Types with only one token

Question

This article seemed to me to suggest that a kind can occur more times than it has tokens, even if the token is a concrete object.

E.g. mister smith is a single token that occurs twice in the list of lottery winners.

The article seems to base the sense of that claim based on sequences

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

But what if a type has [and necessary must have] only one token - can a sequence be constructed of a type with just one token?

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.

i.e. does this idea of "sequence" allow for types with one token and if so why or why not?

Answer 1

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.

Answer 2

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.