for example, is a trope; and the red of a
certain fire extinguisher is another trope. Both are real entities belonging to
the outside world. When we say that two objects, the tomato and the fire
extinguisher share the same property, this is just an equivocal way of saying
that they have tropes that are precisely similar to one another. When we
say that these two objects have the same color, this is also an
equivocal way of saying that they have tropes of colors that are precisely
similar, that is, they have a qualitative identity (of one thing with
another) instead of numerical identity (of one thing with itself).
The relationship of precise
similarity between tropes is internal to them and primitive, since it excludes
explanation or analysis: it is a fact of the world, to which we are so used
that we find it difficult to perceive. Our cognitive device makes us able to
identify such similarities, and because of this we can apply the same
predicative term to two or more tropes and say the same of many... The idea of
considering property-tropes as spatiotemporally localizable is, if we consider
more carefully, perfectly intuitive. They are the immediate objects of our
perceptual attention. They are also objects of selective attention: I can focus
my attention on the roar of the stormy sea, on the forms of its waves, on its
gray color… and when attentive to its gray color I am not thinking about the
gray color itself, but in that particular gray, in that tropes I am seeing now.
Furthermore, insofar as we see tropes as spatiotemporally localizable
properties, then not only sensible qualities can be classified as tropes, but
much more, in fact, anything with exception of particulars.
Consider the forces of nature: electromagnetic forces, as much as weak and
strong forces, are spatiotemporally localizable, even if only in a very diffuse
way. The curvature of space-time in the proximity of massive bodies
characteristic of a gravitational field is also a trope, since it can also be
seen as spatiotemporally localizable in a diffuse way. Also, events, like the
performance of a concert are tropes, or made up of tropes of sounds, since
spatiotemporally localizable. Psychological properties, like personalities, are
also spatiotemporally localizable in human brains. The spin of an electron is a
trope, since localizable, not less than the rotation of a galaxy.
Our paraphrases of B*1, B*2, and
B*3 become clarified if we present them in terms of tropical properties:
B**
1.
That
tomato and that fire extinguisher have color-properties that are precisely
similar to each other.
2.
Color-properties
precisely similar to the color-property of this object here (say, a
tomato) were found in many other objects.
3.
All
of John’s character trait properties and all of Mary’s character
trait-properties are precisely similar.
The theorization about tropes can be extended in explaining the nature of
individual objects such as stones, chairs, people, planets... A physical object
like the book you have in your hands is nothing more than a specially
organized bundle of tropes: tropes of shapes, white and black tropes,
tropes of solidity, flexibility, etc.
In this way trope theory also
provides an answer to the old metaphysical problem of the substance. For some
theories of the substance, such as Locke’s, the substance is a bare
substratum, an unknowable “we know not what”, which is a repository of
properties.
The difficulty is that postulating the existence of something that in principle
cannot be known seems to be something profoundly contradictory. How can we say
that something exists in the absence of an experience that allows us to infer
its existence? Locke believed that there was a reason for such inference in the
fact that we need to have subjects for the predication of properties, as we
cannot begin by predicating properties of the properties themselves.
However, trope theory makes this
postulate unnecessary. A concrete object, the book you have in your hands, is a
more or less organized bundle of tropes, and it is such a
combination of properties that serves as the subject of predicates. These
properties have the property of being compresent – a technical term that
means that they are as much co-spatially located as also co-temporally located.
In this way, we could think of a material object – to use a Wittgenstein
metaphor – as a kind of artichoke, whose leaves represent its properties. It
may seem that after we defoliate the artichoke something will be left: the
artichoke-in-itself, its substance. However, nothing, in
fact, will be left since the artichoke consists only of its leaves.
This is a point with which I
cannot agree fully. It seems to me that there is a trope common to all material
objects, something common to a book, a stone, a galaxy, and an electron, which
allows us to unify them as material objects, which is what physicians call rest
mass. In physics mass is the resistance to the acceleration, that all
bodies have, inclusive sub-atomic particles. But mass can increase with the
velocity of the body relative to the observer, reaching the infinite if the
object could reach the speed of light. Rest mass, on the other hand, is defined
as the inertial mass that an object has when it is at rest within an observational
system,
and we can see this as a complex trope, namely, as a property common to all
material objects, from subatomic particles to galaxies, distinguishing them
from energy since energy had no (or almost no) rest mass. If it is so, then
rest mass is also a trope, but an essential trope distinguishing material
objects. Hence, it seems reasonable to introduce the trope-property of rest
mas to unify all material objects, since it is also spatiotemporally
located.
It seems better than the resource to some shameless naked substances.
One objection against the idea
that material objects are associations of tropes is Cris Daly’s remark that if
a material object is made up, say, by the tropes T1, T2,
T3… Tn, and we unite them by means of a trope of compresence Tc,
we have a new trope, which will form a new bundle that would need a new
compresence trope Tc1 able to unite the whole, which would produce a
regression ad infinitum.
The answer to this objection is
not difficult. A
predicative singular sentence has the form Fa. There cannot be any gap between
the predicate F and the singular term a. If there were a gap, we would need a
relational property linking F with a so that we could write it as the (strange)
form FRa. But then we would have no reason to deny new relational properties
linking F and R and R and A, building the (still stranger) form FR1RR2a,
and so indefinitely. In other words, if there were intermediate links between subject and predicate, we would be
forced to stuff our world with an infinite number of intermediary links. It is
for a similar reason that Cris Daly’s argument does not work. To show this,
suppose that we call an object formed by T1, T2, and T3
with the name m. We could formalize the
sentence “The m has compresence” as Tcm. Or, if we wish to be more perspicuous, we can
formalize Tcm as “Tc{T1,
T2, T3}”. What these examples clearly show is that since
the compresence is a property that is predicated from the whole bundle of
tropes constituting the singular object, there cannot be any gap between them
and, consequently, no intermediary link like Tc1, Tc2…
should be required.
An objection against the idea that
bundles of tropes can be objects of predication is that in this case, every
predication would become tautological: to say that this ball is red becomes
redundant, because the subject of the predication, the ball, consists of a
combination of tropes that already includes red as a constituent.
The answer to this objection is in principle very easy: we must distinguish the
tropes that essentially constitute the identity of a physical object from the
tropes that adhere to the physical object in a contingent, inessential way.
Only a combination of tropes essential to the individuation of an object
functions as a subject of predication. That is why when we say, “This ball is
round”, we say something that is tautological: admitting that ‘This ball’ is
the subject of the predicate, we predict from it a trope that belongs
essentially to it; but the same does not happen when we say that the ball is
red or that it is a rubber ball.
This same distinction between the
tropes or combinations of tropes that essentially constitute a material object
and those tropes that are inessential to its constitution makes it possible to
respond to the objection that the trope theory does not allow understanding
processes of change in concrete particulars. They change without failing to remain
the same insofar as what changes is not the combination of tropes that
essentially constitute it, but the tropes that make it up contingently.
Let us now come back to our main
problem, namely, how trope theory could explain our apparent statements about
abstract universals. Philosophers such as D. C. Williams have suggested that in
the place of abstract universals property terms would represent sets of
tropes (precisely) similar to each other.
These sets of similar tropes could be called tropical universals
because, although they do not have the nature of a realist universal, they
preserve its function. For instance, the sentence “This tomato has a red color”
could be paraphrased as “This tomato-cluster of compresent tropes has a
red-trope that belongs to the class of red-tropes that belongs to the class of
color tropes”.
There are well-known difficulties
with this solution. One of them is that similarities must be similar to one
another. If precise similarities are tropes, by being similar two similarity
tropes generate a higher-order trope of similarity in a process that can be
repeated indefinitely producing an infinite regress of new similarity tropes.
Another problem is that as a class of tropes as the functional Ersatz
for the realist kind of universal must be able to increase and decrease in the
number of members, and it does not seem that universals are variable in size.
Moreover, sets themselves are considered by many abstract objects.
To circumvent these problems, I
propose translating the classic solution that empiricists like Berkeley and
Hume gave to the problem of universals in terms of tropes. As Berkeley noted,
if I use a blackboard to prove that the sum of the sides of a triangle is 1800,
I do not need resort to an abstract triangle to convince that this proof is
correct and can be generalized to any triangle.
Accordingly, the grasp of a what may be called the tropical universal
would demand an operation of the mind based on one only trope, say, a red trope
that I have in my mind. Now, to grasp the tropical universal redness,
all that I need to be able is to consider the trope of red that I have in my
mind or any other trope precisely similar to the first trope. More
formally, considering a trope T, in order to reach the tropical universal of T,
all that we need is to consider any trope T arbitrary chosen, which I take as a
model and call T*, and consider T* or any other trope that is precisely similar
to T*. Thus, we can define a tropical universal of T as follows:
A tropical universal of T (Df.) = an
arbitrarily chosen T* or any other trope precisely similar to T*.
That is, to conceive the universal of a trope, we must first be able to
represent this trope and then, be able to conceive of the possibility of any
trope precisely similar to that trope. In doing so we are already conceiving
the tropical universality through that trope.
We can figure out this simple
procedure in a clearer way setting it outside the mind. Imagine a student of
painting who is caring a model of an unusual color, say, amaranth red, with
him. He intends to verify if there are patches of amaranth red in the pictures
present in the room. He compares his piece of amaranth red with the others.
After some time, he has found a set of amaranth reds in the studio. Now, to
understand or grasp the universal in the sense I am proposing is nothing but to
be able to realize such a procedure, not behaviorally, like the student, but
mentally, appealing to memory. Nothing more is required. It is true that such
conceivable procedures (behavioral or mental) imply the existence of some class
of similar amaranth red tropes, a possibly with an immense number of members.
Moreover, it is also possible to generate classes of higher-order tropes of
precise similarity, and we really appeal to a higher-order precisely similarity
when we consider the concept of precise similarity. However, nothing of the
kind concerns neither the student nor we, when we identify the color of a
picture and say, “this color is amaranth red” in the sense in which we mean
“This color is (a property-instance of) amaranth red”, which assumes that we
have the capacity to compare qualitatively identical patches of amaranth red.
If we think in this way, we do not
need to be worried in having in our minds perhaps an infinite multitude of
patches of amaranth red as the referent class of the conceptual word in our
heads, a class to be called the universal amaranth red. In fact, this is an
error perpetuated by the logicians who have too hastily identified the
reference of a concept word with a class of objects or properties or tropes,
and it is the long tradition of identifying the referent of a conceptual word
with a class that made many philosophers go astray, including Williams, in his
attempt to identify the universal with a sum or class of precisely similar
tropes.
Another point concerns our ordinary
concept of property. The word ‘property’ has in my view two senses. In the
first sense, it means a trope, a spatiotemporally localizable property like the
red of the walls of the Kremlin. In this sense it means a trope, a so-called property-instance.
For example, if I am for the first time in the Red Square, point to the wall
and say: (i) “This wall is painted red”. Here I simply mean the property-trope
of being red that I am looking. In the second sense, however, I mean the
tropical universal. When I say to others, (ii) “The Kremlin is painted red”. I
typically mean something more general. I mean: (iii) “The Kremlin has a trope
of red T that is precisely similar with any trope T of red like the trope of
the red model T* that we have in our minds”. In this sense, I also speak about
the redness of the walls of the Kremlin. It is true that I could have in
mind: (iv) “The Kremlin has a trope of red T that belongs to the class
of tropes precisely similar to T”, since this can be deduced from (iii). But
since I do not even know this class, this information is of secondary
importance.
We can ask how to interpret more
abstract concepts in terms of tropes. Consider the concept of existence. We
cannot see the existence of a book, but we see that the book exists. One needs
at least one (warranted) applicability of a conceptual rule to assert
existence. Consider: “The Moon exists”. Calling Moon ‘M’ and the functor of
existence E, the logical analysis will be Ex (Mx) & (y) (My → y = x), which
means that the concept of Moon of the earth has the higher-order tropical
property of applicability to precisely one object.
Still, a problem would be that of
formal entities like numbers. For the trope theory, they should not be abstract
entities, but something constructed out of tropes. For applied arithmetic, this
is easy: the two horns of a caw are in some way there since I can see the caw
and count her horns. But what about abstract arithmetic? For instance, the
abstract number 2? Well, in this case we will need to handle numbers as
constructed tropical universals. We can define the number 2 as a set
of tropes of double (warranted) applicability (symbolized as ‘a’). The
definition of the constructed tropical universal of the number 2 can be stated
as follows:
The number 2 (Df) = a selected model of a
higher-order set-trope of located countable applicability {a, {a, a}}* or any
further higher-order set-trope of located countable applicability that is
strictly similar (equinumerous) to {a, {a, a}}*.
The great advantage of this approach – in the case it shows to be feasible
– is that it with a simple stroke solves the old problem of the applicability
of arithmetic to our physical world, since it also belongs to it.
Only the carpet tip of the trope
theory has been raised here and, if my amendments are correct, it would be
necessary to defend it in a much more careful and detailed way than it has been
done so far. The construction of an adequate theory of tropes is an open
undertaking that allows us to refrain from the appeal of concepts with
questionable intelligibility, particularly because it more correctly analyses
the concept of property as it is really used in our common language. As a
result, it would finally provide the true descriptive ontology instead of the
wishful-thinking ontologies provided by all previous ones.
The present distinction realism/nominalism is
sometimes confused with the distinction realism/idealism about the external
world. The first is an ontological distinction (i.e., relative to
what is, to what exist) about the most fundamental entities that constitute
reality; the second is an epistemological-metaphysical distinction about the
nature of what experience gives to us, realism stating that we know an external
world independent of the mind and distinct from it, while idealism states that
the outside world is in some way mental in nature.
See Gilbert Ryle’s
article “Systematic Misleading Expressions”, Meeting of the Aristotelian
Society, London 21/3/1936.
In its radical form of one category ontology the
theory of tropes was born with D. C. Williams’ article, “The Elements of Being
I”, Review of Metaphysics 7, 1953, pp. 3-18 and 171-92. A good selection
of papers from Williams, including his locus classicus article, can be
found in The Elements and Patterns of Being: Essays in Metaphysics, ed.
A. R. J. Fisher (Oxford: Oxford University Press 2018). Later attempts to
develop William’s ideas make them in my view unnecessarily weak, Keith
Campbell’s Abstract Particulars (Oxford: Oxford University Press 1990) inclusive. My
intent here is to preserve Williams radically empiricism and his genially simple
insight.
Daly: "Tropes", in D. H. Mellor and A. Oliver (eds.): Properties
(Oxford: Oxford University
Press) 1997,
p. 157.
See M. J. Loux: Metaphysics: a Contemporary
Introduction, London 1998, p. 103.
A more extended explanation can be found
in Costa,
“Natural Numbers as Tropes”, WAB Archives, v. 41, pp. 65-68, 2019.
