24 ideas
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
| A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
| A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
| A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
| A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
| From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
| 13134 | We negate predicates but do not negate names [Westerhoff] |
| Full Idea: We negate predicates but do not negate names. | |
| From: Jan Westerhoff (Ontological Categories [2005], §88) | |
| A reaction: This is a point for anyone like Ramsey who wants to collapse the distinction between particulars and universals, or singular terms and their predicates. |
| 13124 | Categories can be ordered by both containment and generality [Westerhoff] |
| Full Idea: Categories are usually not assumed to be ordered by containment, but also be generality. | |
| From: Jan Westerhoff (Ontological Categories [2005], §02) | |
| A reaction: I much prefer generality, which is responsive to the full picture, whereas containment seems to appeal too much to the orderly and formalised mind. Containments overlap, so we can't dream of a perfectly neat system. |
| 13117 | How far down before we are too specialised to have a category? [Westerhoff] |
| Full Idea: How far down are we allowed to go before the categories become too special to qualify as ontological categories? | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: A very nice question, because we can't deny a category to a set with only one member, otherwise the last surviving dodo would not have been a dodo. |
| 13116 | Maybe objects in the same category have the same criteria of identity [Westerhoff] |
| Full Idea: There is an idea that objects belonging to the same category have the same criteria of identity. This view was first explicitly endorsed by Frege (1884), and was later systematized by Dummett (1981). | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: This approach is based on identity between equivalence classes. Westerhoff says it means, implausibly, that the resulting categories cannot share properties. |
| 13118 | Categories are base-sets which are used to construct states of affairs [Westerhoff] |
| Full Idea: My fundamental idea is that 'form-sets' are intersubstitutable constituents of states of affairs with the same form, and 'base-sets' are special form-sets which can be used to construct other form-sets. Ontological categories are the base-sets. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: The spirit of this is, of course, to try to achieve the kind of rigour that is expected in contemporary professional philosophy, by aiming for some sort of axiom-system that is related to a well established precise discipline like set theory. Maybe. |
| 13125 | Categories are held to explain why some substitutions give falsehood, and others meaninglessness [Westerhoff] |
| Full Idea: It is usually assumed of ontological categories that they can explain why certain substitutions make a statement false ('prime' for 'odd'), while others make it meaningless ('sweet' for 'odd', of numbers). | |
| From: Jan Westerhoff (Ontological Categories [2005], §05) | |
| A reaction: So there is a strong link between big ontological questions, and Ryle's famous identification of the 'category mistake'. The phenomenon of the category mistake is undeniable, and should make us sympathetic to the idea of categories. |
| 13126 | Categories systematize our intuitions about generality, substitutability, and identity [Westerhoff] |
| Full Idea: Systems of ontological categories are systematizations of our intuitions about generality, intersubstitutability, and identity. | |
| From: Jan Westerhoff (Ontological Categories [2005], §23) | |
| A reaction: I think we might be able to concede this without conceding the relativism about categories which Westerhoff espouses. I would claim that our 'intuitions' are pretty accurate about the joints of nature, and hence accurate about these criteria. |
| 13130 | Categories as generalities don't give a criterion for a low-level cut-off point [Westerhoff] |
| Full Idea: Categories in terms of generality, dependence and containment are unsatisfactory because of the 'cut-off point problem': they don't give an account of how far down the order we can go and be sure we are still dealing with categories. | |
| From: Jan Westerhoff (Ontological Categories [2005], §27) | |
| A reaction: I don't see why this should be a devastating objection to any theory. I have a very clear notion of a human being, but a very hazy notion of how far back towards its conception a human being extends. |
| 13131 | The aim is that everything should belong in some ontological category or other [Westerhoff] |
| Full Idea: It seems to be one of the central points of constructing systems of ontological categories that everything can be placed in some category or other. | |
| From: Jan Westerhoff (Ontological Categories [2005], §49) | |
| A reaction: After initial resistance to this, I suppose I have to give in. The phoenix (a unique mythological bird) is called a 'phoenix', though it might just be called 'John' (cf. God). If there were another phoenix, we would know how to categorise it. |
| 13123 | All systems have properties and relations, and most have individuals, abstracta, sets and events [Westerhoff] |
| Full Idea: Surveyed ontological systems show overlaps: properties and relations turn up in every system; individuals form part of five systems; abstracta, collections/sets and events are in four; facts are in two. | |
| From: Jan Westerhoff (Ontological Categories [2005], §02) | |
| A reaction: Westerhoff is a hero for doing such a useful survey. Of course, Quine challenges properties, and relations are commonly given a reductive analysis. Individuals can be challenged, and abstracta reduced. Sets are fictions. Events or facts? Etc. |
| 13115 | Ontological categories are like formal axioms, not unique and with necessary membership [Westerhoff] |
| Full Idea: I deny the absolutism of a unique system of ontological categories and the essentialist view of membership in ontological categories as necessary features. ...I regard ontological categories as similar to axioms of formalized theories. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: The point is that modern axioms are not fundamental self-evident truths, but an economic set of basic statements from which some system can be derived. There may be no unique set of axioms for a formal system. |
| 13119 | Categories merely systematise, and are not intrinsic to objects [Westerhoff] |
| Full Idea: My conclusion is that categories are relativistic, used for systematization, and that it is not an intrinsic feature of an object to belong to a category, and that there is no fundamental distinction between individuals and properties. | |
| From: Jan Westerhoff (Ontological Categories [2005], Intro) | |
| A reaction: [compressed] He calls his second conclusion 'anti-essentialist', but I think we can still get an account of (explanatory) essence while agreeing with his relativised view of categories. Wiggins might be his main opponent. |
| 13135 | A thing's ontological category depends on what else exists, so it is contingent [Westerhoff] |
| Full Idea: What ontological category a thing belongs to is not dependent on its inner nature, but dependent on what other things there are in the world, and this is a contingent matter. | |
| From: Jan Westerhoff (Ontological Categories [2005], §89) | |
| A reaction: This is aimed at those, like Wiggins, who claim that category is essential to a thing, and there is no possible world in which that things could belong to another category. Sounds good, till you try to come up with examples. |
| 14082 | No sortal could ever exactly pin down which set of particles count as this 'cup' [Schaffer,J] |
| Full Idea: Many decent candidates could the referent of this 'cup', differing over whether outlying particles are parts. No further sortal I could invoke will be selective enough to rule out all but one referent for it. | |
| From: Jonathan Schaffer (Deflationary Metaontology of Thomasson [2009], 3.1 n8) | |
| A reaction: I never had much faith in sortals for establishing individual identity, so this point comes as no surprise. The implication is strongly realist - that the cup has an identity which is permanently beyond our capacity to specify it. |
| 13129 | Essential kinds may be too specific to provide ontological categories [Westerhoff] |
| Full Idea: Essential kinds can be very specific, and arguably too specific for the purposes of ontological categories. | |
| From: Jan Westerhoff (Ontological Categories [2005], §27) | |
| A reaction: Interesting. There doesn't seem to be any precise guideline as to how specific an essential kind might be. In scientific essentialism, each of the isotopes of tin has a distinct essence, but why should they not be categories |
| 14081 | Identities can be true despite indeterminate reference, if true under all interpretations [Schaffer,J] |
| Full Idea: There can be determinately true identity claims despite indeterminate reference of the terms flanking the identity sign; these will be identity claims true under all admissible interpretations of the flanking terms. | |
| From: Jonathan Schaffer (Deflationary Metaontology of Thomasson [2009], 3.1) | |
| A reaction: In informal contexts there might be problems with the notion of what is 'admissible'. Is 'my least favourite physical object' admissible? |