relations with their location and other objects; for otherwise, congruent
geometric figures would be counterexamples.
While the identity of indiscernibles could be viewed as a principle to be tested
by the instruments and methods of physics and other natural sciences, Leibniz
also used the term ‘identical’ in the sense of identity of meaning, without always
making the difference explicit. He distinguished between contingent propositions
and identical proposition. The truth of the former depends on the existence of
things, whereas the latter are based on the principle of contradiction. Any
sentence of the form is an identical proposition, and any sentence of the
form is a contradiction, regardless of any empirical examination.
Taken as a metaphysical principle, the identity of indiscernibles has far-reaching
consequences. Most importantly, it is an argument against atomism, more
precisely, against atoms as they were comprehended in Leibniz’s time, i.e. the
smallest hard immutable elements of matter. Without going into the many critical
assessments of Leibniz’s ideas, suffice it here to point out that the logic of
identity is deeply interconnected with our understanding of the natural world.
Because it is rigid and narrow, the notion of ‘identity of indiscernibles’ is quite
clear and logically consistent. However, there are many other ways of asserting
identity. A question that arises is whether identity is a relation between objects or
between signs (predicates, proper names) and objects. Consider the principle of
intersubstitutivity. If x 1 is identical with x 2 then x 1 can be substituted by x 2 in all
possible worlds.
For example, let x 1 be ‘Aristotle’ and x 2 ‘the son of Nicomachus’:
(3) Aristotle is the son of Nicomachus.Since Aristotle was indeed the son of Nicomachus, whatever is true of Aristotle
should also be true of the son of Nicomachus. The difference between (2) and (3)
is that (2) tells you nothing, while (3) tells you something about the world which
you may or may not have known. (2) is an analytic sentence, true by virtue of its
form, whereas (3) is a synthetic sentence, contingent on the correctness of the
information it conveys.