There is one further worry associated with the Determinate Universal Theory, namely, the problem of missing, uninstantiated determinates. This is especially a problem for those who embrace the Dretske-Armstrong-Tooley account of natural laws as relations among universals. A law like Newton's law of gravity specifies the force of attraction between any two bodies, given their masses and the distance between them. The law would seem to apply to any quantity of mass whatsoever, including quantities that are never in fact realized in the actual world (there must be such values, if there will have been only finitely many actual bodies, given that the number of possible mass-values is infinite). Many metaphysicians (including, for example, David Armstrong) are reluctant to posit uninstantiated universals, but if all universals are determinates, it seems that either there must exist some uninstantiated universals or the laws of nature are radically gappy, failing to apply to an infinite number of actually-unrealized values.
10.4.2 Simple Intensity Theory
As we have seen, Determinate Universal Theory has two major drawbacks. It can't explain why distinct determinates of the same determinable exclude each other, and it can't explain why the same set of real numbers can be used to order different kinds of quantities (like mass, charge, distance, and so on). It has to treat the applicability of the same real numbers to different quantitative determinables as a brute coincidence.
We could instead suppose that whenever a substrate S instantiates a determinate quantity q1, S in fact jointly instantiates a pair of universals, a determinable Q and an intensity I. On this account, the very same class of intensity-universals is instantiated by each determinate quantity, no matter what determinable it belongs to. In other words, the same set of intensity-universals would qualify instances of MASS, CHARGE, ENERGY, FREQUENCY, and so on. Real numbers would then be relations between pairs of intensities, with intensities forming a single ordered domain.
Suppose that two substrates S1 and S2 are characterized by determinate mass-quantities m1 and m2 and that the ratio between m1 and m2 corresponds to the real number r. According to the Simple Intensity Theory, in such a case S1 is connected to both the mass-determinable universal M and to some intensity-universal I1, and S2 is connected to M and to some intensity-universal I2. The pair of intensities I1 and I2 stand (in that order) in the real-number relation r. (A similar account can be given by Trope Nominalists, replacing the universals with tropes.)
10.2T.2 Simple Intensity Theory (Nexus version). Positive real numbers are relations between intensities, and intensities are universals that are jointly connected to particulars and determinable universals.
10.2A.1T.2 Simple Intensity Theory (Trope-nexus version). Positive real numbers are binary relations between intensity-tropes, and each intensity-trope is jointly attached to both a substrate and a trope of some determinable.
Realists who are Relational Ontologists (treating instantiation as an extrinsic relation, not involving parthood) will have to complicate their theories by introducing a new, three-place instantiation relation (x jointly instantiates y and z) in addition to the ordinary, two-place relation that connects particulars to single universals. Similarly, Modifying Trope Theorists will have to postulate a new kind of modification relation, one in which two tropes cooperate in modifying their associated concrete particular.
What about Realists and Trope Theorists who embrace Constituent Ontology? On these views, a particular instantiates a property by actually containing a universal or a trope as a part. What would it mean for two universals (or two tropes) to be a joint part of the particular? We will have to suppose that particulars have an internal compositional structure—that they are not simple bundles of properties. If a particular x has a determinable quantity Q with intensity I, then there must be a special part of x that contains both Q and I as parts. In addition, suppose x also has a different quantity Q2 with a different intensity I2. Then x will have two complex parts, y1 and y2, with y1 composed of Q and I, and y2 composed of Q2 and I2. At the same time, we will have to deny that any combination of parts of x compose a further part. For example, we will have to say that x has no part composed of just Q and I2, and no part composed of just Q2 and I.
The situation is somewhat simpler if we have bare particulars in our ontology, in addition to universals or tropes. Then we could suppose that, if x is a bare particular with two quantities at two different intensities, x simply belongs to two different bundles, one containing the first quantity and its intensity, and the second bundle containing the second quantity and its intensity. The bare particular can tie all of these diverse bundles together into a single thick, ordinary particular.
THE SELF-APPLICABILITY OF REAL NUMBERS There is another simplifying modification that can be made to Simple Intensity Theory. Rather than assuming that there are two classes of universals, the class of intensities and the class of real numbers, the two classes can be identified. The identification is possible because pairs of real numbers stand in ratios that are themselves real numbers. If r1 and r2 are real numbers, then there is a real number r3 = (r1/r2), which can be thought of as the intensity of the greater-than relationship between r1 and r2. In other words, GREATER-THAN is itself a determinable universal that is instantiated by ordered pairs of numbers to varying intensities. We can identify the real number 2 with that intensity that is jointly instantiated (along with the GREATER-THAN determinable) by the ordered pair of real numbers 2 and 1. That is, given a set of real numbers and a set of intensities, there is a natural way of mapping each on the other. This gives us good grounds for supposing that there is in fact only one set of universals.
We can use facts about physical parts and whole to identify the intensity that is the number 2. Suppose that a massive object x with determinate mass m2 has two separate parts, each of which is equal in mass to m1. We know then that m2 is twice (in terms of mass) m1, and so the intensity of the GREATER-THAN relation between m2 and m1 must be equal to the number 2. Once we have identified the number 2, we can identify other intensities in a principled way. For example, the number 4 is that intensity that stands in the GREATER-THAN relation to the number 2 with an intensity that is itself equal to 2.
The class I of quantitative intensity-universals is ordered by the asymmetric relation GREATER-LESSER. This relation itself is modified by the very same class I of quantitative intensities. That is, if intensity-universal I1 is greater than intensity-universal I2, then these universals (taken in the not-less-than/not-greater-than order) jointly instantiate the GREATER-LESSER universal and some intensity-universal I3, where I3 is another member of the class I.
If we assume that the intensity-universals are related to one another by a primitive additivity relation of the form Ii + Ij = Ik which satisfies the laws of commutativity and associativity, then we can identify each of the universals in I with a unique real number. Suppose, for example, that I′ + I′ = I′′. Then the intensity I* of the GREATER-THAN relation between I′′ and I′ can be identified with the real number 2.7
How does Simple Intensity Theory explain the four basic facts about determinates and determinables? First, Simple Intensity Theorists must suppose that the nature of each determinable universal is such that it can only be instantiated jointly with some intensity. Second, it follows immediately that a substance cannot possess a determinate property without instantiating the corresponding universal, since possessing the determinate property consists in jointly instantiating the determinable and some intensity. Third, Simple Intensity Theorists must again suppose that it is either a brute metaphysical necessity or a law of nature that no substance can instantiate two different intensities in conjunction with the same determinable. Finally, the ordering of the real-number intensities explains the similarity ordering among determinates.
Thus, Simple Intensity Theory resolves one of the problems faced by Determinate Universal Theory. It can explain why the same system of real-numbered relations is realized in many different families of properties. However, it shares w
ith Determinate Universal Theory the second cost since it cannot explain the mutual incompatibility of different intensities.
Simple Intensity Theory helps somewhat with the problem of uninstantiated determinates and gappy laws. So long as each intensity is realized in conjunction with some determinable quantity or other, the fact that specific masses or volumes are unrealized won't matter. However, given that there seem to be only finitely many determinable quantities, it seems almost certain that infinitely many intensities will be uninstantiated anyway.
There is one further cost to Simple Intensity Theory: it entails that there are metaphysically natural units of measurement, in addition to all of the familiar, conventional ones. For example, there is some specific mass that is the property of having the determinable mass to degree 1 (the metaphysical gram). These metaphysical measures are empirically undetectable, and the positing of such empirically inaccessible facts comes at some cost.
10.4.3 Composite Intensity Theory
On the final theory, Composite Intensity Theory, there are once again both determinable universals and intensity-universals, but each intensity-universal is of the form: HAVING SOME QUANTITY TO AT LEAST DEGREE D. Thus, there is no universal corresponding to the having of exactly one degree of intensity, but there are intensity-universals of the following kinds:
(IU1) HAVING INTENSITY TO AT LEAST DEGREE 0.1.
(IU2) HAVING INTENSITY TO AT LEAST DEGREE 1.
(IU3) HAVING INTENSITY TO AT LEAST DEGREE 3.
etc.
Moreover, each determinate of the form HAVING INTENSITY TO AT LEAST DEGREE D contains as proper parts all of the determinates of the same form for any D* < D. Thus, IU2 above contains IU1 as a proper part, and IU3 contains both IU2 and IU1 as proper parts.
A substance's having a determinate quantity of mass, such as exactly 3 grams, corresponds to its instantiating the determinable MASS together with the intensity-universal IU3 and not instantiating MASS together with any intensity-universal that contains IU3 as a proper part. In other words, the determinate properties of a substance correspond to the maximal intensities of each quantity that the substance instantiates.
10.2T.3 Composite Intensity Theory. Positive real numbers are intensities, each positive real number contains all smaller positive real numbers as proper parts, and the having of a determinate property consists in the joint instantiation of a determinable universal and an intensity.
Composite Intensity Theory can explain why two determinates of the same determinable are logically incompatible and why the same system of real numbers applies to many different determinables. It is logically impossible to instantiate two different intensities of the same determinable, since one intensity must contain the other, and a determinate property corresponds to whichever intensity is maximal. Second, the same system of intensities is realized together with each of the various quantitative determinables—at least, for those scalar quantities having a natural zero: mass, charge, volume, momentum, temperature, entropy, and so on.
Composite Intensity Theory can also resolve, to a large extent, the problem of missing determinates and gappy laws. If an intensity of degree D is contained by any actual substance, then every intensity of degree less than D will also be contained, since they will all be proper parts of the universal DEGREE AT LEAST D. We will still be missing all of the very large intensities, intensities so large that nothing in fact realizes them in conjunction with any determinable. But the class of actual intensities will not be riddled with gaps or holes.
10.5 Conclusion and Preview
In Part III (Chapters 7–10), we have developed various views of the nature of facts and substances, with an eye toward understanding the relationship between properties and these different types of particulars. We turn next to further questions about substances. In particular, we will consider whether there really are any such things as substances, and if there are, how many there are. This is taken up in Chapter 11, on Nihilism and Monism. Later, we consider whether substances are anything more than collections of sensory properties, that is, whether there really is a physical world. This is Chapter 13, on Solipsism and Idealism.
Notes
1. A symmetric relation R is one such that Rxy entails Ryx. The relation of identity and that of being the same size as are paradigm symmetric relations. Non-symmetric relations aren't symmetric. See below for more.
2. There are similar properties of symmetry that can be defined for ternary (three-term), quaternary (four-term), and other relations with still larger numbers of terms. Most of our subsequent discussion will apply to such relations as well.
3. This would involve extending the plural logic of George Boolos (1984) to cover not only unordered but also ordered pluralities. Boolos provides a logic that quantifies over, not just individual things (corresponding to singular pronouns like ‘it’), but also over pluralities (corresponding to plural pronouns like ‘they’). We can build on Boolos's theory by supposing that some plural variables and pronouns stand not just for some things but for some things in a particular order. If, for example, we say (concerning Texas and Oklahoma) that they stand in the north to south relation, we mean by ‘they’ not just the two states but the two states taken in the Texas-first and Oklahoma-second order. This seems to make sense, even if we don't suppose that we are talking about some further single entity, like an ordered pair (as defined in set theory).
4. Lewis rejected his own suggestion as “too bizarre to be taken seriously”, but bizarreness, like beauty, is in the eye of the beholder. Amphibian theory is an attractive approach to structural universals for Constituent Ontologists.
5. This distinction was first introduced by W.E. Johnson (1921–1924). For further developments, see Prior (1949), Searle (1959), and Johansson (1989).
6. There is some controversy about whether sensory qualities like color or taste satisfy this condition. There are cases in which something can at least appear to have two different colors at the same time.
7. We can give a similar account for vector quantities, like velocity, acceleration or momentum (that is, quantities with both a scalar measure and a direction). We could suppose there to be, for example, a single universal of VELOCITY, instantiated by velocity tropes. Each velocity trope is attached to its substrate by a trope-nexus, which in turn instantiates some vector-intensity universal. The set of vector-intensity universals would form a three-dimensional space of vectors. Relations between vectors (comparing their scalar value and direction) would themselves have vector-intensities, which would enable us to identify a plane of vector-intensity universals that constitute the complex numbers. Each complex number is itself a vector, and so the intensities of the comparative relation between complex numbers are themselves complex numbers (vectors in the same plane).
Part IV
The Nature of Reality
11
Nihilism and Monism
In this chapter, we look at two big questions: does anything exist, and, if so, does more than one thing exist? In Section 11.1, we consider the possibility of Nihilism, that nothing exists, (‘nihil’ is Latin for ‘nothing’) and its alternative, Aliquidism, that something exists (‘aliquid’ is Latin for ‘something’). This will lead us into an investigation of the point of positing existing things. Then, in Section 11.2, we look at the debate between Monists, who believe in only one thing, and Pluralists, who believe in many. We consider both radical and more moderate forms of both Nihilism and Monism, including, for example, Priority Monism—the view that there exists only one fundamental thing.
11.1 Nihilism and Aliquidism
We have discussed the issue of universals and particulars, including views according to which the world contains both universals and particulars (UP-Realism 7.1T.1T), only particulars (almost all forms of Nominalism 7.1A) or only universals (Classical Bundle Theory 9.1T.1T.1A). Before moving on, we need to address a still more fundamental question: why think there is anything at all? We will use the terms Nihilism and Aliquidism to re
present the two options:
11.1T Aliquidism. Something exists.
11.1A Nihilism. Nothing exists.
Why think that Aliquidism is true? Common sense, of course, suggests that many things exist, things like rocks, trees, clouds, rivers, people, artifacts, numbers, features and aspects, thoughts, propositions, facts, events, states of affairs, times, and regions of space. In addition, René Descartes produced a celebrated argument, the Cogito, which purports to prove that at least “I” (the maker of the cogito argument, whoever that happens to be) exist. We could express the Cogito (‘cogito’ is Latin for ‘I think’) in the following form:
I think that some things exist.
Either I am right or I am mistaken.
If I am right, then some things exist.
If I am mistaken, then I am thinking something false.
If I am thinking anything at all (whether true or false), then I (the thinker) exist. (Cogito ergo sum, in Latin.)
Therefore, at least one thing exists.
Descartes's Cogito depends on two things. First, when I am thinking something, it seems to me that I am thinking something. Second, it is impossible for the skeptic to convince me that I am wrong about this, since if the skeptic were to succeed, I would come to think that I am mistaken in my thought, which still entails that I am thinking that very thought. Thus, the appearance to me of my own present thought is incapable of being defeated by any skeptical challenge.
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