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|Empty Set||I 374
Empty set/Bigelow/Pargetter: Problem: how we transfer the plural essence to them.
Solution: "rival theory" about which universals constitute sets.
Thesis: Sets result from the relations of coextensiveness between universals. That is, a set is what is shared by coextensive universals. In general: if two universals are not coextensive, they can still have something in common that makes them overlap. This is the set of things that both instantiate ((s) average).
Definition set/rival theory/Bigelow/Pargetter: is then a property of properties. This is something different than the plural essence.
Plural essence/Bigelow/Pargetter: this needs not to be a property of properties, but could be a simple universal that is instanced by individuals. But it can also be instituted by universals, because universals of every level have plural essences.
N.B.: but the fact that it can be instanced by individuals makes the set construction by plural essence to something other than that by coextensiveness.
Definition theory of higher level/terminology/Bigelow/Pargetter: that's what we call the rival theory. (sets of coextensivity).
Advantage: it makes the empty set easier to define.
Empty Set/Coextensive Theory/Bigelow/Pargetter: E.g. Suppose a pair of universals whose extensions are disjoint. These two still have something in common: what all disjoint sets have in common: the empty set. Then we have reason to believe in their existence.
Theory of higher level/Bigelow/Pargetter: can derive plural essences:
Plural essence: E.g. Suppose some things x, y, etc. instantiate a property F, and this in turn instances a property G.
This structure now induces extra properties of the original things x, y, etc., and these properties, although they are instantiated by individuals of lower level, still involve the property of higher level G.
Extra property: here: to have a property of the G-type.
Alternatively: Suppose x has F which again has G. Suppose something else, e.g. z has another property, H, which also has G. We can assume that x has neither H nor G, but z does not have F and not G. Then it follows that x and z have something in common. But this is neither F nor G nor H, but:
Commonality: to have a property that has property G. (As above, the "extra property").
Sets/Bigelow/Pargetter: this can be applied to sets, we say that x, y, etc., instantiate a universal, e.g. F which, in turn, instantiates a universal G.
G: that's what we call provisionally a set.
Set: is a better candidate for the "extra property" than a property of properties.
Definition element relation/Bigelow/Pargetter: is here simply instantiation.
It is an advantage of our theory that it explains the elemental relationship so simply.
Property of Properties/Bigelow/Pargetter: Problem: is separated by a layer in the type hierarchy. And yet x should also be an element of G. So then element-property could not be an instantiation.
Definition quantities/Bigelow/Pargetter: are then plural essences induced by characteristics of properties.
Definition Empty set/Bigelow/Pargetter: is a property of properties, more precisely: a relation between universals. It is what disjunctive couples of universals have in common. This time, however, no extra property of things is induced two levels below. Therefore, it cannot be constructed as a plural essence.
Nevertheless, the empty set exists. Thus we have all that justifies the infinity axiom.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990