Type–token relations

Pierce's type-token distinction was first described by him in 1906. His first example is the definite article, ‘the’, which appears thousands of times on printed pages, hand-written pages, blackboards, signs, marquees, etc. The visible appearances he calls TOKENS of ‘the’. In fact, he uses ‘the’ in quotes as a common noun for a token of ‘the’, saying that there are ordinarily twenty ‘thes’ on a printed page. The single word ‘the’, which he says is invisible unless “embodied’ in a token, he calls a TYPE.

Peirce’s type-token relation, derived from his distinction, relates abstract types to their concrete tokens; its converse, the token-type relation relates concrete tokens to their abstract types. Following Peirce, we may use the relation-verb ‘embody’ for the token-type relation: every token embodies exactly one type. Naturally, the passive of ‘embody’, namely ‘is embodied by’ would be used for the type-token relation: every type is embodied by many tokens, the exact number varying with time. Peirce’s type-token relation is a one-many relation.

It is clear from Peirce’s words that he intended his relation to apply only in connection strings of characters or sounds, counting the space and punctuation signs as characters. In understanding Peirce, it is important to realize that every type has both written tokens and spoken tokens. Thus, his concept of type is more complex than the one in current use today in logic-related fields, and below in this article, in that, in the current sense, types have only written tokens.In fact, types have been identified with geometric shapes, which would not make sense if types had spoken tokens as Peirce intended.

Many similar relations are found in linguistics, logic, computer science, and other fields where no mention is made of the historical Pierce type-token relation. Several have already been called type-token relations, sometimes no doubt in complete awareness that strictly speaking only Peirce’s relation fully fits the name he gave it.

For example, the schema-instance relation holds between a schema such as the numerical identity schema ‘N = N’, where ‘N’ is a schematic placeholder, and its instances: ‘0 = 0’, ‘1 = 1’, ‘2 = 2’, and so on. There is little danger of confusing the schema-instance relation with the Peirce token-type relation since the schema and its instances are all abstract types in Peirce’s sense. Another example is the universalization-instantiation relation that holds between a universalization such as the Numerical Identity Law ‘for every number n, n = n’ and its instantiations: ‘0 = 0’, ‘1 = 1’, ‘2 = 2’, and so on.

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