[FOM] Type-Occurrence-Token

[A.P. Hazen]

[In response to John Corcoran's most recent posting.]

    Charles Parsons has written, somewhere, of the "obscure notion" of 
an OCCURRENCE.  Put on hold any doubts you might have about the 
type/token distinction.  (If you don't have any nervousness about 
THAT distinction, go read David Kaplan's "Words" ["Aristotelian 
Society Supplementary Volume" 1990, pp. 93-199], or maybe Peter 
Simons's "Token Resistance" ["Analysis" (the philosophy journal of 
that title, not the math or psychotherapy ones!) v. 42 (1992), pp. 
195-203].  But for the length of this posting, assume the type/token 
distinction.)

    An occurrence of a symbol in a TOKEN expression is a token: in 
writing "cat" you will produce a physical object-- a thin, 
discontinuous, film of ink on part of your whiteboard, say-- and the 
occurrence of "a" in that token of "cat" will be a smaller physical 
object.  But what is an occurrence of a symbol (or, more generally, 
an expression) in a TYPE expression?  It's not a type.  ("Proof": 
There are two occurrences of "b" in "rabbit," but the symbol "b" is a 
single type.)  It's not a token.  ("Proof": just as it makes sense to 
talk of formulas too long ever to be written down -- formula-types 
which will never have tokens -- it also makes sense to talk of 
occurrences in such a formula of terms that are too long ever to be 
written down.)

   Suspicion: "occurrences," if taken seriously, are a third 
ontological category, distinct from both types and tokens!

   I know of little published literature on this.  There's an article 
by Linda Wetzel in "Journal of Philosophical Logic" v. 22 (1993), pp. 
215-220.

   I suspect it is a deep philosophical question: perhaps connected to 
that of the status of objects "in" a structure which arises in 
"structuralist" approaches to the philosophy of mathematics.  (Cf. 
Charles Parsons, "The structuralist view of mathematical objects," in 
"Synthese" v. 84 (1990), pp. 303-346.)  What is surprising is how 
LITTLE it seems to worry actual workers in logic and foundations. 
The explanation, at least in part, seems to be that the concept of an 
occurrence doesn't really do much work, even in the elementary 
metatheory of symbolic logic where one would expect it to.  In 
informal contexts (like: classrooms) we talk about the occurrences of 
a variable in a formula, and which ones are free and which ones are 
bound and....  But in formal, technical, work only the relational 
notion of a variable HAVING OCCURRENCES in a formula is used.  The 
"Syntax" chapter of Quine's "Mathematical Logic" illustrates the 
phenomenon.  (Quine, by using a first-order language of elementary 
syntax,  manages to be fully rigorous and also to keep fairly close 
to the ... umm, grammar? ... of informal discussions  of syntax.)  A 
(quite artificial) notion of an occurrence of a substring in a string 
is defined, but it is subsequently USED only to define the relational 
concepts.

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Allen Hazen
Philosophy Department
University of Melbourne