Abstract
It has been argued above that it is essential for a universal the nomological relations into which they enter. Even more, it has been argued that there is just a unique nomic network of necessary existence. Nevertheless, some have maintained that the conditions of identity for universals that result from these ideas are incoherent. It is explained here that this problem of incoherence can be dealt with, but assuming that the nomic structure is ontologically prior to the universals that enter in it. Universals are nodes in this unique necessary nomic structure.
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Notes
- 1.
More precisely, according to a combinatorial conception of modality, a causal power is grounded on modal facts about possible causal connections (see § 63). In a ‘causal’ conception of modality, on the other hand, the order of grounding is the inverse. Metaphysical possibilities about causal connections are grounded on causal powers (§ 64).
- 2.
The modal principle is [□p → p] and is an axiom characteristic of normal modal systems of type T. Suppose that [□p] in the world w. Then, according to the standard semantic clauses of modal logic, in all the worlds accessible to w, p is the case. If the accessibility relationships were not reflexive, that is, if w were not accessible from w, then there would be no guarantee that p would be true in w. Part of the minimum content of metaphysical necessity is that what is necessary is effective. Another way to see why accessibility relationships should be reflexive is that otherwise, it would not be the case that ab esse ad posse valet consequentia, that is [p → ◊p]. Indeed, if p is true in w, then—given reflexivity—it is possible that p because at least there is a possible world accessible to w in which p is the case, that is, w.
- 3.
The transitivity of the accessibility relationships guarantees the validity of the modal principles [□p → □□p] and [◊◊p → ◊p]. If [□p → □□p] were false, then it should be the case that [□p] and [◊¬□p]. Given [□p] it trivially follows that [◊□p], so it would be contingent that [□p].
- 4.
More precisely, there could be several classes of possible worlds, each of which would satisfy the condition that all and only the possible worlds of the class would be accessible to each other. Assuming several different equivalence classes of possible worlds would be very unlikely if the modality in question is metaphysical modality. A modal logic in which accessibility relations satisfy these conditions of reflexivity, symmetry, and transitivity in accessibility relationships is called an “S5-type logic”.
- 5.
More precisely: [□∀X□∀Y□ ((X = Y) ↔ ∀C∀x ((X confers C to x) ↔ (Y confers C to x))]. The variables ‘X’ and ‘Y’ have as range universals; the variable ‘C’ has as range causal powers; the variable ‘x’ has as range objects.
- 6.
The situation is very different when it comes to particular objects or tropes. The particular character of a particular is to be this, something to which—in principle—an ostensive indication can be made. A universal, on the other hand, appears phenomenologically as a ‘quality’ that by itself is multiply instantiable.
- 7.
Following an analogy already presented, the ‘causal role’ works for a universal such as the ‘description’ of a particular object, while the quidditas would function as a ‘proper name’ of an object. For radical quidditism, it is contingent for a universal—whose identity is given by its quidditas—what causal role it is satisfying. Here, on the other hand, the causal role is necessary for a universal, but different universals—whose identity is given by their quidditates—could satisfy the same causal role, although no universal could have a causal role different from the one it possesses.
- 8.
It is a tough question to answer what is such cardinality. It depends, among other things, on the general conception of metaphysical modality. If one supposes, for example,—from a combinatorial perspective—that spacetime is a continuous structure and that each point of spacetime can be filled or empty, then the cardinality of the modal space of possible worlds should be the cardinality of the whole power set of the continuum. Assumptions just a little more complicated—and realistic—make these accounts vary dramatically.
- 9.
And it is, also, doubtful. The modal operators of necessity and possibility are dual among themselves so that these equivalences must be valid: [□p ↔ ¬◊¬p] and [◊p ↔ ¬□¬p]. It seems reasonable to suppose that satisfying these equivalences is part of the minimum content of the notion of ‘metaphysical possibility’. But if this is so, then the problems for the intelligibility of metaphysical necessity will also infect the metaphysical possibility.
- 10.
Bird notes that this objection of regress or circularity has been proposed in several ways, not all of which have the same systematic interest. Some have argued that assuming structural identity conditions would generate an incoherence (see Bird, 2007, 132–133). Others have argued that structural identity conditions would make universals unknowable (see Bird, 2007, 133–135). These other forms of objection will not interest for the reasons that Bird himself exposes in the cited passages.
- 11.
Recall that the ordered pair <x, y > is defined as {{x}, {x, y}} à la Kuratowski.
- 12.
An automorphism on the graph G that maps each node of G on itself is a trivial automorphism. What is interesting here are non-trivial automorphisms, of course, in which at least to one node of G is assigned a different node.
- 13.
Suppose, by hypothesis, that a nomic network N* is integrated by the universals U1, U2, and U3. It is true, for example, that U1 depends on U2 and U2 depends on U1. Here it happens that: (i) the asymmetry of dependence is violated, and (ii) by transitivity of dependence, it will turn out that U1 will be dependent on itself. A repair of (i) requires relaxing asymmetry. A repair of (ii) requires either relaxing irreflexivity or relaxing transitivity. Recall that a relation R is said “asymmetric” if it is necessary that, if Rab, then it is not the case that Rba. It is symmetric when it is necessary that if Rab then Rba. It is non-asymmetric when it is not necessary that, if Rab, then it is not the case that Rba. Something analogous is true for reflexive, irreflexive and non-reflexive, transitive, intransitive and non-transitive relationships.
- 14.
A sample of these trends is that in graph theory, as we have seen, lines, arcs, or sides are defined as sets of nodes. A set is ontologically dependent on its elements so that defining lines in this way presupposes—in some way—the ontological dependence of lines on the nodes that these lines connect.
- 15.
The ‘reduction’ of universals to the nomic network, for that matter, is difficult to understand, since there is a plurality of universals and only one nomic network. It only makes sense to complete an identity statement flanked by an expression designating a plurality, by another expression that also designates a plurality.
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Alvarado, J.T. (2020). Identity Conditions for Transcendent Universals. In: A Metaphysics of Platonic Universals and their Instantiations. Synthese Library, vol 428. Springer, Cham. https://doi.org/10.1007/978-3-030-53393-9_10
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