Signs and objects
In everyday discourse, we use sentences of this synthetic form all the time,
connecting two expressions with ‘is’. The identity relation thus asserted has
three properties. It is reflexive, symmetric, and transitive.
Reflexive:
(2) Aristotle is Aristotle (identical with himself).Symmetric:
(4) If Aristotle is the son of Nicomachus, then the son of Nicomachus is
Aristotle.For transitivity we need another predicate, for instance, ‘a student of Plato’s
academy’, and arrive at the following.
Transitive: and
(5) If Aristotle is Nicomachus’ son, and Nicomachus’ son is a student of
Plato’s academy, then Aristotle is a student of Plato’s academy.It is well known that Aristotle joined Plato’s academy, but only when he was 16
or 17 years old. Is (5) therefore false? If so, the substitution test would be
imperfect.
This raises the question of ‘sense and reference’, the title of a famous treatise by
logician Gottlob Frege (1848–1925), Über Sinn und Bedeutung, which deals
with the problem of identity of meaning as it comes to bear when considering
sentences such as (2), (4), and (5). If a and b are objects—Aristotle and
Nicomachus’ son, respectively—it is unsatisfactory (paradoxical) that a is
sometimes identical with b, and sometimes not. Frege argues that, therefore, an
identity statement ‘ ’ must be conceived as a relation holding between signs
rather than objects:
Nobody can be forbidden to use any arbitrarily producible event or object as a sign for something. In