This article suggests that a type can occur more times than it has tokens. e.g., the type mister smith has a single token but occurs twice in the list of lottery winners. The article bases that on sequences

the same person occurs twice in the sequence of New Jersey million dollar lottery winners, remarkably enough. If we think of an expression as a sequence, then the air of mystery over how the same identical thing can occur twice vanishes

Can such a sequence be constructed of tokens that cannot repeat? If not, then it seems that type can't have further types within it

[sequences] distinguish tokens of types from occurrences of types whenever types of things [can] have other types of things occurring in them.

which is exactly what the idea of a "sequence" is meant to resolve.

If lottery winners are barred from playing the lottery again, then the type mister smith can only occur once, in his token. That seems to suggest that each of those lottery winners are tokens of types that do not occur abstractly (lottery winners), but only occur in - and are only - their tokens.

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    As types need, to be understood as a type, more than one token, I think that saying "'my death' is a type" just is a wrong use of the term "type". It can be a token of the type "important events in your life" or "the deaths in your family". So every type could be expressed as a sequence consisting of its tokens, there is nothing obscure behind that thought. I think the difference of token and occurance, which is described in the article, is overseen too easily.
    – Philip Klöcking
    Oct 12 '15 at 21:11
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    @jobermark: thanks ;) And: ocurrance and token. Even if your death can occur only once, there can be several, if not infinite possible tokens thinkable. And only in this sense your death can be a type of them.
    – Philip Klöcking
    Oct 12 '15 at 22:15
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    Another point is that "sequence" does not have to mean "sequence in time", as you seem to take it. A sentence is a sequence, as well as of letters as of words. A sequence may be all tokens of a type put in a line (all STRINGs of an ARRAY, as an example out of programming, or all variables of a vector in mathematics - in these examples only the type occurs and the tokens only as part of the type). And as I take the article, having the background of linguistics, it is probable that "sequence" is meant in this broader sense.
    – Philip Klöcking
    Oct 13 '15 at 8:39
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    You really aren't improving your chances of getting this question answered by cluttering up the comments with rants about downvoting which have been discussed extensively elsewhere already per your request.
    – user2953
    Oct 13 '15 at 10:20
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    @PhilipKlöcking You are missing a distinction in addition to the one between occurrence and token, that is reference. Even a single occurrence can appear in a sequence at multiple places, that does not mean it has multiple instances. Dead is dead, even if we cannot agree what that means (brain death, no breath, etc...) There is only one there, there, even if we continue discussing it forever after, making more and more references.
    – user9166
    Oct 13 '15 at 14:32

The Type-Token distinction is a modern counterpart of the old universal-particular one.

In some field, like linguistic, it is a useful tool; in general, it has the same problems of the old one.

If we consider the mathematical sequence of numbers <0,1,0,1> this is an "abstract" entity : the two occurrences of 1 are not two tokens (like two different "handwritings" of the same word "one") but exactly two occurrences of the same (abstract) object 1.

Regarding the fact that :

Even a concrete object can occur more than once in a sequence — the same person occurs twice in the sequence of New Jersey million dollar lottery winners, remarkably enough. If we think of an expression as a sequence, then the air of mystery over how the same identical thing can occur twice vanishes.

it seems to me that there is a little confusion; if we consider a written list of the New Jersey million dollar lottery winners, then we have a sequence of names, and it is correct to say that we can have multiple occurrences of the same name (an "abstract" ?).

But if we try to "build" the sequence of the men that won the New Jersey million dollar lottery we cannot arrange them with a multiple occurrence of e.g. John Smith; there are no possible "tokens" of him to be used twice in the sequence.

Thus, it seems to me, the individual John Smith is a particular, and it has little meaning to speak of it as a type or as having occurrencs.

The name "John Smith", as a word, is a type, and thus it may have multiple tokens or occurrences.


Perhaps this is just the Computer Scientist in me talking, but all of this is sheer grammatical effusion resolved readily by a single mathematical notion -- that of equivalence relation.

First the computer science:

A sequence is not a collection, it has more structure than a collection, and less direct possession of its contents. The things in a list aren't tokens of the listed items, and they aren't even occurrences of the listed items, they are references to occurrences of the listed items.

Your death is the single token of its type, and its single occurrence. But the sequence, since it does not, in itself, allow for counting, makes for irrelevant obfuscation. There can be a sequence which just lists the same occurrence of the same token of a single type over and over again, (e.g. "Me, me, me, me...!")

If there is a list of all the Names of God, they all reference God, who (in the general notion of big-G God) has a single token and occurrence.

Sequences may separate tokens in a type, but they do not properly enumerate them, or we would have to say that the entries in the collection of all the names of God somehow create copies of God.

So while there can only be a single collection of the tokens of a type with a single token, there can be many sequences.

Now the math:

The less direct ownership implicit in a sequence and resolved in a collection avoids contention over ownership that leads into confusion, and ultimately paradox. The issue is that the distinctness that is required to reduce a sequence to a collection is not always present and safe to rely upon.

A collection requires an equivalence relation that needs to be proved consistent, if we are going to enumerate distinct elements correctly.

If what you mean by 'contain' is to have all the same tokens, and potentially then some more, the single-type can clearly contain no other types, if it can be collected.

And finally an answer to the question as asked:

Yes, there are types with just one token. But the notion of sequence has nothing to do with it. You need another, slipperier concept, involving equivalence, to reasonably establish the number of tokens of a type.

(That is why we have classes that are not sets, and why category theories can still handle them, while type and set theories cannot necessarily do so. The essential binding force here is not elements or exemplars, it is the clarity of the identity relation.)

And a (kind of weak) example to indicate why this concept is truly essential:

The type of God is not introduced pointlessly. It is a singleton type that cannot be collected. The definition itself implies uniqueness, but two images of God seldom agree. People constantly choose for God to have different essential contents, and those using one set find that their God must have qualities forbidden the other's God. He does not equal Himself, and there is not an equivalence relation between the named versions of him.

In that, the type of God seems to have clear subtypes. There are essentially Christian-ish perspectives upon God (who loves us), essentially Deist-ish perspectives upon God (who has left us to our own devices), essentially Pantheistic perspectives upon God (who loves us as his aspects love us and hates us as his aspects hate us), etc. One might think that these cannot reference the same thing. But they reference a thing with a single, consistent definition and are derived from that definition.

The dual definition of a contained type from the one determined by tokens, is that a type contains those types that limit the original type via additional constraints. By that notion, these versions of God do seem to be subtypes. In this type, additional constraints somehow break down a single token into multiple types, violating our intuition of what a constraint does, but in a way that still retains meaning.

So we have proof that if a type cannot be collected, even if we all agree it only has one occurrence and one token, it may still have subtypes.

  • Let us continue this discussion in chat.
    – user9166
    Oct 14 '15 at 12:27
  • btw i have access to the article where Wetzell explain the idea of sequence, and it seems that she allows them to consist of just one occurrence, but it's not immediately totally clear - and i lack the patience for anything today :/
    – user6917
    Oct 14 '15 at 16:43
  • (I'm not your downvote) but please take downvote discussions to meta. People are not required to indicate why they downvoted though it is potentially helpful. If you think you're beginning systematically targeted, (1) there's an algorithm that finds and corrects much of that and (2) you can raise it on meta. And by doing so a moderator can try to look into it
    – virmaior
    Oct 21 '15 at 2:33

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