# Drawing a distinction between a 'type', the set of it's tokens

If we take an abstract 'type' like 'man', this type sort of defines the required characteristics to be 'a man', however what is the difference between the type 'man' and the group/set of 'men'?

For example if we take the real numbers, how does the 'reals' differ from the idea of 'real number', are they the same?

How come we use 'man' to sometimes refer to the set of all humans, I assume this is an element of natural language, so refers to the set of tokens of the type 'human' and not the type 'man'?

When does a property refer to the 'type' and the set? If we have a set, do we make a distinction for a property that the tokens share and a property of the set, the set itself, or property of the 'type'?

If I have a property like 'men have beards' is that describing a property of the 'type', a property of the 'set' or a common property of the tokens?

• Indeed the truth-conditional semantics of sets and types would be almost synonymous if you only understand them in terms of traditional logic without computational procedure. For example in mathematical type theory logic and recursion are completely subsumed by its various type-forming rules such as sum type, inductive type, dependent type in Martin-Löf type theory combining the benefits of both the material set theory such as ZFC and categorical structural set theory such as ETCS whose propositional or judgmental elements have no proof computational info, while any element of a type has... Commented Oct 9, 2022 at 5:43

Lot's of questions, eh?

If we take an abstract 'type' like 'man', this type sort of defines the required characteristics to be 'a man', however what is the difference between the type 'man' and the group/set of 'men'?

This is straightforward. The type 'man' is a collection referred to by the word 'man'. A set of all members of the type 'man' is an extension. This is in distinction to determining members of a collection by intension. If 'Socrates' is in 'man', then 'Socrates' is a token or instance in this context and could be listed in the extension. In set theory:

man := {Socrates, man1, man2, ...}

What is the difference? Well, if you take type to mean a general collection of more specific instances, nothing. Why the synonym? Charles Sanders Pierce was one of the founders of semiotics, and had an interest in languages and needed terminology to differentiate between the idea (man) and multiple instances of the word in the sentence "A man is not a man, until he decides to be a man." Here, there is man-as-type, and man-as-token. Presumably, man refers to the same idea (isn't polysemous) but is written three times. In computer science, we use the term token in exactly the same way.

how does the 'reals' differ from the idea of 'real number', are they the same?

'Reals' is a synonym for 'real number'.

How come we use 'man' to sometimes refer to the set of all humans, I assume this is an element of natural language, so refers to the set of tokens of the type 'human' and not the type 'man'?

Words start with meaning, and then often are used in different senses to differentiate experience. This is the polysemy I referred above. There's a lot of research into how it happens, but the basics are, one group of people uses it one way, another a different way, and then both people just accept there are two senses. In literature, they're usually indicated with the subscript. Thus, if you open up the OED to 'man' you'll see definitions man1, man2, man3, ...There are some words that dozens of distinctions depending on the place and the people. Both etymology and historical linguistics study this sorts of differentiation of meaning.

If I have a property like 'men have beards' is that describing a property of the 'type', a property of the 'set' or a common property of the tokens?

Excellent question. First, the property is an intension: {x : x has a beard}. Intensions are used as rules as opposed to relying on lists. Now, the question is when you say man, are talking about a collection of real entities or a the language used to label the real entities. Here, things get philosophically tricky and you're entering metaontological and ontological discourse. Intuitively, we can differentiate between man and 'man'. This is called use-mention distinction. Man without delimitters is generally presumed to refer to the being with physical existence, while the 'man' with delimitters refers to the language. What's confusing is they're both language: the former is language to talk about the physical being, and the latter is language to talk about the language referring to being. It's meta. In fact, it's metalanguage.

All of this stuff traces its origins back to the problem of universals, and if you study your philosophy, you'll find that men like Quine, Carnap, Socrates, Plato, and others have very different views.

When you are talking about types of real physical objects, there really isn't much call to refer to the set of the existing tokens of that type. Statements like "men have beards" is presumably not just about the men who exist today, but also the men who existed yesterday and the men who will exist tomorrow. In fact, it also refers to possible men: "if it was a man who murdered Mildred, then he had a beard", and fictional men: the Count of Monte Cristo had a beard.

Therefore most such sentences have to be understood as referring either to the type or to the set of all possible members of the type. In normal language, it doesn't matter which one you are referring to so there probably isn't a real answer. In formal language, of course, you would specify.

In addition, there are statements specifically like "men have beards" which are true, but not true of each individual member of the set of all possible men. There are men who do not have the potential to grow facial hair. To be true then, the sentence has to be construed as asserting a general fact about a type rather than a universal fact about a set, so in this case, even in casual conversation, you have to assume that the sentence is about the type.

In mathematics there is no distinction between existing and possible members of a set, in a statement like "all real numbers have a square root", it doesn't matter whether you are referring to the type or the set; however, there are cases where it does matter.

I'm going to take a bit of liberty with the notion of a type, but here is an example: "the odd numbers between three and seven are the same as the prime numbers numbers between three and seven". Here, the sentence is only true if you are talking about the members of the named sets and not the named sets themselves. The type "odd number between three and seven" is clearly not the same type as "prime number between three and seven".

In the case where we use the name of a type but are speaking of the members of that type, we are speaking extensionally. Mathematics is generally extensional, so in mathematics, you are usually talking about the the set.