quinta-feira, 18 de maio de 2017


 Advanced draft for the book Philosophical Semantics to be published by CSP in 2017.

Appendix to Chapter III

Trope Theory and the Unbearable Lightness of Being OriginalEla provém da consideração de que na definição da existência do pensamento não entra em questão a mente singular que o tem, nem a pessoa na qual ele ocorre.

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…), e.g., the redness of this sofa, the hardness of that wall, this property of a shirt being made of cotton… Many traditional philosophers, however, would say that these things cannot really be properties in the most proper sense of the word. For in this strict 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 ontological realism is opposed to the ontological starting point of the common people and even of our own modest common sense. Common sense begins by considering as prototypical examples of properties the spatio-temporal properties directly given to us in perceptual experience, only afterwards considering those properties that are in some way derived from perceptual experience. The contemporary ontology that sustains this commonsensical view is called trope theory. Properties are for trope theorists spatio-temporally 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 spatio-temporally 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 development 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 the 1950s did an Australian philosopher named D. C. Williams have the bold idea of assigning tropes metaphysical pride of place as the only fundamental ontological building-blocks of the universe.[1] His central aims were to use the notion of trope 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 our methodological perspective – which gives primacy to established knowledge (modest commonsense plus science) – seems the most plausible.

1. Introducing Tropes
First, what are tropes? Although simple tropes are primitives and as such cannot be intrinsically defined, they can be elucidated as being properties individually located in space and enduring in time, whereby properties are here understood in the ordinary sense of p-properties. As such, p-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 smell of a particular daisy at a certain time and the snorting of a particular rhino trying 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 what 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, an emotion, a purpose, a love affair or an act of contrition or a piece of impudence (called by Williams mixed tropes); and they can be events like a smile, a sneeze, an election, a cold snap, triangles, circles, shapes, a 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 varied, as in the case of dispositional and psychophysical traits. This is the case of Socrates’ psychological character, of biological properties like that of a certain cat being a mammal, of social properties like that of India being a democratic country. Im all these cases they are in some way spatio-temporally located, even if dependent on concrete physical things.
   Tropes contrast with what I prefer to call individuals: things that are seen as unique, and are referred to by singular terms like ‘this daisy,’ ‘that rhino,’ ‘the Golden Gate Bridge,’ ‘Socrates’ and ‘India’. In the standard case they are 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, of a cloud in the sky. Moreover, not all tropes are individuals in the considered sense. There are complex tropes like a performance of Beethoven’s 5th Symphony, which consists of only one class of sound-tropes and can be designated by means of a predicative expression, as in the statement ‘The orchestra has performed the 5th Symphony.’ These tropes are complex and varied in kind. Finally, there are indirectly accessible derivative tropes. This is the case of the fundamental physical forces: in order to have a clue about them, we need to begin by experiencing our more modest perceptible quality tropes. This is so even if from a physical perspective we can ask whether our common tropes are not in a sense grounded on them (cf. Campbell 1990, Ch. 6).
   As particulars, tropes have identity conditions. As such, I propose an ontological condition (a) followed by a linguistic indicator (b):

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

So understood tropes contrast mainly with material objects referred to by means of nominative expressions, particularly proper names.
   The linguistic indicator (b) has a guiding function: as spatio-temporally located properties normally related to individuals, tropes are usually designated by means of predicative expressions. In statements beginning with demonstratives like ‘This is a daisy’ or ‘There is the Golden Gate Bridge’ it is preferable to take the nominal terms (indexicals) ‘there’ and ‘this’ as referring to spatio-temporal regions, as localizing rules for the identification of the daisy and the bridge, which justifies the non-application of the linguistic requirement (b) to its predicates, which are individuals and not tropes. As we already saw, the spatio-temporal location is often a complement to the individual’s identification.
   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 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 possibly even the same duration of its brown color. Does this mean that this brown and this smooth are the same trope? No, since they are different kinds of tropes, accessed through different perceptual ways and conditions.
   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 elucidating (cf. Campbell 1990: 136-7). Because of this, and again drawing on common sense and our ordinary language, it seems better to say that the unity of a trope – which we can rightly call a property – is better established by the natural limits of its spatio-temporal continuity, 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 ordinary language 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 spatio-temporally located. And the same could be said about the more complex color, size and forms of a human body. 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, while maintaining its spatial continuity.
   I mention all these 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, allowing the constitution of a conceptual system with a suitable amount of precision. Moreover, many complex tropes (e.g. social tropes) can be highly dispersed in space and time. This makes their boundaries still less determinate.
   Since tropes are any spatio-temporally situated properties, they are also existent particulars. 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 spatio-temporally locate a trope or a set of tropes. Moreover, tropes are said to have proper existence, though I must disagree with 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; but the blue of the sky depends on the existence of the atmosphere made up of gases, and the colors of the rainbow depend on the existence of droplets of water, both of them constituted by tropes. Tropes do not have an independent, but 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 continuous existence of some tropes or clusters of tropes... Keith Campbell, disagreeing with D. C. 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 interdependent existence. If we hold this view, 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 spatio-temporally located, though often only in a rather vague way, and since relations are designated by means of dyadic or polyadic predicative expressions, it seems that relations are tropes, even if their existence is subsidiary to the existence of their relata. There are different kinds of relations with different strengths and I cannot develop this point here. Particularly distinguished 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 (1990, Ch. 5.15). The relational predicate ‘x causes y’ is not between the objects stone and window, but between cause, as the throwing (of a stone), and the effect, as the breaking (of the window). Cause and effect are here events that can be designated by means of predicates (in ‘The stone was thrown’ and ‘The window was broken’), being therefore tropes according to our identity conditions. Moreover, we call the causal relation internal, since we define an internal relation as a relation that exists as a consequence of the existence of their relata, given adequate conditions. A clearer case of an internal relation is that of strict similarity between two tropes. 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. It may not be easy to admit, but strict similarity is also not just predicatively designated; it is also spatio-temporally 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, even if grounded on them.
   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.[3] I will argue against this objection by first noting 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 regression, 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 senseless! The lack of sense becomes clearer when we replace the symbols with words and see that we fail to give any sense to these new relations. It does not make sense to say ‘The Earth is related with its roundness,’ instead of saying ‘The Earth is round.’ Hence, it is better to see the link between subject and predicate as a ‘non-relational tie’ (Strawson 1959, part II, Searle 1969: 113), or as something like the invisible link of a chain, to use Wittgenstein’s metaphor. They are not tropes but pseudo-additions – now in the true sense of the word. Thus, my view is that we do not need to postulate FRa in order to explain Fa.[4] 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 friendship is a relation. 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 can we apply the same general term to many different individuals; and ontologically stated, it consists in the question of how it is possible that many different individuals can share the same property. Traditional realist philosophers suggested 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 ‘Platonic’ or the ‘Aristotelian’ version 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. This answer was never satisfactorily rescued from unsolvable difficulties.[5] 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 relation between these abstract entities and our cognitive minds. Finally, 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 a Platonic realm.
   This contrast leads us to the suspicion that only a disposition originating from the ideological pressure of some mystical belief could lead to a committed Platonic solution. It exemplifies the consolation of what a Nietzschean philosopher would call a ‘world of beyond’ (the Überwelt). Philosophers are particularly susceptible to this; they are unworldly creatures and it may be a true temptation for them to set their minds to see properties in such an idealized way.
   The Aristotelian solution is an attempt to bring Platonic ideas from the heaven of ideas (topos hyperuranios) to the concretness of the earth. However, this seems an impossible middle way. For him universals exist in the visible world, so that if there were no world there would be no universal. Now it seems completely impossible to understand how the universal can preserve its unity if its only reality is in being multiply instantiated by entities belonging to the real world.[6]
   Dialectically opposed to realism is 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. Similar views are counter-intuitive, being justly nicknamed ‘ostrich nominalism.’ A more sophisticated form is class-nominalism: a predicative expression designates the class 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. One alternative is to suggest that a predicative expression designates the classes of individuals to which the predicative expression applies in all possible worlds (Lewis, 2001: 51). This liberates us from a strong identity of extensions of different general terms because there are possible worlds where some animals with kidneys have no hearts and vice versa, etc. 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 conceivably 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 class 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 class of red tropes; and when we say ‘Red is a color,’ we mean that the class of all tropes of red (universal-R) is included in the class of all tropes of color (universal-C).
   However, there are problems with this view. First, there is a problem with the notion of class; if we see a class as an abstract object, it seems that we are abandoning the great advantage of trope theory. Second, there is a problem with size: a class can become larger or smaller; but a universal cannot change its size, for it has no size. 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 color-likeness, and say that anything else is to be called a color-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 argued before. It must be 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 in the case an identity between differently located things (differing from numerical identity as 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 (e.g., wavelengths between 450 and 495 nanometers).
   Now, according to the kind of reasoning suggested by Russell, in order to construct the class 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 by appealing to the abstract idea of identity, it must be by appealing to another trope of qualitative identity 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, such a multiplication of tropes of likeness or strict similarity seems overwhelming to our finite minds. I believe, however, that we can overcome this difficulty very easily, dispensing with classes. Concerning this, we are 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.[7] According to this view, we can symbolize as T* any trope that is used as a pattern or model. With the help of this approach, we can define the universal in a disjunctive way as:

Universal (Df) = Any 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 pattern trope, T* in no way needs to remain always the same. On the contrary, one can choose any trope strictly similar to a chosen T* and then use it as T* in order to make new comparisons. Moreover, what we normally know of T* is some recollection in our memory.[8]
   Accepting this definition, we do not need to use sets 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 the definition does not require us to ask how many tropes are qualitatively identical to T*. 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 (as retained in the person’s memory) or any other similar trope. When he utters the sentence, ‘Red is a color,’ he means that any trope strictly similar to Tr* is also a Tc* (a color trope) or anything similar that in a looser way is strictly similar to Tc*, as the wider paradigm of a 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 schema will typically take the form {T1 = T*, T2 = T*… Tn = T*}. In other words, as long as all we need to do to get a universal is to compare any trope with a chosen model trope T*, there is no need to compare similarities with similarities generating similarities of similarities. Russell’s problem would not arise because our definition makes universals potentialities instead of actualities.
   Furthermore, we can also construct the universal ‘strict similarity’ requiring that some chosen trope Ts* (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 schema will be: {Ts1 = Ts*, Ts2 = Ts*… Tsn = Ts*}, where Ts* can always remain the same. This means that we have second-order strict similarity tropes referred to by the strict similarity signs 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 can refer to an infinite number of higher-order strict similarity tropes in this way.
   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 is to 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. A similar consequence results from Platonic realism. As H. H. Price noted (1953 Ch. 1): the idea of the ideas constantly used in Plato’s doctrine of ideas is a second-order idea. But Plato stops with the idea of the ideas, not because he must, but simply because there is no explanatory advantage in going further, considering, for instance, the idea of the idea of the ideas. In the same way, we can find no explanatory soundness in going beyond the trope of precise similarities among first-order tropes.[9]
   Finally, it is worth noticing that strict similarity is not a trope like others. To begin with, it is what one could call 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 colors. Campbell suggested that strict similarity is only a supervenient pseudo-addition that does not add any being to what already exists (1990: 37). However, if we take seriously our identifying condition for tropes, the fact of being an internal relation does not make strict similarity a quasi-trope or a non-trope. 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 the condition for the existence of a (simple or complex) trope is its spatio-temporal location, established by the application of its denoting predicative expression, we can argue that similarity is also spatio-temporal, 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, the likeness would be somewhere in this place, which may include myself, but not in a distant place. My home 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 home or the Taj Mahal disappears, the color likeness also disappears, which means that the similarity also exists in time. On the other hand, when someone considers similarities 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 located. As great as this distance may be, it remains ridiculously minuscule 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. Even if this spatial relation is internal, depending on the existence of its relata, it can be classified as a trope, since it satisfies our identifying condition for tropes as spatio-temporally 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 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 Newton’s theory of absolute time and space, the answer should be in the affirmative: Newton saw space and time as 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 disappeared. On the other hand, according to the relational view defended by Leibniz, space could be constructed by means of relations, which can easily be extended to time. In the latter case, space and time could not exist in themselves, because being constructed of relations they require the existence of material objects. Both answers have always been controversial, and the discussion has been intensified by contemporary physics.
   It seems there is a chance of explaining space and time relationally in terms of tropes, if we begin with a modest commonsensical approach. It is clear that people began to consider space by thinking of relations such as above, under, in front of, behind. We can localize an object x as being twice as far above object y as is object z.’ Time could be defined relationally, by means of relations like earlier, present (simultaneous with the act of observation) and later. One can say that event x occurred three times later that event y in relation to event z.’ Moreover, in order to make measurements, the ordinary man appeals to regularities as patterns: a foot to measure distances in feet, a day to measure periods of days… These are our naïve concepts of space and time. All these relations should be tropes, since they are spatio-temporally located. However, since quality-tropes and material objects are spatio-temporally located entities, it seems that we would end in circularity: space and time would be defined as relations of spatio-temporally located properties.
   The answer to the circularity objection in this very modest approach could be that space and time are constituted by a network of relations among entities that can be quantitatively compared. For instance, consider the following rough description of the Southern Cross: star c is seen twice as far below the smaller star b than 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 spatial relation could be located in the spatial network and because of this be defined as a trope. Likewise, we could locate the terms of these relations as tropes or clusters of tropes (the same holds for time-relations).
   Nonetheless, it is an entirely open question how such rough intuitive notions could be developed and extended in order to comprehend the sophisticated and often controversial theories of contemporary physics.

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 and obscure concept of substance understood as some hidden substratum of properties. For the trope theorist, the material object turns out to be like an artichoke consisting only of its leaves, which are the tropes.
   The key-concept here is that of compresence (also called concurrence, togetherness, etc.), which can be understood as the sameness or quasi-sameness of the spatio-temporal 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 a certain part of space, leaving aside when each of them is placed in this part. 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-existent.
   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. According to Ernst Tugendhat, a material object can be identified by means of an indexical added to a sortal predicate,[10] as in the statement ‘This is a chair’ (1983, Ch. 9). Now, the necessary tropes are those typically specified in the definition of the sortal. Thus, ‘a chair’ is defined as a moveable seat with a backrest, designed to be occupied by only one person at a time. The seat is constituted by a sub-cluster of tropes, the backrest by another, and the conditions that this complex object is moveable and designed to be used by only one person are constituted by dispositional tropes or sequence of tropes that complete the definition. There are also contingent tropes, like those constituting the sub-clusters of armrests or four legs (there are chairs without armrests or without legs); and there are still more casual tropes associated with a chair, like its color, the relation with a 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 not very dissimilar way. Gold is defined as an element with the atomic number 79, a yellow, dense, 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 the objects referred to by Tugendhat’s 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.). Simons admits the possibility of variations: a concrete object formed only by kernel-tropes, etc.
   A precise definition is difficult, if not impossible. A stone, for instance, is a material object that can be composed of very different materials, having few tropes to individualize it, with the exception of a tight connection of form, hardness, solidity, weight, volume, and color, all of them compresent. However, based on this bunch of properties, sometimes combined with a fixed spatio-temporal location, 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 in a complex and varied trope, as it seems. These tropes seem to have compresence, since they are all located where Socrates is. Moreover, they could be designated by a sortal predicate delimiting the spatio-temporal location of Socrates (‘Here comes Socrates again with his uncomfortable wisdom!’). Finally, they can have a kernel: the ‘peculiar core of Socratic wisdom.’ But they are not a material object, not even an individual, insofar as in principle others could 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 – as its 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. 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:

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

 This already excludes Socratic wisdom, the rainbow and the holomorphic projection. But liquids, although they are material entities, do not have a specific form or solidity, unlike a stone, a tree, a table. For example, water takes the form of its container, and water can be added to a given amount 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, and its droplets have minimal resistance to pressure. But what about individuals and supposed material entities like viruses or atoms and the hypothetical strings in string theory?
   My final condition bases itself upon the assumption that our commitment to modest common sense does not exclude science. We can refine our 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 material body (an individual) is broadly defined as its resistance to acceleration when force is applied (an idea accepted in both Newton’s and Einstein’s mechanics).[11]
   I conclude that in an inevitably vague characterization, having some inertial mass, some size… and compresence of its definitional tropes seems to be necessary for identifying a material object. This excludes, for instance, electromagnetic, gravitational, weak and strong forces, which are better seen as tropes. However, one cannot generalize this result to any individual. Consider 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.
   A more technical difficulty arises from the alleged fact that the idea that particulars are clusters of tropes is vulnerable to a regression argument parallel 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 against 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 spatio-temporal delimitations that remind us of the cases of form and duration. They are sui generis tropes, since they behave 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 (or that a co-location is co-located or that a co-temporality is co-temporal or even that identity is identical). Concurrence is a sui generis non-self-predicating limiting trope, requiring no new trope of concurrence to warrant its own co-location and co-temporality together with other tropes. Strict similarity is also a sui generis trope, because one cannot say of the strict similarity between T1 and T2 that it is strictly similar without falling into nonsense.
   As I have noted, not all individuals are material objects. Social entities like the British Parliament and historical entities like the Battle of Hastings are in themselves not material objects. They are complex structures made up of tropes, mental tropes like intentional states depending on material entities to be spatio-temporally located, even if only in a vague way. Since these tropes are unique and identified by nominal terms, they are particulars.
  What should we say about individuals that are formal entities like numbers? They do not seen to consist of tropes, since they do not seem to be spatiotemporal. However, there is room for doubts. The empirical world is made up of quantities. Would the number three exist if the world did not exist? Though this is a queer question, the tendency is to answer in the negative. For Locke the numbers belong to the ideas of primary qualities (= properties = tropes), which are solidity, shape, size, motion and… number. If you consider one chair and abstract all its properties, the chair will not exist anymore and we let to have one chair. If tropes are characterized as spatio-temporal properties, it seems that the ten fingers of my two hands are in some way here and now and that the ten players of the American Liga baseball team can take a flight and move to another city (cf. Maddy 1990: 87). Even thausend grains of wheat scattered in the wind are spatially scattered and temporally located. Thinking in this way, I am not denying Frege’s account of numbers as properties of concepts or even as ‘objects’. As he wrote, the attribution of existence is the negation of the number zero. The concepts of number and existence are related. In fact, I wish to consider ‘the property of being a number’ and ‘the property of existing’ as higher-order tropes because, like tropes, they are in some way spatiotemporally located. And I believe that when I say ‘This is one chair’, the numeral one is located between me and this one chair, as much as the existence in the utterance ‘This chair exists.’ These things cannot be in the outer space or in the primeval times or nowhere. In applied arithmetic numbers – I call them in these cases numerals – are used to count empirical objects. We learn numbers by counting material objects: ‘There are two apples and one pear in the basket, totalizing three fruits.’ In this case, the ascription rule of the predicate ‘…is a fruit’ has shown its effective applicability to three distinct objects, attributing physical existence to each of them. In my view (contrasting with Frege) the concept is a rule, what means that the attribution of existence is here the second-order property of effective applicability of a conceptual ascription rule. In a similar way, I suggest that a numeral is the higher-order property of the countability of lower-order conceptual rules that we apply to the entities. For instance, I can say (i) ‘There are hats with three corners’ meaning by this that the conceptual rule for the identification of some hats has the countable property of being applied to three corners. Moreover, this countable property or trope can be abstracted in the form of a von Newmann set. Hence, I can instead of (i) say (ii) ‘There are hats with {{}, {{}}, {{},{{}}} corners.’
   But what about the natural number in itself, as it is used in pure arithmetic, abstracted from its application in counting objects? Is this, as Frege wished, an abstract object like a Platonic universal? Or is this a further Platonist illusion? I think we can derive the abstract concept of number from the councept of numeral, of the countability of a concept. As we have seen above, numerals should be tropes, since they are originally spatiotemporally located properties of properties determined by respective conceptual rules. Consequently, regarding the universal, that is, the number in abstraction of its being properties of conceptual rules, we can try to define it by applying again our disjunctive model. In this case, it is conceivable that the number two would be a disjunction between a class instantiated by a chose higher order trope pattern/model of effective applicability of some conceptual rule (e.g., the bull horns) or any other strictly similar class – one could also say, ou any other equinumerous class. Thus, calling the numeral one the unity trope formed by the applicability-trope of an ascription rule, we can (following von Newmann) symbolize zero {0} as the empty set {}, one as {1} or the set that contains only the empty set {{}}, two as {0, 1} or {{}, {{}}}, three as {0, 1, 2} or {{}, {{}}, {{},{{}}}… This definition is insufficient, since it applies only to one applicability of a conceptual rule, even if it remaind indetermined, being therefore a conceptual higher-order property. This is different from the old Fregean-Russellian view of a number as the class of all classes of the same kind. But all we need to do in order to get the number as object, that is, the ‘abstract’ universal called number three, for instance, is to apply the same procedure we have applied to get universals from our usual tropes, namely:

Number 3 (Df.) = set of tropes {{}, {{}}, {{}, {{}}},* or… any further set of tropes that is strictly similar (or equinumerous) to {{}, {{}}, {{}, {{}}}*.

In this sense the number as an universal is the higher-property (trope) of the enumerability of a certain set of tropes (determined by a concept) or of any set of tropes strictly similar to the first one – a higher-order property-trope that may be found scattered in any spatio-temporal world.
   Assuming a definition like that, we neither stumble over Russell’s set paradox nor are limited to particular instances or directly committed differentiating concrete features. If a similar answer proves to be viable, one could conclude that even the abstract world of mathematics is made up of some sorts of thin higher-order tropes. Such tropes, like others, would be situated at the peak of a building whose genetic-epistemic foundations are our more feasible perceptually given quality-tropes, so that numerical tropes can be univocally named in this way are also dispersed over the whole world and able to be meta-predicatively designated. Trope theory turns Platonic realism upside down.
   Much of what I have written here is speculative, demanding a great deal of work and refinement. I could do no more than to offer a sketch of what seems to me the most plausible way to deal with the one category ontology that will 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 article in the Stanford Encyclopedia of Philosophy.
[2][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 very convincing, since the conventionally charged intromission of epistemic subjects is inevitable in any conceptual application.
[3] This objection is related to Bradley’s argument that reality is an indivisible unity because there can be no ontologically real relations (see Maurin 1992: 134 f.). But the answers would be similar.
[4] In Russian there is no proper verb for the copula. One says something like ‘Me nice’, ‘Me angry’… Thus, they seem to be less susceptible to such concerns.
[5] 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.
[6] Though by tradition labelled ‘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 form-substance-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. (See Shields 2007, Ch. 6.6; Copleston 1993, vol. I: 306)
[7] 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, section 12.) See also the more sophisticated but also less clear view of David Hume (1738, Book I part 1, section VII).
[8] 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, comparing these patterns with any found trope. The templates can have all the tonalities of a single color. They may call the possibilities resulting from the comparisons the universal of a color-trope.
[9] See Anna-Sofia Maurin, 2007. As she remarks, in a vicious infinite regress a considered statement (trigger) is dependent on the subsequent steps, while in a virtuous regress the subsequent steps depend on the considered statement, which makes them unnecessary.
[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] 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.