|Extensionality, philosophy: (also extensionality principle, extensionality thesis) an attempt to make the language distinct by taking complete sets of denoted objects as the meaning of the referring words. See also extensions, intensions, extensional language, ambiguity, propositional attitudes.|
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Set/Identity/Bigelow/Pargetter: a set will change when it receives an additional item or loses one. Logical form: extensional axiom:
(y)(N(y e x) v N ~ (y e x)).
Everyday translation: "Either something is an element of a given set in all possible worlds or in no world. It cannot be element in some worlds, "but not in others".
Principle of Predication/Bigelow/Pargetter: this is an instance of this principle. (See 3.2).
It applies to quantities, but not to universals. A universal is only a set if the extensional axiom applies.
Essential properties/Bigelow/Pargetter: have a very similar character as sets.
Universals: also have essential properties.
Sets: for them, the set of elements is essential. ((s)> Association?).
Sets/Bigelow/Pargetter: are universals. Their essential property, the extensionality is a reflection of the determining essences of universals.
Elemental relation/Bigelow/Pargetter: acts in both directions. All elements taken together could not exist without simultaneously constituting this set. The individual elements could of course. Therefore, belonging to a set is not an essential feature of an element, taken alone ((s) of an individual?).
Definition plural essence/plural essence/Bigelow/Pargetter: is then the essential that affects all elements of a set simultaneously.
It always affects a plurality of things. "To be one of this group of things".
Extensionality axiom/Bigelow/Pargetter: does not yet ensure the existence of sets. This is achieved by the comprehension scheme.
Comprehension scheme/comprehension/abstraction scheme/Bigelow/Pargetter: asserts that for each description there are a lot of things that fulfill this description. (Possibly the empty set). This is one of the dramatic examples for the resulting of ontological conclusions from semantic assumptions.
Formally. Let (x) be an open sentence, then
(Ey)(x)((x e y) equi (x)).
Problem: fortunately or unfortunately the comprehension scheme contains a contradiction: ((s) E.g. possible instance of the schema: "The set of objects that do not belong to a set.
Priest: (1979) concludes that some contradictions are true.
Comprehension scheme/Bigelow/Pargetter: but is not valid because of the contradiction.
Ontology/Bigelow/Pargetter: the decision about what exists should precede the semantics. The semantics can then modify them.
Therefore, we should not expect that the comprehension scheme is valid.
BigelowVsComprehension Scheme: E.g. Suppose a general description, which we want to call an "open-ended type". Perhaps there is a property to be one of the things shared by many of these things that fulfill this description. But then there may be many other things that fulfill the description that do not have the property of being one of these things. There may be things that do not have the property, but they meet the description.
E.g. It may also be that the property of the form "be one of these things" is fulfilled by some, but not by all, things which fulfill the description. (?).
VsComprehension scheme/Zermelo-Fraenkel/ZF/Bigelow/Pargetter: Zermelo-Fraenkel propose a replacement for the comprehension scheme: Separation:
Separation/separation axiom/Bigelow/Pargetter: with the help of other axioms, it entails the existence of sufficient sets for the purposes of mathematics. Especially for the reduction of geometry to the number theory. ...+...
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990