Kinds and Classes

[math] class   set

extension-of/ scope-of

unary predicate

kinds [MB3 143ff]

A kind T = k(R) is determined by a set RProp of [substantial] properties, ie., is the intersection of the scopes S(P) of properties PR (IOW, of the classes determined by properties PR) [MB 143f]. Let p(T) be the properties shared by the members of kind T. Theorem 3.4: A kind is determined by the properties shared by its members: T = k(p(T)) [MB3 149]. An intersection of kinds is a kind. But the union of two kinds is not necessarily a third kind [MB3 149] - because for Bunge there are no properties designated by the disjunction P \/ Q of corresponding predicates => predicates/attributes denote properties.

property  extension-of/scope-of     [onto] classes = property scopes A class C is (a subset of the set Thing of things that is) the scope S(P) of some [substantial] property P [MB3 140]. The scope S(P) of the [unary?] substantial property P is the set of things possessing it: S: Prop -> 2Thing such that xS(P) if x possesses P [MB3 140]. Obviously, any class C = S(P) is a kind C = k({P}). OTOH, following Postulate 3.5, if R is finite then k(R) is a class defined by the property designated by the conjunction of the predicates for properties in R [MB3 144]: k({ P1, ..., Pn }) = i=1...n S(Pi) = P1 /\ ... /\ Pn Postulate 3.5 [MB3 140]: «The intersection of two classes, if nonempty, is a class: For any two compatible properties P, QProp there is at least a third property RProp such that S(R) = S(P) S(Q)». - He must mean the property P /\ Q

Natural kinds (aka. species, nomological kinds) «When laws are made the fundamentum divisionis of a set of things, the resulting kinds are maximally natural - or, in Aristotelian jargon, accident is then unlikely to prevail over essence. The outcome is a set of natural kinds of species. Such sets ... are typical of the advanced modern science in contrast to purely empirical knowledge and to descriptive science. ... Things have then properties of two kinds: the laws characteristic of their natural kind, and idosyncratic properties.»



algebra

Instance: The algebra of property-sets & kinds (thing-sets)

«Both the set of properties of a thing and the set of all the properties shared by all the members of an arbitrary set of things, far from being unstructured sets, are ideals.» [MB3 142]. Principal ideal (Theorem 3.1): The collection p(S(P)) of properties of all members of a class S(P) defined by property PProp is the same as the set of the properties preceding P: p(S(P)) = { QProp | Q < P }        Property precedence [MB3 80]: «If P and Q are substatial properties (i.e. members of P) then P preceeds Q iff P is more common than Q, ie., P < Q = S(Q) S(P).» This is equivalent to: «P preceeds Q iff P is necessary for Q, i.e., P < Q = (x)(x S => (x possesses Q => x possesses P)).» «Note that p(S(P)) is the principal ideal of PProp, i.e. (P)Prop» [MB3 142]. Antitonic p: 2Thing -> 2Prop (Corollary 3.4): The greater the set of things the fewer properties they share [MB3 143]: For T,SThing, TS => p(S)p(T) Antitonic f: 2Prop -> 2Thing: For R,SProp, RS => k(S)k(R) [MB3 147]. Symmetry between p and k (Theorem 3.2): For RProp and TThing, Rp(T) <=> Tk(R) [MB3 148]. p·k [and by symmetry also k·p] is a closure operator (Theorem 3.3) [MB3 148]:   (i) Rp·k(R);   (ii) RS => p·k(R)p·k(S);   (iii) p·k(p·k(R)) = p·k(R) Isomorphy, «the correspondence between kinds and p·k-closed ideals is one to one» (Theorem 3.5) [MB3 149]: The lattice of kinds is isomorphic to the dual of the lattice of closed ideals.