sexta-feira, 11 de maio de 2018

# TROPE THEORY RIGHTLY UNDERSTOOD

This is a DRAFT for the book PHILOSOPHICAL SEMANTICS to be published in 2018 by CAMBRIDGE SCHOLARS PUBLISHING 

Trope Theory Rightly Understood



Any possible world and, of course, this one, is completely constituted by its tropes.
—D. C. Williams

‘Could you show me some properties (qualities, characteristics…) of the things around us?’ Asked in this way, any normal person would surely point to a few nearby objects, naming their properties (qualities, characteristics…), such as the redness of this sofa, the hardness of that wall, the property of a shirt of being made of cotton… Many traditional philosophers, however, would say that these things cannot really be properties in the true sense of the word. For in this true sense, properties are abstract entities, universals accessible only to our intellect, not to our senses.
     This comparison suggests that the ontological starting point of traditional realism, particularly in the form of Platonism is opposed to the ontological starting point of ordinary people and even of our own modest common sense. Common sense begins by considering as prototypical examples of properties the spatiotemporal properties directly given to us in perceptual experience, only afterward considering those properties that are in some way derived from perceptual experience. The contemporary ontology that shares this commonsense view is called trope theory. Properties are for trope theorists spatiotemporally located entities called ‘concretized properties,’ ‘particularized qualities,’ ‘individual accidents,’ ‘quality-bytes,’ ‘abstract particulars’ or simply ‘tropes.’ According to trope theory, universal properties should follow from the ontological building blocks that are the spatiotemporally particularized properties or p-properties called tropes, and not the other way round.
     One reason for the importance of trope theory resides in the fact that since the rise of nominalism already in the Middle Ages, this might turn out to be the only really groundbreaking advance in ontology. Although the concept of trope as a particularized property has been known at least since Aristotle, only in 1953 did an American philosopher named D. C. Williams conceive of the bold idea to assign tropes metaphysical place of pride as the universe’s only fundamental ontological building-blocks.[1] His central aims were to use the notion of tropes to solve (or dissolve) the traditional problem of universals and to explain the nature of concrete particulars. In fact, pure trope-theory is a one-category ontology. Because of this, my hunch is that the theory of tropes is so revolutionarily simple in its fundamentals that it could produce an upheaval in ontology similar to that caused by the introduction of new physicalist theories to solve the mind-body problem in the second half of the twentieth century.
     In what follows, instead of doing the hard work of discussing different versions of trope theory, I will take the easier and more direct route of outlining the view that from my assumed methodological perspective seems more plausible, namely, a methodology that gives primacy to established knowledge (Ch. II, sec. 5).
1. Introducing Tropes
First, what are tropes? Although tropes (or properties) considered as simple cannot be intrinsically defined, they can in my view be clearly characterized as follows:

Tropes (Df): are properties localizable in space and enduring in time, regardless of their vagueness.

As such, these particularizing properties can be identified as the empirical designata of predicative expressions. The most obvious tropes – fundamental from a genetic-epistemological perspective – are those accessed by direct perceptual experience, like qualities. Examples of quality-tropes are the yellowness of this sofa, the heat of that stove, the smell of a particular daisy at a certain time and the song sung by a particular blue whale to attract a female. Other tropes would be the red color of the Golden Gate Bridge, its weight, hardness, form, height above sea level… These are all that we could call external (third-personally accessible) physical tropes. However, tropes can also be internal; they can be psychological properties, like a feeling of pain, sorrow, love, and pleasure and even a whole mind, insofar as not understood as a thinking substance (Williams 1953 I: 17). They can be partly internal and partly external like a belief, emotion, purpose, love affair, act of contrition or expression of impudence (called by Williams mixed tropes); and they can be events like a smile, sneeze, election, cold snap, triangle, circle, shape or bodily form (Williams 1953 II: 171 f.). We can prove the reality of tropes by considering that they can be removed, like the color of a cloth (Campbell 1998: 352) and can be objects of selective attention (Loux 2002: 86): gazing at the ocean, one can alternately concentrate on its color-tropes, the form-tropes of its waves or their sound-tropes.
     Simple tropes appear in combination with other tropes, and some conglomerates of different kinds of tropes are highly complex and multifarious. This is the case of biological properties like that of a certain cat being a mammal. This is the case with some psychological properties like Céline’s idiosyncratic personality. And this is also the case with social properties like that of India being a democratic country. If I say that India is a democracy, ‘being a democracy’ is a property-trope dependent on the country, the individual entity called ‘India,’ though this trope is surely a very complex one. And there are complex and diversified cultural properties like the socio-cultural traits emphasized in the ancient Spartan state. In all these cases, the tropes are in various ways spatiotemporally located, and they are properly referred to with predicative expressions (they are at least logically repeatable).
     Tropes contrast with what I prefer to call individuals: these are things that are seen as unique and non-repeatable and are referred to by singular terms like ‘this daisy,’ ‘that blue whale,’ ‘the Golden Gate Bridge,’ ‘Socrates’ and ‘India.’ In the standard case they are what can be called ‘material objects’ and, as we will see, nothing but compositions of tropes. However, some compositions of tropes are individuals without being material objects. This is the case of a rainbow or of a cloud in the sky. And there are individuals that are constituted by the absence of tropes, for instance, a particular shadow.
     Moreover, there are complex tropes like a performance of Beethoven’s Fifth Symphony, which are homogeneous in the sense that they consist of only one kind of trope comprising a great diversity of sound-tropes. They can be designated by means of a predicative expression, as in the statement ‘The orchestra performed the Fifth Symphony.’ Considering that the Fifth Symphony can be performed over and over by many different orchestras at different times and in different places, it is clear that it is better classified as a repeatable complex homogeneous trope and not as an additional individual; moreover, it is dependent on an orchestra (an individual) to be performed, while individuals (e.g., the Vienna Philharmonic Orchestra) are relatively independent in their uniqueness.
     Finally, one can consider the existence of indirectly accessible, derivative tropes. This would be the case of fundamental physical forces: in order to have a clue about them, we need to begin by experiencing our more modest perceptible quality tropes. The fact that these forces are indirectly accessible is only a contingent one (some birds navigate using the earth’s magnetic field). This is how things are, even if from the perspective of physical science the origins could be reversed.
     As particularized properties, tropes have identity conditions. As an attempt to clarify this, I propose an ontological condition (a) followed by a linguistic indicator (b):

Tropes are identified:
(a) By their spatiotemporal existence to the extent that they display sufficient continuity over space and time and are amenable to certain direct or (often) indirect experiential ways and conditions of access, and
(b) By being linguistically designated by predicative expressions of singular statements whose nominal terms refer to individuals.

So understood, tropes contrast mainly with individuals such as material objects referred to by means of nominative expressions, particularly proper names.
     The linguistic indicator (b) has a guiding function: as spatiotemporally located properties most properly linked with individuals, tropes are usually designated by means of predicative expressions like ‘…is red.’ This isn’t always so straightforward: in statements beginning with demonstratives like ‘This is a daisy’ or ‘There is the Matterhorn’ it is preferable to take the nominal terms (indexicals) ‘this’ and ‘there’ as referring to spatiotemporal places, as localizing rules for the identification of the individuals daisy and mountain, which justifies the non-application of the linguistic requirement (b) to its supposed predicates, whose owners are in fact individuals and not tropes. Better to analyze these sentences relationally as ‘<This place is> where <a daisy> is located,’ and ‘<That is the place> where <the Matterhorn> is located,’ sentences in which the predicate designates the property-trope ‘x where y is located.’
     Regarding the ontological condition (a), I have something more to say. Consider the following example: the pair of shoes I am wearing is brown. The right shoe’s property of being brown can be seen as a trope, since it displays continuity and is located on my right shoe, and the left shoe’s property of being brown can be seen as another trope since it displays continuity and is located on my left shoe. Because these shoes have different spatial locations, we can regard them as displaying two tropes of the color brown. And because of the relatively homogeneous continuity of the right shoe’s color, this color can be said to be only one trope – a (located) property. The smoothness of my left shoe is also a trope that has the same location, homogeneity and maybe even the same duration as its brown color. Does this mean that this brown and this smooth are the same trope? No, since they are accessed through different forms of perception and under different conditions. This is the most natural way to identify properties, although there is much more to be considered on this point.
     To the further question of how much my left shoe’s trope of brown can be subdivided, one possible answer would be: into as many unities as we can distinguish. However, since depending on perceptual distance and acuity we can distinguish different amounts, this does not seem to be very helpful (Cf. Campbell 1990: 136-7). Because of this, and again drawing on common sense and natural language, it seems better to say that the unity of a trope – which we can rightly call a property – is usually better established by the natural limits of its spatiotemporal continuity and what is considered as being the same, disregarding its possible divisions. Thus, for instance, the whiteness of a wall would be a myriad of tropes if any visible point of whiteness were considered a trope; but considering a trope of whiteness to be a continuous whole, we are not only being economical but also following the usual linguistic practice. Indeed, we would rather say that this wall ‘has the property of being white’ than that it has a myriad of punctiform properties of whiteness. The size and form of the wall, on the other hand, also deserve to be called tropes, since they can be spatiotemporally located. A related question concerns the duration of tropes. How long will my left shoe’s brown trope last? A reasonable answer is: it will probably survive no longer than my left shoe. A trope lasts as long as it remains essentially the same, maintaining its spatial continuity.
     I mention all these seemingly trivial things because hasty considerations can easily give rise to attempts to discredit identity conditions for tropes, for example, by pushing precision beyond its contextually reasonable bounds. The vagueness of our identity conditions for tropes is as much a direct consequence of the way we experience the world as of the way the world is supposed to be under our assumed practices, enabling us to define a conceptual system with a suitable degree of precision. Moreover, many complex tropes (e.g., socio-historical tropes) can be highly dispersed in space and time. This makes their boundaries still less determinate.
     Since tropes are any spatiotemporally situated properties, they are also existent particulars. This is because existence – as we will see later in this book – can be seen as the effective applicability of a predicative ascription rule to at least one thing. By asserting existence we assume a need to spatiotemporally locate a trope or a set of tropes. Moreover, tropes are said to have proper existence, though I must disagree with Keith Campbell’s view that their existence is independent (1998: 353). He gives as examples the blue of the sky and the colors of the rainbow. However, the blue sky above must be identified against the landscape below, and the colors of the rainbow are intransitively related one another and form an arc against a certain background, and all these things are, according with our definition, tropes. Therefore, I would prefer to say that tropes have rather an interdependent existence.
     Are spatial forms and duration in time tropes? Well, these things cannot be found without being associated with tropes, a shape with a color, a volume with a weight, a duration in time with the continued existence of some tropes or clusters of tropes... Campbell, disagreeing with Williams, did not consider forms as tropes because of their dependence upon other tropes (Campbell 1998: 360-361).[2] However, as I noted above, his examples are inadequate: tropes have to be always to some extent interdependently considered. If we hold this view together with our definition of a trope as any spatiotemporally localizable property, we can see forms and durations as limitations in space and time respectively. They would arise from limitations imposed by standard quality-tropes. Hence, it seems that we could view forms and durations as kinds of tropes. Let us call them limiting tropes.
     Another question is whether relations are tropes. Since relations are spatiotemporally located, though often only in a rather vague way, and since relations are designated by means of dyadic or polyadic predicative expressions, it seems clear that relations are tropes, even if their existence is subsidiary to the existence of their relata. Although there are different kinds of relations with different strengths, particularly important is the causal relation. For instance: ‘The throwing of a stone broke the window.’ As Williams and Campbell have noted, a causal relation should be analyzed as a relation between tropes (Campbell 1990, Ch. 5.15). The relational predicate ‘x causes y’ is not between the objects stone and window but between a cause, such as throwing (a stone), and an effect, such as breaking (a window). Cause and effect are here located events associated with different individuals, which can be represented by means of statements (‘The stone was thrown’, followed by ‘The window was broken’), being all made of tropes according to our identity conditions. It is doubtful if a causal relation is internal. We define an internal relation as a relation that exists as a consequence of the existence of their relata, so that if the relation does not exist the relata will be different. But a trope-event x will only be a cause of y if the right contextual conditions are added, what must be extrinsic to the relata. A straightforward case of an internal relation, however, is that of strict similarity between two tropes, which I understand as a relation of qualitative identity. For instance, ‘The blue of this ocean is like the blue of the sky above it.’ Once these two blues are given, the similarity follows. Moreover, it may not be as easy to admit, but the relation of strict similarity is also not just predicatively designated; it is also spatiotemporally located: it is in-between and not out there. Therefore, it should also be classified as a relational trope, even if subsidiary to its relata. Like causality, strict similarity is in this way a dependent trope.
     One objection to the idea that relations are tropes could be that if relations are tropes then the relational trope and its relata must be related by a new relational trope, and so on ad infinitum. We can argue against this objection by first noticing that the same problem comes up again in a stronger form in the case of one-place predications. In other words, if a refers to an individual and b refers to another individual, and there is a relation aRb so that this relation produces an infinite regress, then the same should be true of a one-place predication of the form Fa, as in the statement ‘The Earth is round.’ That is, we would need a relation R to relate the object referred to by the nominal term ‘the Earth’ and the trope of roundness designated by the predicate ‘…is round,’ symbolizing it as FRa. Being related to the relata F and a, this relation R would require two new relations ‘FR1RR2a’, and so on ad infinitum. But this seems preposterous! The strangeness becomes clearer when we replace the symbols with words and see that we fail to give a sense to these new relations. It does not make sense to say ‘The Earth is related to its roundness,’ instead of saying ‘The Earth is round.’ Hence, it is more reasonable to see the link between subject and predicate as what some philosophers called a ‘non-relational tie’ (Strawson 1959, part II, Searle 1969: 113), something like the invisible link of a chain, to use Wittgenstein’s metaphor. They are not tropes but pseudo-additions in a literal sense of the word. Thus, we do not need to postulate FRa in order to explain Fa.[3] And if this seems obviously true of the monadic links represented by singular predicative sentences, there is no reason not to extend this result to the relations said to produce a regress. After all, relations must be seen as linked with their relata in the same way as non-relational properties are linked with their objects. To see this clearly, consider the following example: (i) ‘Socrates is a friend of Plato.’ Since friendship is a relation, one would be entitled to replace sentence (i) with (ii): ‘Socrates has a relation of friendship with Plato,’ which still says the same thing by being interpreted as specifying that the kind of relation is that of friendship. But if we try to go ahead, deriving from (ii) the sentence (iii) ‘Socrates relates himself to his relation of friendship, which is itself related to Plato,’ which is an instantiation of aR1RR2b, we again wind up speaking nonsense.
2. Tropes and Universals
The theory of tropes is important because it promises a parsimonious solution for at least two perennial ontological problems: the problem of universals and the problem of concrete individuals.
     I begin with the problem of universals. Linguistically stated, this problem consists in the question of how we can apply a single general term to many different individuals; ontologically stated, it consists in the question of how it is possible that many different individuals can share the same property. Traditional realist philosophers supposed that the only possible solution to this problem is to postulate that a general term refers to a universal understood as an abstract entity (existing ante rem or even in rebus, according to the ‘Platonist’ or the semi-Platonist ‘Aristotelian’ versions of realism respectively) that in some obscure way can be instantiated in many individuals.
     For the Platonic realist, we can think and see that this rose and that strawberry are red because they instantiate or exemplify the idea (universal) of redness (‘red-in-itself’). For Plato, the world was real only insofar as it instantiates ideas. However, this view was never satisfactorily rescued from unsolvable problems.[4] After all, universal properties must be non-empirical abstract objects accessible only to the intellect. This duplicates the world: we have our empirical world and a world with an infinite number of abstract entities whose intelligibility is highly questionable and for which we have no identity criteria. Moreover, the realist is left with unsolvable problems of how to explain the supposedly causal relation between these abstract entities and our minds. Finally, as we already noted, if you ask a layman where properties are, he will answer by pointing to the blue of the sky, the hardness of a table, the softness of jelly… and not to an otherworldly Platonic realm.
     This contrast leads us to the suspicion that only a disposition originating from the pressure of some mystical or quasi-mystical belief could lead to a committed Platonic solution. It exemplifies the consolation of what a Nietzschean philosopher would call a ‘world of beyond’ (Überwelt). Philosophers are particularly susceptible to this sort of thinking; they are to some extent unworldly creatures, and it may be a temptation to adjust their minds to see properties in such an idealized way.
     The Aristotelian solution was an attempt to bring the Platonic archetypal ideas down from their heaven (the topos hyperuranion) to the concreteness of the earth. However, this seems an incoherent middle way. For him universals exist in the visible world so that if there were no world there would be no universals. Now it seems completely impossible to understand how the universal can preserve its unity if its only reality consists in being multiply instantiated by entities belonging to the real world.[5]
     Dialectically opposed to realism was nominalism. According to the philosopher Roscelin (XI century), called the originator of nominalism, a universal is a mere flatus vocis (emission of a sound), since a general term has no designatum. This and similar counter-intuitive views were justly nicknamed ‘ostrich nominalism.’ A more sophisticated form is the contemporary set-nominalism: a predicative expression designates the set of individuals to which it applies. This is less counter-intuitive than strange. One problem with this view is that predicative expressions with the same extension – like ‘…animals with kidneys’ and ‘…animals with hearts’ – must mean the same thing since they form the same set. One alternative is to suggest that a predicative expression designates the sets of individuals to which the predicative expression applies in all possible worlds (Lewis, 2001: 51). This liberates us from the objection of identities of extensions of different general terms because there are possible worlds where some animals with kidneys have no hearts and vice versa… However, it also leads to implausibility, like accepting the reality of merely possible worlds and assuming the existence of unicorns.
     As the solution to the problem of universals by means of realism is too obscure and by means of nominalism is too implausible, trope-theory appears to be the safest lifeboat. To solve the problem of universals by appealing to tropes, we need to introduce the idea of similarity, or resemblance or likeness between tropes, which possibly could be understood as a kind of relational trope. Philosophers like D. C. Williams (1953 I: 9) and Keith Campbell (1998: 358) saw universals as classes or sets of precisely similar tropes.
     Thus, the universal ‘red’ refers to the set of all tropes of red, which are unified by the fact that these tropes all have the internal relation of being precisely similar one with the other. For Williams, when we say, ‘This rose is red,’ we mean that this rose has a red trope that belongs to the set of red tropes; and when we say ‘Red is a color,’ we mean that the set of all tropes of red (universal-R) is included in the set of all tropes of color (universal-C).
     However, there are problems with this view. First, there is a problem with the notion of set or class; if we see a set as an abstract object, it seems that we are abandoning the great advantage of trope theory. Second, there is a problem with size: a set can become larger or smaller; but a universal cannot change its size, for it has no size. It does not help to appeal to an open set, since even open sets also have their sizes, though unknown and also variable… Third, we can develop objections of regress concerning precise similarities based on Russell’s criticism of Berkeley’s and Hume’s nominalism. According to Russell, two patches of the same color have a relation of color-likeness that seems to be a universal or abstract idea… It is true that a nominalist can decide to consider applying the same analysis to color-likeness, considering it a particular. But then he will face the following problem:
We may take a standard particular case of colour-likeness, and say that anything else is to be called a colour-likeness if it is exactly like our standard case. It is obvious, however, that such a process leads to an endless regress: we explain the likeness of two terms as consisting in the likeness which their likeness bears to the likeness of two other terms, and such a regress is plainly vicious. (Russell 1994: 111-112)
To offer a more detailed explanation, I begin by assuming that likenesses or strict similarities are also tropes, as I have assumed before. It must be a case of what I prefer to call ‘strict similarity,’ because mere similarity or resemblance or likeness lacks transitivity: If trope T1 is only similar to trope T2, and T2 is only similar to T3, then it is possible that T3 is not similar to T1. The solution is to appeal to strict similarity understood as the same as qualitative identity, which is the case of an identity between differently spatiotemporally located things (differing from numerical identity, which is the identity of a thing with itself). Qualitative identity does not need to be perfect: our cars are both yellow, but your car’s color is faded. We must, however, establish a corrigible limit to the differences. Corrigible differences are usually found within the range of a concept’s applicability (e.g., turquoise blue and cobalt blue are both called blue) insofar as we have a correction criterion (in the case of blue it is what we identify as corresponding to wavelengths between 450 and 495 nanometers).
     Now, according to the kind of reasoning adopted by Russell, in order to construct the set of strictly similar tropes, we need to know that a first trope of identity is like a second trope of identity. But how do we know this? Well, since it cannot be known by appealing to the abstract idea of identity, it must be by appealing to another trope of qualitative or strict similarity. Since the same question can be posed regarding the strict similarities between these strictly similar tropes, it seems clear that this leads to a kind of pyramidal infinite regress.
     Russell would see this regress as plainly vicious. Even if this is not the case, I see this as a pseudo-problem born from the wrong solution. And the reason why I think so is because this seems not to be the real way in which we conceive universality. In fact, we can overcome Russell’s objection in a much easier way, simply by dispensing with his fixation on classes. The much better way I propose to build universals only from particulars is inspired by just the kind of treatment that particularist philosophers like Berkeley and Hume gave to ideas or impressions in order to ensure their unity. In its plain form, the insight is clearly expressed by George Berkeley in the following passage:

...an idea, that if considered in itself is private, becomes general by being made to represent or be in the place of all other particular ideas of the same type. ... a private line becomes general by being made a sign, so that the name line, which considered absolutely is private, to be a sign is made general.’ (1710, Introduction, sec. 12)[6]

Following a similar line of thought, we can symbolize as T* any trope that we wish to use as a pattern or model. Then we can define the universal in a disjunctive way as:

Universal (Df.) = A given trope T* or… any further trope T that is strictly similar to T*.

To explain this definition better, we must note that used as a model trope, T* in no way needs to remain always the same trope. On the contrary, one can choose any trope T strictly similar to a chosen T* and then use it as a new T* in order to make new comparisons. Each speaker is free to use his own T* as a model to build the universal. Moreover, what we normally know of T* in real life is only some recollection in our memory.[7]
     Accepting this definition, we do not need to appeal to sets or classes of strictly similar tropes or some mereological sum to explain universality since the definiens covers any trope strictly similar to T*. The problem of size disappears, since how many tropes are qualitatively identical to T* is a matter of indifference. When a person utters the sentence ‘This rose is red,’ he means that this rose has a trope of red Tr1 that is identical to some trope of red Tr* taken as a pattern (recalled in the person’s memory) or any other strictly similar trope. When he utters the sentence, ‘Red is a color,’ he means that any trope strictly similar to Tr* is also a Tc* or any other trope strictly similar to Tc*, as the wider pattern of the color trope. Finally, Russell’s problem also disappears, since we don’t need to compare one identity trope with another, but only the tropes T1, T2,… Tn individually with some chosen trope T*. Instead of possibly generating an infinite pyramidal regress, the sequence of our comparisions will take the form T1 = T*, T2 = T*… Tn = T*, without any need to consider the totality of T’s. In other words, as long as all we need to do to get a universal is the ability to compare any given trope with our chosen model trope T*, there is no need to compare similarities with similarities, thereby generating further similarities of similarities. Russell’s problem does not arise because our particularist definition makes universals mere potentialities instead of actualities.
     Furthermore, we can also construct the universal ‘strict similarity’ requiring that some chosen trope Ts* (a model trope of strict similarity) is taken as a standard and allowing it to be compared with any other trope of strict similarity strictly similar to Ts*. Our sequency of comparisions would be Ts1 = Ts*, Ts2 = Ts*… Tsn = Ts*, where Ts* can remain the same while other tropes of strict similarity are changing. This means that we have second-order strict similarity tropes referred to by the third-order strict similarity signs ‘=’ occurring between Ts1 and Ts*, between Ts2 and Ts*, and so on – call them Tss1, Tss2, etc. Thus, in order to make reference to the universal composed of these strict similarities of strict similarities, we need to appeal to a standard trope of strict similarity of strict similarity Tss*, and it is easy to predict that we could in principle refer to an indefinite number of higher-order strict similarity tropes by taking this ascending path.
     Would this be a vicious regress? I don’t think so. For nothing prevents us from stopping where we wish, insofar as we see no reason for going further – a point that can be understood in terms of explanatory demand. If we do not see any explanatory advantage in going further, we can simply stop where we choose, which is not possible with vicious infinite regresses. A similar consequence results from Platonic realism. As H. H. Price noted (1953, Ch. 1): the idea of ideas constantly used in Plato’s doctrine of ideas is a second-order idea. He also needs to consider the idea of the idea of ideas in his dialogues. But then he stops, not because he must, but simply because there is usually no explanatory advantage in going further. In the same way, we can find no explanatory soundness in going beyond the trope of precise similarity between two other tropes.[8]
     Finally, it is worth noting that strict similarity is not a trope like others. To begin with, it is what we have called a dependent trope: it depends on the existence of things considered alike. Color-similarity, for instance, is an internal relation depending on the existence of color-tropes. Campbell suggested that strict similarity is only a supervenient pseudo-addition that does not add any being to what already exists (1990: 37).
     Nonetheless, if we take seriously our identifying condition for tropes, the fact that we are dealing with an internal relation does not make strict similarity or even higher-order strict similarities quasi-tropes or a non-tropes, as some theorists think. As already noted, the identity condition for the reality of similarities as tropes is satisfied, even if distinguishing strict similarity from other more primary kinds of tropes. If an essential condition for the existence of a (simple or not, homogeneous or not, external or not) trope is its spatiotemporal localizability, established by the application of its ascriptive predicative expression, we can argue that similarity is also spatiotemporal, though in a broad way. For example: when I consider the strict similarity between the colors of two shoes I see in a store window, this likeness would be somewhere in this place, which may include myself, but not in a distant place. My house and the Taj Mahal have a color-likeness: both are white. Nevertheless, I can swear that this likeness is situated on the planet Earth and not on the surface of the sun. Moreover, if my house or the Taj Mahal disappear, the color likeness also disappears, which means that the similarity also exists in time. Furthermore, when someone considers the similarity between the form of our Milky Way galaxy and the form of the Andromeda galaxy, this coarse-grained qualitative identity must have to do with the total distance between them, which is still localizable. But as great as this distance may be, it remains insignificant if compared with the immensity of the cosmos.
     Problems for the theory of tropes do not stop here. What about other spatial relations? For example, the Golden Gate Bridge is (on the average) 67 m. above sea level. Certainly, this spatial relation is there and can even be measured. And this relation is located in space and time, enduring as long as the bridge exists and the average sea level does not change. This spatial relation isn’t internal, insofar as it is independent of the relata only. This makes easier to classify it as a trope, but it is not because of this that it satisfies our identifying condition for tropes as spatiotemporally localizable entities.
     But what about space and time in themselves? Normally we admit that only tropes and space-time exist. Even in realist ontologies, a separate existence of space and time was never seriously questioned. However, could space-time in some way consist of tropes or something derived from tropes? Imagine that all the world’s objects and properties disappeared. Would space and time remain? We have the intuitive tendency to answer in the negative. However, according to a Newtonian theory of absolute time and space, the answer should be in the affirmative: space and time would be individual-like entities. Space would be like a great container with material objects within it and would not cease to exist even if all the matter and energy ceased to exist and disappeared. On the other hand, according to the relational view originated from Leibniz, space could be constructed by means of relations, and this conception can easily be extended to include time. In the latter case, space and time could not exist in themselves, because being constructed of relations they require the existence of the relata (not necessarily material things). Both answers have always been controversial, and the discussion has been intensified by contemporary physics.
     The attempt to explain absolute space and time in terms of tropes seems to be condemned to failure. If space as a whole is a trope, it cannot be located in space, and the same holds for time, contradicting our definition of tropes. However, it seems there is a good chance of explaining space and time relationally in terms of tropes if we begin with a modest commonsense approach. It seems clear that in primeval times people understood space by thinking of relations such as above, below, in front of, behind, inside and outside. We can localize an object x as being twice as far above object y as is object z. Originally time would also be relationally understood, by means of relations like earlier, present (simultaneous with the act of observation) and later. One can say that event x occurred three times as long ago as event y in relation to event z. Moreover, in order to make measurements, the plain man appealed to regularities as patterns: a foot to measure distances in feet, a day to measure periods of days… And one could with the aid of these regularities calculate speeds in order to conclude, for example, that Pheidippides could run more than 160,000 Greek Steps in one day before dying of exhaustion. This is how our usual concepts of space and time worked and still works in everyday life, where they do not demand a further explanation. The main point here is that all these relations should be tropes since they are also spatiotemporally located. However, since quality-tropes and material objects are also spatiotemporally located entities, it seems that we would end up in circularity: space and time would be defined as relations of spatiotemporally located property-tropes and objects as clusters of property tropes.
     The answer to the circularity objection in this modest commonsense approach is that space and time are constituted by a network of spatiotemporal relations among spatiotemporal entities that can be quantitatively compared. For instance, consider the following rough description of the Southern Cross against the horizon: star c is seen twice as far below the smaller star b than b below star a, while stars d and e are seen on opposite sides of b and (approximately) at the same distance from b as a is from b. With a similar approach, any particular spatiotemporal relation, for instance between a and b, could be located in the spatiotemporal network and because of this could be defined as a trope. And the same could be said of the individual star b as a spatiotemporally located cluster of tropes.
     Of course, it is an entirely open question how such a rough commonsensical view could be developed, extended and transformed in order to comprehend the sophisticated and often controversial theories of contemporary physics. However, nothing could be more distant from the truth than to commit the naïve mistake of believing that the above account is so primitive and superficial that it could effortlessly be dismissed based on the discoveries made by modern science.[9]
3. Tropes and Concrete Particulars
The second major problem is that of constructing concrete individuals by means of tropes. For D. C. Williams, a material object is a set or sum of different conjoined tropes (1953: 11 f.). The advantage of this view is that it enables us to abandon the old, obscure concept of substance understood as some hidden substratum of properties. For the trope theorist, the material object turns out to be a kind of artichoke consisting only of its leaves, which are tropes.
     The key-concept here is that of compresence (also called concurrence, togetherness, etc.), which can be understood as the sameness or near-sameness of the spatiotemporal location of tropes. The concept of compresence can easily be analyzed as composed of two other concepts: co-location and co-temporality. The co-location of tropes is their joint location in space, leaving aside when each of them comes to be located. Thus, two persons who take turns sleeping in the same bed can be said to be co-located in this place. The co-temporality of tropes is their simultaneous existence during the same time interval. Thus, my friend Magda and I are co-temporal, though not co-located, since we are very distant in space. The compresence of tropes arises only when they are co-located and co-temporal.
     A naïve but instructive objection to the view according to which concrete objects are clusters of tropes is that if it is true, then all predication turns out to be tautological: the utterance ‘This chair is yellow’ would be tautological, because yellow is predicated of a subject that already has the trope yellow as a constituent (Loux 1998: 103). This objection is easy to refute. We just need to distinguish necessary from contingent tropes. As has been pointed out, a material object can be identified by means of an indexical added to a sortal predicate, as in the statement ‘This is a chair’ (Tugendhat 1983, Ch. 9).[10] Now, the necessary tropes are those typically specified in the definition of the sortal. Thus, ‘a chair’ is defined as a non-vehicular seat with a backrest, designed to be occupied by only one person at a time. The seat is constituted by one sub-cluster of tropes, the backrest by another, and the conditions that this complex object is non-vehicular and designed to be used by only one person at a time are constituted by dispositional tropes, variations and alternations of tropical relations that complete the definition. There are also contingent tropes, like those constituting the sub-clusters of armrests or four legs, since there are chairs without armrests and chairs without legs; and there are still more variable tropes associated with a chair, like its color, the relation to a certain person sitting on it, its distance from a table… The concept of a chair is one of an artifact. But we can consider natural kinds in a similar way. Gold is defined as an element with the atomic number 79, a dense, yellow, precious metal. However, its having a determinate atomic number is a necessary trope, though gold does not have to be yellow or even considered a precious metal, since these are contingent tropes.
     Peter Simons gave a helpful answer to the question of the nature of material objects by pointing out that they should not be seen as an unstructured cluster of compresent tropes. A material object is typically made up of a nuclear kernel of necessarily interdependent tropes giving a foundation to an accidental halo of contingent tropes. The halo-tropes can be replaced by tropes of other kinds, but the kernel-tropes cannot (they can be approximated to sortal predicates). A consequence of Simons’ view is that the halo-tropes are specifically founded on the kernel-tropes, while the kernel-tropes only generally found the halo-tropes (1994: 376 f.). Moreover, Simons accepts the possibility of variations: a concrete object formed only by kernel-tropes, etc.
     Here a much more precise definition seems to be simply impossible. Stones, for instance, are material objects that can be composed of very different materials, having few tropes to individualize the object-kind stone, with the exception of hardness, solidity, weight, volume, and color, all of them compresent. However, based on this cluster of properties, often combined with spatiotemporal determinations, we are already able to re-identify the stone as the same one.
     Unhelpfully, compresence and kernel-tropes are still not enough to define material particulars. Socrates’ wisdom is a dispositional property consisting of a very complex property-trope, as it seems. These tropes appear to have compresence, since they all seem to be located where Socrates is. Moreover, they could be individuated by a sortal predicate delimiting the spatiotemporal location of Socrates (‘There comes Socrates again with his inconvenient wisdom!’). Finally, they can have a kernel: the ‘peculiar core of the inconvenient Socratic wisdom.’ But it is not a material object, not even an individual, insofar as it is said to belong to the individual Socrates and others could in principle, at least, share strictly similar qualities of Socratic wisdom. A common rainbow is constituted by co-located and co-temporal tropes of colors and forms – the seven colors of the spectrum – jointed together in a structured kernel, but it is less than a material object. The holographic projection of a teacup also has a proper compresent set of colors and forms. They belong to its kernel as an individual. But despite having colors, spatial extension, and form, it is no material object.
     One strategy to deal with this problem is to add to the core of compresent tropes some tropes necessary for the identification of our typical material objects like:

 volume,
 form,
 hardness or solidity (measured by resistance to pressure),
 weight (depending on the presence of a gravitational field),
 mobility in space…

This already excludes the property of Socratic wisdom and individuals like the rainbow and the holographic projection. But liquids, although they are material substances, do not have a specific form or solidity, unlike a stone, a tree or a table. For example, water takes the form of its container, and additional water can be added to a given quantity of water, increasing its volume. In a frozen state or as water vapor it ceases to be liquid. Resistance to pressure can be lower or higher. The water in a glass is already a material entity and an individual, though not properly a material object, since it lacks definite form, is not solid and has only limited resistance to pressure. A cloud has a low level of materiality: its droplets have minimal resistance to pressure and it has no fixed and necessarily defined form. And what about supposed material entities like bacterias, viruses, atoms quarks or hypothetical super-strings?
     My final condition is based on the already discussed assumption that our commitment to modest common sense does not exclude science.[11] We can refine the idea of hardness or resistance to pressure by proposing that a necessary trope constitutive of the core of any physical object is a derived trope that physicists call inertial mass. In physics, the inertial mass of a body is broadly defined as its inertial resistance to acceleration when forces are applied to it (an idea accepted in both Newton’s and Einstein’s mechanics[12]). This seems to me the most pertinent characteristic of what we call matter. Energy also has mass, but it isn’t inertial mass.
     I conclude that in an inevitably vague characterization, having the expected inertial mass, some size… and compresence of its definitional tropes would be necessary for singling out a material object. This excludes electromagnetic, gravitational, weak and strong forces, which are better seen as tropes. However, one cannot generalize this result to any individual. Consider the cases of a cloud, a rainbow, and a shadow. Consider the case of a crowd or the British Empire. These individuals do not form a material object or a physical body. Unlike material objects, a crowd and the British Empire are composed of tropes that are at least partially grounded on material, not tightly connected physical entities. And a historical tropical event like the Battle of Hastings was a spatio-temporal tropical event, not a material object. They are all complex structures made up of tropes, including mental tropes like intentional states and depending on material entities to be spatiotemporally located, even if only in a vague way. Since these tropical entities are independent and unequal and identified by nominal terms, they are individuals (Ch. IV, sec. 7).
     A more technical difficulty arises from the alleged fact that the idea that particulars are clusters of tropes is vulnerable to a regression argument analogous to the third man argument used against the abstract objects assumed by a Platonist ontological view. Thus, suppose that a concrete particular were constituted only by the tropes T1, T2, and T3. Since the relation of concurrence could not be an abstract entity, it must be a trope. Call this relation Tc. In this case, it seems that we need a new concurrence for T1, T2, T3, and Tc, which will be Tc’, and so on infinitely (Daily 1997: 158).
     My proposal to answer this objection takes a form similar to what realist philosophers have applied in defense of their own abstract properties. Compresence is made up of co-location plus co-temporality, which are spatiotemporal delimitations that remind us of the already considered cases of form and duration. They are all dependent relational tropes that must be considered sui generis, behaving somewhat like Platonic ideas with their resistance to self-predication. In other words: although you can meaningfully say that this red is red, and even that this triangle is triangular, you cannot meaningfully say that a concurrence is concurrent. Concurrence is a sui generis non-self-predicating limiting trope. Strict similarity is also a sui generis non-self-predicating dependent relational trope because one cannot say of the strict similarity between T1 and T2 that it is strictly similar without rising questions like: ‘what would strict similarity be similar to?’
4. Formal Tropes
What should we say about formal entities like natural numbers? Numbers are often seen as Platonic or semi-Platonic universals. And they would not be tropes since they do not seem to be spatiotemporal. However, this isn’t so uncontroversial! Much of our empirical world is made up of countable things. Would the number 3 exist if the world did not exist? Though this is an odd question, the tendency is to answer in the negative. For an empiricist like Locke, the number would be a primary quality (a trope), together with solidity, extension, figure, motion or rest, which are accessed by diverse senses and should remain the same independently of the perceiver (1690, Book II, Ch. VIII). Indeed, I can perceive one, two, even six things at a glance and these seem to be spatiotemporally located tropes; and some savants are able to perceive hundreds of things at a glance. To use an example borrowed from Penelope Maddy, it seems that the ten fingers of my two hands are in some way here (1990: 87). It seems that even a thousand grains of wheat scattered in the wind remain spatially and temporally located, though in a diffuse way. And if the insufferable rock band called ‘The Fevers’ flies from São Paulo to Rio de Janeiro, it seems that the number of their members has also moved. However, it is important to note that these trope-numbers are dependent on countable entities of our choice.
     One can associate this dependency with Frege’s account of numbers as properties of concepts, since as he has taught us, things to be counted must be first conceptualized. The question ‘How many?’ only makes sense if followed by a conceptual expression. For instance, if the concept is of the fingers of my hands, they are ten, but if the concept is of my hands, they are only two. The property of the concept of those grains of wheat scattered in the wind is that there are a thousand. And the movable property of being five is a property of the concept of The Fevers. Moreover, as Frege famously wrote, the attribution of existence is the negation of the number zero (1892, sec. 54).
     So it seems that the concepts of number and existence are related. In fact, one can suggest that the property of existing and the property of being a number are higher-order tropical-properties because, like tropes, they are in a vague way spatiotemporally located: this black spot on the carpet exists here and now and not somewhere outside in a remote time. And it seems plausible that when I say ‘This is my one and only nose,’ ‘these are my two hands,’ ‘these are my ten fingers,’ the number one I am applying is located where my nose is, the number two is where my hands are, and the same with my ten fingers. The naïve error would be only to confuse these ethereal tropes with those qualities primarily constitutive of the nose, the hands, and the fingers. Indeed, numbers, as much as the existence of things, do not seem to be in outer space or in ancient times or in the solely intelligible realm of abstract ideas.
     These considerations seem to be valid for applied arithmetic, insofar as numbers are first used to count empirical objects. After all, we learn numbers by counting material things: ‘There are two apples and one pear in the basket, totaling three pieces of fruit.’ In this case, the ascription rule of the predicate ‘…fruit in the basket’ was applied to three distinct objects, attributing physical existence to each of them and showing in the process of counting that the rule has the higher-order trope-property of being applicable three times in an additive way.
     In the view defended in this book (See Ch. IV) a concept is a rule, which means that the attribution of existence is here the second-order property (or trope) of a dispositional first-order conceptual rule (always understood as a trope) of being satisfied by at least one thing. And in a similar way, an applied natural number would be the second-order property (or trope) of a dispositional first-order conceptual rule (or trope) of being satisfied by means of an idealized counting procedure, where counting originally results from the distinguishable applications of a first order conceptual rule to things like material objects or events or qualities attributing existence to them n times...[13]
     Using Fregean devices it is easy to formalize this suggestion using only countable tropical applications of (tropical) concepts and the (tropical) concept of existence. The affirmation of the number 0 is the negation of existence.[14] Thus, using V in place of the conceptual expression ‘moons of Venus,’ we can symbolize the idea that there are 0 moons of Venus as ~Ǝx (Vx), saying that the conceptual rule expressed by V isn’t applicable at all. Using E to symbolize the conceptual expression ‘moons of Earth,’ we can symbolize the idea that there is 1 moon of Earth as Ǝx [Ex & (y) (Ey → y = x)]. Here E is applied only once. And using M to symbolize ‘moons of Mars,’ we can symbolize the idea that there are 2 moons of Mars as Ǝx [(Mx) (My) & (x ≠ y) & (z) (Mz → (z = x) v (z = y))]. Here M is applied twice. It is the application of a tropical ascription rule for two and only two moons of Mars.
     Above we considered first order conceptual tropes together with higher order existence tropes and applied numbers as higher order numerical counting-tropes. However, I think we can separate or abstract the numerical trope from these other concepts. We can do this by representing these tropes of countability by means of localizable sets. Thus, I propose that we can represent the 0 in ‘the moons of Venus’ as the located non-countability (non-applicability) of a concept symbolized by ~a. Instead of the 1 of ‘the earth’s moons’ we can speak of a set that has as its only member a located higher-order applicability trope or {a}. Instead of the 2 of ‘the Mars’ moons,’ we can speak of a set that has two located higher-order numerical tropes as members, as follows: {a, {a}}. In this way we can represent an applied number 3 by the localizable set {a, {a}, {{a}}} and so on. Note that this 3 has the right complexity by containing {a, {a}} (=2) and {a} (=1). But the fundamental point here is that we are explaining applied numbers by means of spatiotemporally localizable sets of countability-tropes and by convention the null set. The set of Mars moons numerical tropes is spatiotemporally located in our solar system and not in the Andromeda galaxy or in the origin of time. And such sets are not Platonic or sub-Platonic entities!
     At this point, one can object that we have until now explained only natural numbers applicable to things. One could, however, instead point out that what really matters is the number of abstract arithmetic, the universal independent of its satisfaction by countable material objects or events. The suggested construction has indeed this limitation since it represents only one number among many identical numbers. The natural number 3, formulated as {a, {a}, {{a}}}, is a triad and not what is common to all triads, namely, the abstract universal three, the three-in-itself. Indeed, the only way to represent what is common to all triads seems to be the appeal to a Russellian set of all sets of the same kind, which has its own shortcomings like the axiom of infinitude, overpopulating our world with an infinite number of objects.
     However, I think that in the same way as we have constructed universal quality-tropes without appealing to abstract sets, we can also construct universal number-tropes without appealing to abstract sets. I think we can derive the universal concept of number, the number-in-itself, from our spatiotemporally located tropes of counting. As we have seen above, an applied number can be understood as a trope, since it is spatiotemporally localizable as a second-order property of a potential conceptual rule resulting from its at least ideally countable applications. Consequently, in order to account for the universal as a set of equinumerous sets of applied numbers, we can appeal again to our disjunctive model.
     In this case, for instance, it is conceivable that the number 2 in itself would be a disjunction between a located dispositional higher order trope-set of countable applications used as a model (e.g., the number 2 in the statement ‘I have 2 hands’) or any other strictly similar (equinumerous) located set of countable tropical applicabilities. Now, in order to get the number 2 as the ‘abstract universal,’ the ‘two-in-itself,’ all we need is to apply to the separated set of tropical applicabilities the same procedure we have applied to get universals from our usual quality-tropes. For instance:

Number 2 (Df.) = a located model set of tropes of countable applicabilities {a, {a}}*, or any further located set of tropes of countable applicabilities strictly similar (equinumerous) to {a, {a}}*.

In this sense, the number as a universal (or ‘abstract entity’) can be defined as:

The higher-order property of a conceptual rule of being a located set of tropes of (at least ideally) countable applicabilities taken as a model or of any higher-order located set of tropes of (at least ideally) countable applicabilities strictly similar (equinumerous) to the first one.

Note that such constructed universals remain empirical since they are higher-order disjunctive property-tropes that can be found scattered across our whole spatiotemporal world. This makes graspable why something abstract like mathematics applies to the empirical world.
     Assuming a definition like that, we neither stumble over controversial infinite sets of objects (as in Russell’s definition) or over pure sets (as in von Newmann’s and Zermelo’s definitions) nor remain unintentionally limited to particular instances or directly committed to any differentiating concrete feature (as in naïve empiricist views). The conclusion is that even the abstract world of arithmetic (hence, mathematics) is made up of some sort of thin higher-order tropes. Such tropes, like some others, would be situated at the peak of a building whose originating genetic-epistemic foundations are our more feasible perceptually given quality-tropes, so that numerical tropes that can be univocally named in this way can also be seen as dispersed over the world and able to be meta-predicatively designated. Finally, I would not be surprised if even logical properties were susceptible to similar treatment!
     Now one could object: aren’t such formal properties not too thin to be tropes? A dependent trope like a conceptual rule might be a thin trope. But a trope that is dependent on other possible dependent tropes will be still thinner so that formal tropes are simply too thin to be real tropes! However, isn’t it a foolish prejudice to reject tropical properties only because of their thinness? There is no quasi-trope.
5. Conclusion
In this section, we have seen how trope theory can turn Platonic realism upside down. Much of what I have written here is speculative, still requiring a great deal of additional work and refinement. In this short space, I could do no more than offer a sketch of what seems the most consequent and plausible way to deal with the one-category ontology chosen to play a central role in this book.






[1] This groundbreaking work was D. C. Williams’ paper ‘The Elements of Being’ (1953), because he was the first to propose constructing the whole world using only tropes as elementary building blocks. The most relevant attempt at a systematic development of trope theory remains in my view Keith Campbell’s book, Abstract Particulars (1990). Since then, the discussion devoted to this view has grown steadily. For access to the literature, see Anna-Sofia Maurin’s, 2013.
[2] In his book on tropes, Campbell writes, ‘because boundaries in space need to be drawn rather than revealed it is perhaps best to view individual specimens of each of the shapes as quasi-tropes rather than as genuine tropes.’ (1990: 91) This argument is not forceful since a conventionally charged intromission of epistemic subjects is inevitable in any conceptual application.
[3] In Russian, there is no proper verb for the copula. One uses expressions like ‘Me nice’, ‘You beautiful’… Thus, it seems that Russian speakers are less susceptible to such worries.
[4] Plato was the first to see some main difficulties of the doctrine in the first part of his dialogue Parmenides. Others were added by Aristotle in the Metaphysics (book VII) and by later critics.
[5] Although traditionally labeled ‘Aristotelian’, this is the most simplistic interpretation. More sophisticated interpretations tend to see Aristotle as identifying his forms (ideas) as ‘this so-and-so,’ the species building the substantial form or essence of the individual (to be distinguished from its matter). According to medieval interpreters, such a form cannot really be a universal; consequently, it is a work of the intellect to abstract the universal from the particular, so that it exists only post rem. (Copleston 1993, vol. I: 306; see also Shields 2007, Ch. 6.6)
[6] See also the more sophisticated but also less clear view of David Hume (1738, Book I part 1, sec. VII).
[7] We can imagine circumstances in which people are unable to retain memories of the color-trope, but bring with them templates with patterns T* of this color-trope, so that they can compare these patterns with any trope they come across. Moreover, the templates can have the most varied shades of a single color, say, blue. They may call the possibilities that might result from their comparisons ‘the universal of a blue color-trope.’
[8] As Anna-Sofia Maurin remarks, in a vicious infinite regress a considered statement (trigger) is dependent on the subsequent steps, while in a virtuous infinite regress, the subsequent steps depend on the considered statement, which makes them unnecessary (2007).
[9] There is no prima facie reason to believe that a relational view cannot in principle be made compatible with general relativity theory.
[10] Tugendhat defines a sortal as a predicate that has criteria for the spatial delimitation of the object, allowing us to distinguish what does or does not belong to it.
[11] J. L. Austin objected that terms like ‘material object’, ‘material thing’ and ‘sense-data’ do not originally belong to our ordinary language (Austin: 1962). Against this, we can only repeat that there are gaps left unexpressed by ordinary language, later filled by new philosophical terms (See Ch. II, sec. 6 of this book; see also Grice 1989: 227).
[12] As is well-known, the reason why according to relativity theory a body cannot reach the speed of light is that at this speed its mass would become infinite, requiring infinite force to accelerate it.
[13] Of course, there are large numbers that are uncountable for us. But they remain at least ideally countable. And they can be seen as later extensions that can be calculated by means of symbolic manipulation alone.
[14] I argue for a higher-order view of existence in chapter IV, sec. 11 to 19.

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