The expression problem According Leibniz's thesis of universal expression, each substance expresses the whole world, i.e. all other substances, from its particular, unique viewpoint, or, in other words, the complete individual notion of a substance-the comprehensive notion which includes, in internal terms, everything truly attributable to a substance-is such a universal expression of, or 'involves,' every other substance, or, as Leibniz also frequently states, from any given complete individual notion one can 'deduce' or 'infer' or 'demonstrate,' or from it 'follows,' or it 'leads to,' (all truths about) the whole world. 1 On the other hand, in Leibniz's
There is a certain popular argument, deriving from Ruth Barcan and Saul Kripke, from the conjunction of the Principle of the Indiscernibility of Identicals (PInI, for short) and the Principle of the Necessity of Self-Identity to the Thesis of the Necessity of Identity. My purpose is to show that this argument does not work, at least not in the form it is often presented. I also give a correct formulation of the argument and point out that PInI is not even needed in the argument for the necessity of identity.
Leibniz, it seems, wishes to reduce statements involving relations or extrinsic denominations to ones solely in terms of individual accidents or, respectively, intrinsic denominations. His reasons for this appear to be that relations are merely mental things (since they cannot be individual accidents) and that extrinsic denominations do not represent substances as they are on their own. Three interpretations of Leibniz's reductionism may be distinguished: First, he allowed only monadic predicates in reducing statements (hard reductionism); second, he allowed also 'implicitly relational predicates' such as 'loves somebody' (soft reductionism); third, he allowed also 'explicitly relational predicates' such as 'loves Helen' (nonreductionism). Hard reductionism is problematic with respect to Leibniz's doctrines of universal expression and incompossibility (among other things). Nonreductionism, in turn, faces insurmountable problems with Leibniz's doctrine of self-sufficiency and internal identification of substances, as well as with that of individual accidents. The remaining option, soft reductionism, standing between the other two interpretations, arguably avoids at least some of their problems.
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