I 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 view each (created) substance is internally individuated, self-sufficient and independent of other (created) substances,² and the complete individual notion of a substance is such that it respects this requirement. Leibniz also says, with an apparent intention of pronouncing his doctrines of expression and self-sufficiency at the same time, that each substance is 'as if a world apart,' or like a 'multiplication' or 'concentration' of the world.³

What may be called Leibniz's expression problem is now, how to reconcile these views with each other, that is, how a substance that (somehow) 'expresses' or 'involves' the whole world (i.e. every other substance), even in the sense that the whole world can be 'inferred' from its complete individual notion, can be self-sufficient, independent, internally individuated and 'as if a world apart' - for does not this self-sufficiency seem to block the expression of others. The purpose of this paper is to give an exact account of this tricky problem of universal expression, an account that retains substances' self-sufficiency under the constraint that the whole world is to be obtained from complete individual notions. More precisely, the aim is to provide a detailed construction of complete individual notions, within the confines of which it can be shown, explicitly, how truths about (or 'involving') all other substances are derivable from a given complete individual notion (i.e. how it is a universal expression) even though this complete individual notion is given in terms that adequately respect internal individuation, self-sufficiency and independence. It will also be shown how the explication of universal expression to be given accounts for Leibniz's thesis of universal change, i.e. the view that any change in any substance is reflected as a real, internal change in each and every other substance. In the final section I shall address, briefly, various potential objections to the account given.

II Complete individual notions, denominations, and monadic ascriptions

On the basis of substances' self-sufficiency, it is (as indicated above) plausible to ascribe to Leibniz the view that the complete individual notion of a substance, although involving every other substance, or being a universal expression, is as such an accurate or proper way of determining that substance, respecting its self-sufficiency, independence and internal individuation. For example, even assuming that 'Helen loves Paris' is true, Helen's complete individual notion does not really contain the notion expressed in 'loves Paris' for this would violate the self-sufficiency requirement (and, furthermore, would leave us with the question of Paris's individuation). In the interest of fixing terminology, and keeping clearly in mind the difference between what is strictly speaking in a complete individual notion, viz., notions that respect self-sufficiency, and what can somehow be inferred or deduced from it, let us say that a complete individual notion consists in the former notions (or that these are in a complete individual notion) while involving also the latter.

How, then, should we understand the distinction between those notions that 'respect self-sufficiency' and those that do not? It seems natural to suppose that this distinction coincides with that between intrinsic and extrinsic denominations.⁴ An intrinsic denomination of a substance, such as (the one expressed in) 'is tall' in 'Helen is tall,' is exclusively about that substance (Helen) in the sense of involving no other substances, or representing that particular...