Appendix to chapter 3
DRAFT FOR THE BOOK 'PHILOSOPHICAL SEMANTICS' (Cambridge Scholars Publishing, 2017)
Appendix ch. 3
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 yellowness of this sofa, the hardness of that wall, this property of my shirt of being made of cotton… Many traditional philosophers, however, would say that these things cannot really be properties in the strict 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 place of traditional ontological realism is opposed to the ontological starting place of the common man, and indeed of our own common sense in general. 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 in its pure form sustains this view is 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 be consequences of the ontological building blocks that are the spatio-temporally singularized properties called tropes, and not the other way round.
The importance of the theory of tropes resides in the fact that after the development of nominalism already in the Middle Ages, this may turn out to be the only really ground-breaking advance in ontology. Although the concept of the trope as a singularized property has existed at least since Aristotle, only in the 1950s did an Australian philosopher named D. C. Williams conceive the bold idea of assigning tropes metaphysical pride of place as the most fundamental ontological building-blocks. His central aim was 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 methodologically modest common-sense perspective seems the most plausible.
1. Introducing Tropes
First, what are tropes? Tropes can be elucidated as being properties individually located in space and enduring in time, whereby properties must be understood simply as empirical designata of predicative expressions. The most fundamental tropes, from a genetic-epistemological perspective, are those that are 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… Tropes can be psychological properties, like feelings of pain, sorrow, love and pleasure (Williams, 1953: 17). We can prove the reality of tropes by considering that they can be objects of selective perception: looking at the ocean, one can concentrate alternately on its color-tropes, the form-tropes of its waves or their sound-tropes. Tropes usually appear combined with other tropes, and some conglomerates of different kinds of tropes are highly complex and to some extent dispositional. This is the case of Socrates’ psychological character, of biological properties like that of a certain cat of being a mammal, of social properties like that of India being a democratic country; they are all in some way spatio-temporally located, even if dependent on concrete physical things.
Tropes differ from what I prefer to call individuals: things that must be unique and are referred by nominal terms like a daisy, a rhino, the Golden Gate Bridge and Socrates. Most of them are material objects. But some compositions of tropes are individuals, though not material objects. This is the case of a rainbow someone has pointed to and Beethoven’s 5th Symphony, which each consist of only one class of tropes and is referred to using singular terms. Finally, there are derivative tropes like the fundamental physical forces, even if from a purely physical perspective we can speculate as to whether all other tropes aren’t to some extent grounded on them (see Campbell 1990, ch. 6).
Like all particulars, tropes have identity conditions. I suggest an ontological condition (a) followed by a linguistic requirement (b):
Tropes are identified by their (a) spatiotemporal existence to the extent that they display continuity over time (are continuous) and are amenable to certain direct or (mostly) indirect experiential ways and conditions of accessing them and (b) by being linguistically accessible by means of the predicative expressions of singular statements.
So understood tropes contrast mainly with material objects referred to by means of nominative expressions, particularly proper names.
The linguistic requirement (b) has a guiding function: as spatio-temporally located properties, tropes can be designated by means of predicative expressions. Regarding the ontological condition (a) there is more to say. Consider the following example: the pair of shoes that 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 color of the right shoe, this color can be said to be a 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 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 perceptually accessed through different perceptual ways and conditions of access.
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. But since according to perceptual distance and acuity we can distinguish different amounts, this does not seem to be very elucidating (see Campbell 1990: 136-7). Because of this – and again drawing on common sense and ordinary language – it seems better to say that the unity of a trope – which we can rightly call a property – would be established by the natural limits of its spatio-temporal continuity as being the same, disregarding its possible divisions. Thus, for instance, the whiteness of a wall could be considered a myriad of tropes if any visible point of whiteness were considered a trope; but considering a trope of whiteness to be the whole of its continuity, we are not only being economical but also following ordinary language practices, for 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, are spatiotemporally well delimited, deserving to be individuated as tropes, once they are called properties as a point of convention within our usual language-games regarding medium sized dry objects. 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 spatio-temporal 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 assumed practices, allowing the constitution of a conceptual system with the suitable amount of precision. Moreover, many tropes (e.g. social tropes) are highly dispersed in space and time.
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 the trope or the set of tropes. Moreover, tropes are said to have a proper existence, even if unavoidably related to other tropes. This is exemplified by the colors of the rainbow, the sound of the wind, the smell of a daisy, etc. They differ only from other individuals due to their uniqueness.
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 bundle of tropes... Keith Campbell, disagreeing with D. C. Williams, did not consider forms as tropes because of their dependence upon other tropes (Campbell 1981).
However, if we wish to preserve our one-category ontology, tropes are better understood as any spatio-temporally existents designated by means of predicative expressions and not necessarily as independent, primitive or simple proprieties, because these things can vary with the language-game (as we saw, for Wittgenstein the simple and the primitive are relative to the language-game). If we hold that view, a better answer emerges, since we can see forms and durations as limitations in space and time respectively. They would rise from limitations imposed by standard quality-tropes. Hence, it seems that we could view forms and durations as at least dependent 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 polyadic predicative expressions (usually dyadic), 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. The most interesting relation is probably the causal one. For instance: ‘The throwing of a stone broke the window’. As Campbell noted, a causal relation is to be seen as a relation between tropes (1990, ch. 5.15). The relational predicate ‘…causes…’ is not between the objects stone and window, but between the throwing (of a stone) and the breaking (of the window), which are events that can be designated by means of predicates (‘The stone was throw’, ‘The window was broken’), being therefore tropes according to our identity condition. Moreover, the causal relation is called an internal relation, which is defined 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 only predicatively designated but 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, grounded on them.
One objection against the idea that relations are tropes is 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 (Maurin 1992: 134 f.). I will argue against this idea first by appealing to a reductio based on the insight that the same problem comes up again 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, like 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; and this relation R would naturally require two new relations ‘FR1RR2a’ to relate R to their relata, and so on ad infinitum. But this seems absurd! Moreover, these relations seem to be empty and meaningless. If instead of ‘The Earth is round’ we say ‘The Earth is related with its roundness’, we would not be making any sense; the same with ‘The Earth is related with the relation of its relation with its roundness’. Thus, it is better to see the link between subject and predicate as a ‘non-relational tie’ (Strawson 1959, part II, Searle 1969: 113) or like the link of a chain, to use Wittgenstein’s metaphor. They are not tropes but pseudo-additions in the true sense of the word. I conclude that we don’t need to postulate FRa in order to explain Fa. 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 dyadic and polyadic relations said to produce a Bradleyan regress. In my view, 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, consider the following example: (i) ‘Socrates is a friend of Plato’. Since friendship is a relation, a Bradleyan would be entitled to replace sentence (i) with (ii): ‘Socrates has a relation of friendship with Plato’, which still says the same thing by specifying that friendship is a relation. But the Bradleyan would then go ahead, deriving from (ii) the sentence (iii) ‘Socrates relates itself to its relation of friendship relative to Plato’, which is an instantiation of aR1RR2b. But we see that (iii) hardly makes any sense.
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 particulars.
I begin with the problem of universals. Linguistically stated this problem consists in the question of how we can apply the same general term to many different particulars; and ontologically stated it consists in the question of how it is possible that many different particulars 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 version) that can in some way be instantiated in many particulars. Thus, for the realist we say that this rose and that strawberry are red because they instantiate or exemplify the universal of redness (red-in-itself). This solution, which goes back to Plato, has never rescued itself from unsolvable difficulties. After all, universal properties must be non-empirical abstract objects accessible only to the intellect. With this, we are left with two worlds: our empirical world and a world with an infinite number of abstract entities whose intelligibility is questionable and for which we have no identity criteria. Moreover, the realist is left with insoluble problems of how to explain the relation between these abstract entities and the particulars that instantiate them or even with our cognitive minds. On the other hand, if you ask a layman where properties are, he would answer by pointing to the blue of the sky, the hardness of a table, the softness of jelly… and not to a Platonist mood. This contrast leads us to the suspicion that only intensive philosophical training – supposedly originated from the ideological pressure of some mystical belief in what a Nietzschean would call a ‘a world of beyond’ (or Überwelt), a true temptation for unworldly creatures like philosophers – could succeed in conditioning one’s mind to see properties in such an idealized way.
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 could conceivably be understood as a kind of relational trope. Philosophers like D. C. Williams (1953: 9) and Keith Campbell (1981: 477-488) saw universals as classes 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 that 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 a class is seen as an abstract object, it seems that we are abandoning the great advantage of trope theory. And 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 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, we begin by assuming that likenesses or strict similarities are tropes, and that (using ‘=’ to abbreviate ‘strict similarity’) we have the tropes T1 = T2, T2 = T3, T3 = T4, etc. It must be what I prefer to call ‘strict similarity’ because mere similarity or resemblance lacks transitivity. If T1 is only similar to T2 and T2 is only similar to T3, then it is possible that T3 isn’t similar to T1. The solution is to appeal to strict similarity understood as a transitive concept and meaning the same as qualitative identity, which is the identity between different things (differing from numerical identity as the identity of the thing with itself). Qualitative identity does not need to be perfect: our cars are both yellow, but your car’s finish is faded. But the differences must have a limit. Corrigible differences are usually found within the range of a concept (e.g., turquoise blue and cobalt blue are both called blue) as far as we have a criterion of correction (say, wavelengths between 450 and 495 nanometers).
Now, in order to construct the class of similar tropes, we need to know that the first trope of identity is like the 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 a trope of qualitative or strict identity. Since the same question can be posed regarding the strict similarities between these strictly similar tropes, it seems clear that we are becoming bogged down in a kind of pyramidal infinite regress.
Russell considered this regress as plainly vicious. Even if this is not the case, such a multiplication of tropes of likeness or strict similarities seems overwhelming to our finite intellects. I believe, however, that we are able to easily overcome this difficulty, 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. According to this view, the universal could be defined as:
Universal (Df) = Any trope T* taken as a standard/model or any further trope that is strictly similar to T*.
Explaining this definition, we must remark that trope T* used as the standard doesn’t need to be always the same. On the contrary: one can choose any trope strictly similar to T* and use it as T* in order to make comparisons. Moreover, what we normally know of T* is some recollection in our memory.
Accepting this definition, we do not need to take recourse to sets of similar tropes or even to a mereological sum in order to explain universality, since the definiens covers any trope strictly similar to T*. The problem of size disappears, since for the definition it is not a question of how many tropes are identical to T*. When a person utters the sentence ‘This rose is red,’ she means that this rose has a trope of red Tr1 that is identical to a trope of red Tr*, taken as a standard, as retained in the person’s memory. When she utters the sentence, ‘Red is a color,’ she means that some Tr1* is also a Tc (color-trope), and that each trope strictly similar to Tr* is also a Tc by being in a looser way also 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 with the other, but only the tropes ‘T1, T2… Tn’ individually with trope T*. Instead of possibly generating an infinite pyramidal regress, the schema will take the form ‘T1 = T*, T2 = T*… Tn = T*’. In other words, as long as the chosen standard trope T* remains one and the same, there is no need to compare similarities with similarities, in this way reaching similarities of similarities. Russell’s problem would not arise because our definition makes the universals potentialities instead of actualities.
Furthermore, we can also construct the universal ‘strict similarity’ requiring that some 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 one and 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. So, 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 be this 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 to. 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 ideas simply because there is no explanatory advantage in going further, considering, for instance, the idea of the idea of the idea. In the same way, we can find no explanatory advantage in going beyond precise similarities between first-order tropes.
Finally, it is worth noting 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 to be alike; color-likeness, 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 alreay exists (1990: 37). But the fact of being an internal relation does not make strict similarity a quasi-trope or a non-trope, considering our identifying condition of tropes. There are reasons to countenance its reality as a trope, even if distinguishing strict similarity from other more concrete kinds of tropes. First, 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 am looking at 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. But 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 is destroyed, the color likeness also disappears, which means that it 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, and as great as it may be, is ridiculously minuscule compared with the immensity of the cosmos.
The problems for the theory of tropes do not stop here. What about other relations? For example, the Golden Gate Bridge is (on the average) 67 m. above sea level. Surely, 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. 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 of being spatio-temporally localizable.
But what about space and time in themselves? Normally we admit that all that exist are tropes and space-time. Even in realist ontologies a separate existence of space and time was never questioned. However, could space-time be in some way tropes or something derived from tropes? Imagine that all objects and properties of the world disappear. Would space (and time) remain? I believe that we have the intuitive tendency to answer in the negative. However, according to Newton’s theory of absolute time and space, the answer was in the affirmative: space and time were seen 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 this case, space and time could not exist in themselves, because by being constructed of relations they demand the existence of material objects (Alexander 1956). Both answers have always been controversial, and the discussion has become even more complicated due to the theories and discoveries of contemporary physics.
Anyway, aside from the Newtonian view, it seems that there is some possibility that we can explain space and time in terms of tropes. In a unsophisticated commonsensical approach, we could to define space relationally, possibly beginning with relations like above, under, in front of, behind; e.g. ‘object x is located twice behind object y in relation to z’. Time could be defined relationally, by means of relations like earlier, simultaneous, later; e.g. ‘event x occurs three times later that event y in relation to event z’… And we could use regularities as parameters: a foot to measure distances in feet, a day to measure periods of days… However, since tropes are seen as spatio-temporally localized 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 commonsensical approach could be that space is 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 below b and twice as distant as b is from a, while stars d and c are on opposite sides of b and the same distance from b as a is from b. With this 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 for time-relations like before, simultaneous and after). However, it is an entirely open question whether such rough intuitive views could be developed and extended in order to comprehend the sophisticated theories of contemporary physics and their distinct domains.
3. Tropes and Concrete Particulars
The second major problem is that of constructing concrete particulars by means of tropes. For D. C. Williams, a concrete particular is a bundle of tropes (1953: 7 f.). Tropes are spatio-temporally conjoined to form concrete objects. The advantage of this view is that it enables us to abandon the old and obscure concept of substance understood as a hidden substratum of properties. For the trope theorist, the concrete particular turns out to be like an artichoke, which consists 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. We can analyze the concept of compresence as composed of two other concepts: co-location and co-temporality. The co-location of tropes is their joint location in a certain region of space, leaving aside when each of them is placed in this region. 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. 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 bundles 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. All we need is to distinguish necessary from contingent tropes. The necessary tropes are those typically specified in a definition. A chair is defined as a seat with a backrest, designed to be occupied by only one person at a time. The seat is a sub-cluster of tropes, the backrest another, and the fact that this object is designed to be used by only one person is a dispositional sequence of tropes that completes the definition. There are also contingent tropes, like those constituting the sub-clusters of armrests or four legs (there are chairs without armrests or four legs). And there are still more casual tropes associated with a chair, like its color, the relation of a person sitting on it, its distance from a table… The concept of a chair is one of an artefact. But we can consider natural kinds in a not very dissimilar way. Gold is defined as an element with the atomic number 79, being a yellow, dense, and precious metal. But its having a determined atomic number is a necessary trope, though gold does not need to be yellow or dense or even a precious metal, since these are contingent tropes.
Peter Simons gave what seems the best answer to this question by suggesting that a material object should not be seen as an unstructured cluster of compresent tropes. It is typically made up of a nuclear kernel of necessary 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. Consequently, the halo-tropes can be said to be specifically founded on the kernel-tropes, while the kernel-tropes only generally found the halo-tropes (Simons 1994: 376 f.). Simons admits the possibility of a concrete object formed only by kernel-tropes, etc. A precise definition is difficult if not impossible. A stone is a material object that can be composed by very different materials having few things to individualize it except compresence and a tight connection of form, hardness, solidity, weight, volume, colors... among its tropes. But based on this bunch of properties, we can 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 have compresence, since they are located where Socrates is. Moreover, they can have a kernel: the ‘peculiar Socratic core of wisdom’. But they are not a material object, not even an individual, since in principle others could share strictly similar qualities of 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 core, but it is less than a material object. The holographic projection of a teacup also has a compresent set of colors and forms. They belong to its core. 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 that seem to be necessary for the identification of our usual material objects. They are: volume, form, some degree of hardness or solidity (measured by resistance to pressure), some weight (related to presence in a gravitational field) and the possibility to be moved, all related by compresence. This already excludes Socratic wisdom, the rainbow and the holomorphic projection. But liquids, though material objects, 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 more water can be added to an amount of water, changing the 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 material, though not properly a material object, since it lacks a definite form, and it can resist pressure. A cloud has a low level of materiality, but even so its droplets have some minimal resistance to pressure. But what about material objects like viruses or atoms or electrons and the hypothetical strings in string theory?
My proposed answer is based upon the assumption that our commitment to commonsense 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 called in physics mass. In physics the mass of a material body is broadly defined as its resistance to acceleration when a force is applied (as far as I know, this idea is accepted in both Newton’s and Einstein’s mechanics).
We conclude that having mass, some size, mobility and compresence of its central tropes seems to be necessary for identifying the core of a material object and perhaps of any physical object. This excludes electromagnetic, gravitational, weak and strong forces, which are better seen as tropes. But this result cannot be generalized to any individual. Consider individuals as a crowd or the British Empire. These individuals do not form a physical object. Different from material objects, a crowd and the British Empire are composed of tropes that are supervenient to material, not tightly connect physical entities.
Another 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 that triangle is triangular, you cannot meaningfully say that a concurrence is concurrent (or even that a co-location is co-located or that a co-temporality is co-temporal or even that the 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. Also the strict similarity is a sui generis trope, because one cannot say of the strict similarity between T1 and T2 that it is strictly similar, for this would make no sense.
As I have shown, 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 of tropes, mental tropes like intentional states and depend 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 to say about individuals that are formal entities like numbers? They seen not to be made up of tropes, since they are non-spatio-temporal. However, I have my doubts. The empirical world is made up of quantities. Would the number 3 exist if the world didn’t exist? Although this is a queer question, the tendency is to answer in the negative. Perhaps numbers are only a compact way to speak of the numerals used to count empirical objects. We learn numbers by counting material objects: ‘There are three apples and two pears in the basket, totalizing five fruits’. In this case, the ascription rule of the predicate ‘…is a fruit’ has shown its effective applicability to five spatially distinct objects, attributing physical existence to all of them. In this case, the attribution of the number five seems to be the a higher-order property of the ascription rule extracted of its being effectively applicable to each one of the distinct fruits in the basket until the attribution of existence to them all. But what about the number five in itself, abstracted from its application in counting objects, as it is used in pure mathematics? Is this an abstract object like a universal? Or is this also a Platonist illusion? Isn’t it also here only a disjunction between a model of a higher-order property of having five distinct higher-order effective applicabilities of the same rule (five existences) and any precisely similar case of the higher-properties of effective applicabilities? In the positive case, it could be suggested that even the abstract world of mathematics is built up of some sorts of thin higher-order tropes situated at the peak of a building whose genetic-epistemic foundations are our well-known sensorally given quality-tropes, si that numerical tropes are also dispersed around the world and able to be meta-predicatively designated. These are, of course, only speculative divagations! But they serve to give us an idea of the instigating problems that a developed trope theory would have to face.
Much of what I have written here is speculative, demanding a great deal of work and refinement. I could not do more than to offer a sketch of what seems to me clearly the most plausible way to deal with a category that will play a central role later in this book.
 This ground-breaking work was D. C. Williams’ paper ‘The Elements of Being,’ published in the Review of Metaphysics (1953); he was the first to propose constructing the whole world using only tropes as building-blocks. An important attempt at a systematic development of the theory was Keith Campbell’s book, Abstract Particulars (1990). Since then, the discussion devoted to this problematic has steadily grown. For access to the literature, see Anna-Sofia Maurin’s 2013 article in the Stanford Encyclopedia of Philosophy.
 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 the epistemic subject is inevitable in any conceptual application.
 The objection is based on Bradley’s proof that reality is an indivisible unity, because there can be no ontologically real relations.
 I have heard that in Russian there is no proper verb for the copula. Russians say something like ‘Me beautiful’, ‘Me good’… This seems to reinforce the idea of its really pseudo-additional character.
 We can imagine circumstances in which people are unable to retain memories of the color trope T, but bring with them templates with patterns T* of this color trope, comparing these patterns with any found trope and calling the possible effects of this hability the universal of this color.
 As it 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.
 See the discussion of existence in chapter 4, sec. 11-17.