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In (computational) commonsense reasoning, so-called typical elements of sets are used (as described her). I understand why they are useful from the point of view of applied logic but what is their philosophical status? I tried to find an answer in a few books on metaphysics and ontology but couldn't find any discussion of them.

My understanding is this: From a set-theoretic perspective, the axiom schema of comprehension could to extended to include the existence of typical elements—for every set there exists a typical element of it whose principal (extranuclear?) property is that whatever is true of it is also true of any member of the set. From a philosophical perspective, can we say that (1) if sets exist, then typical elements of sets exist and (2) sets and typical elements are distinct (though closely related) entities?

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    Hello. To me it sounds like a semantic trick without any ontological significance. There is no "typical dog". – Ram Tobolski Apr 16 '15 at 17:18
  • I can imagine this being used in sociology, for the study of groups of people. – Keelan Apr 17 '15 at 6:25
  • @tobolski: it occurred to me that a typical element is actually used in a first order expression; for all x in a set A; does this not count as a 'typical' element? – Mozibur Ullah Apr 18 '15 at 9:46
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    I suppose there again in terms of the usual set ontology it doesn't count though as an actual element as such. – Mozibur Ullah Apr 18 '15 at 9:47
  • @tobolski -- that is a variable representing an element. From the POV of set theory these are not elements. Nonstandard Analysis allows a rigorous approach to treating variables representing objects as if they were actual objects. We do this informally all the time, but it can get one in trouble if the object itself might be impossible. You can prove a lot of stuff with a typical element of the empty set, especially by induction. – jobermark Apr 21 '15 at 13:58
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The problem is that "typical elements" actually have to be classes, not set.

If you only allow "typical elements" over normal sets of objects, it works fine. You can get away with treating typical elements as a set. However, consider a set of objects, one element of which is a "typical element." Now we have to consider meta-typical-elements.

At some point, you eventually want to talk about "the set of all typical elements" or something similar. In doing so you create a sort of a tail-chase where the set refers to itself.

This can create all sorts of paradoxes, as famously explored by those like Russel and Whitehead. In the end, the best agreement was that such things were better deemed "classes," which is a step more abstract than sets and have some properties of their own. Classes provided constructions which allowed one to describe such self-referential structures without paradoxes.

The price of this admission is that First Order Logic operates on a set of objects (the Universe). In very imprecise terms, you can't fit a class into a set. Accordingly, it's not as useful.

You can define some "typical elements" in your logic, but you have to be careful not to admit any of the ones which are self-referential in difficult/paradoxical ways. That limits their usefulness.

  • Do you mean categories as in category theory - which seems somewhat puzzling in the context of your answer? I'd also understood the normal terminology for 'collections' that are too large to be sets are called classes? – Mozibur Ullah Apr 18 '15 at 9:49
  • @MoziburUllah Thank you for catching that! I absolutely mixed up my terminology. I have edited the answer to change the wording to "classes." – Cort Ammon Apr 18 '15 at 17:14
  • Recursive function theory allows for lots of tail-chasing in definitions, and is still about sets, not classes. So without a better clue, I don't see where the set vs class distinction means much here. – jobermark Apr 19 '15 at 16:25
  • @jobermark some tail chasing can be allowed with recursive functions, and the results are still sets. However, it is my opinion that it is helpful to know that there are descriptions of things we call "sets" that actually are not sets. It took a lot of mathematicians quite a lot of work to figure that one out, so a budding philosopher might be good to know ahead of time what pitfalls could come from overextending the potential ways to construct a set. – Cort Ammon Apr 19 '15 at 17:09
  • @CortAmmon The answer has pedagogical value, but I think it is false, taken literally. I think your loop reduces to something that can be handled recursively, and is not a candidate for paradoxes. The motivation is that the 'typical element' of a set is captured by a set of propositions and the set of all propositions of a given order is a set, not a 'proper' class. – jobermark Apr 19 '15 at 17:32
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I find this a silly name, since the 'typical element' of any set is never an actual element of that set. But the basic idea captures the dualism between properties and sets. @MoziburUllah is right to point you at mereology, which is a different way of looking at set theory that better leverages this dualism. But it is a little deep for a simple question.

Object-oriented computer people encounter this all the time. By adding a property to an object's description (a member in its class, a field in its memory image), making the object's description bigger, you are making the set of objects described smaller in a perfectly predictable way. Conventions of speech let you talk about the growing description and the (complement of) the shrinking set as "the same thing".

In that sense, 'typical elements' are the 'monads' of sets, in the sense of Nonstandard Analysis, the same way infinitesimals are the 'monads' of the Real numbers. Things like this can be manipulated formally like the real thing, up to a point, in a way that makes things simpler.

Monads are really abstractions that represent collections of restrictions on actual elements -- (2 + epsilon is bigger than 2, but smaller than everything real that is bigger than 2). As such, they are not really elements, just proxies for them. So some "for all" statements apply to them, and some don't. It is not always easy to know which is which. Nonstandard Analysis contains a theorem which tells us (/Los' theorem), but it is not always obvious in informal conversation or even complex instances of normal usage.

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typical elements exist.

As the comment by Tobolski points out there are no such things as 'typical elements'; I'd say that it is an artefact of programming/applied logic to represent the membership clause as an element.

if sets exists then (typical) elements exist

The Ontology of the usual Set Theory has to be granted all at once - ie that there are sets and that there are elements - for what kind of Set Theory would we have if there were sets but no elements? It would mean that all sets would be empty; contra this, suppose we had only elements but no sets, then we can't form sets through the axiom of comprehension which gathers elements together into a set.

This isn't the only Set Theory as a formal theory of Wholes and Parts; there is category theory whose ontology instead of sets and elements is instead sets and functions - except there they are called objects and morphisms; others are mereology and topology.

  • You have to look at the example he gave. "Typical elements" are never elements, they are like the epsilons and deltas in an analysis proof, they abstractly represent a random choice subject to stated conditions. – jobermark Apr 21 '15 at 13:21
  • @jobermark: I did look at the example he gave; it looked like I said like an 'implementation' issue; and therefore as Tobolski pointed don't have ontological status; the same goes for epsilon and deltas wrt to the real-line: the real-line has ontology but the epsilons-deltas do not. – Mozibur Ullah Apr 21 '15 at 13:39
  • If you think otherwise, do you have an online reference where 'typical elements' are defended? – Mozibur Ullah Apr 21 '15 at 13:40
  • See, the comments under the OPs post. – Mozibur Ullah Apr 21 '15 at 13:41
  • In Nonstandard Analysis (which I alluded to in my answer). Typical elements exist like topological modads, or Real infinitessimals exist -- as bound variables acting as members under the theory of /Lo`s (I cannot type the diacriticals on the name) – jobermark Apr 21 '15 at 14:04

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