Type–token relations

Pierce's type-token distinction was first described by him in 1906.^[1] 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.^[2] 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.^[3] 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.

We also have relations of variables to other things. The variable-value relation connects a variable, such as ‘n’ in number theory, to its values, i. e., to the objects in its range—in this case natural numbers: ‘n’ takes zero as a value, also one, two, and so on. The variable-substituent relation connects a variable, again say ‘n’ in number theory, to its substituents, i. e., to the constants that may meaningfully be substituted for it—in this case numerals: ‘n’ has ‘0’ as a substituent, also ‘1’, ‘2’, and so on.

The variable-substituent relation is homogeneous in that it relates types to types. The variable-value relation is non-homogeneous, like the Peirce type-token relation, in that it relates types to non-types.

Further, we have the relation of classes to their members: the class-individual relation as in the writings of George Boole and Alfred Tarski. This relation is non-homogeneous, like the Peirce type-token relation. Perhaps its non-homogeneity prompted people to call it a type-token relation. Similar, but homogeneity is the set-member relation that relates a set to the sets that are its members—as in the set theory due to Ernst Zermelo. The unit set has the null set {} as its only member.

Relations of this general sort are also found in human artifacts. For example, in photography,^[4] the negative-print relation relates a negative to a print. Here the relation is homogeneous in the sense that it connects physical objects to physical objects. Other similar relations relate transparencies to prints, stencils to imprints, typographical types to imprints, and so on. In foundry work, there is the relation of a sand mold to the metal object produced by pouring molten metal into the mold.

Relations resembling the above are found in the works of Plato and Aristotle. For Plato, beauty itself (a form or type) belongs to each beautiful thing (an instance or example) and, in his PARMENIDES, each beautiful thing participates in (has a share in or imitates) beauty itself.^[5] For Aristotle, in his CATEGORIES, the “secondary substance” or class human is predicable of each human, a “primary substance” or individual, and each individual human is a human. Perhaps one of Aristotle’s relations most resembling the Peirce type-token relation is his quality-to-exemplification relation: for Aristotle every white thing has its own individual exemplification of the quality whiteness. Just as “under” humanity there are individual humans, “under” whiteness there are individual whites, so to speak.^[6]

The similarity of many different relations like the above to the Peirce type-token relation warrants their being compared to the Peirce relation. Nevertheless, philosophical and historical clarity might be served by refraining from calling them type-token relations without explicit disclaimer. The expression ‘one-many relation’ would be more suitable if it had not already been saddled with a different meaning.

REMARK: The distinction has been much developed in Linguistics and Philosophy of Language; there are many problems dealing with written tokens and spoken tokens with the possibility of reduplication or different uses of the same token: a written token can be presented in different situations (like the typical "I am out of Office" or "I come back later" or "closed" in front of a shop) and a spoken token can be recorded and repeated. Are these still tokens or something different? Some authors, e.g. Robert Brandom,^[7] speak of the difference between tokens and tokenings, where a tokening is the production or use of a token in a certain context. The written note "I come back later" in front of a shop is a token of the type /I come back later/. But each time you put the written token in front of a shop you have a different tokening. What's the difference? Here the difference is the dependence of the meaning of the token on the context in which the same token is used. "I come back later" means "I come back some time after you have read this token", therefore depends on the time in which the token is read. In pragmatics it is usual to distinguish between "sentence" and "utterance"; this distinction is normally assimilated to the type/toke distinction. But with these kinds of examples it would also be possible to distinguish between sentence-type, sentence-token and utterance. The utterance of a token (that is a tokening) has different meanings depending on the time and place of the use of the token.

References

1. CHARLES SANDERS PEIRCE, Prolegomena to an apology for pragmaticism, Monist, vol.16 (1906), pp. 492–546.
2. JOHN CORCORAN, WILLIAM FRANK, and MICHAEL MALONEY, String theory, Journal of Symbolic Logic, vol. 39 (1974) pp. 625– 637
3. JOHN CORCORAN, Schemata: the Concept of Schema in the History of Logic, Bulletin of Symbolic Logic, vol. 12 (2006), pp. 219–40.
4. Wikipedia: negative (photography)
5. JOHN COOPER (editor), Plato: Complete Works, Hackett, Indianapolis, 1997.
6. JOHN LLOYD ACKRILL, Aristotle’s Categories and De Interpretatione, Oxford UP, 1963.
7. ROBERT BRANDOM, Making It Explicit, Harvard UP, 1994.