Aims

This chapter’ deliberates on explaining and clarifing the principles of Leibnizian metaphysics by means of the resources afforded by the formalisms of modern symbolic logic and the instrumentalities of modern value theory.

A common practice in philosophical exegesis is to explain how historical figures treated our issues in their terms of reference. Here, however, we follow the reverse course of treating their issues in our terms of reference, in deploying the expository resources of modern, logic, semantics, and valuation theory upon Leibniz's philosophical proceedings.

This line of argument is to be typical of the present treatment of philosophical history throughout the book.

Possibilities, State of Affairs, Circumstances

First, some conceptual preliminaries must be addressed.

Possibilities are states of affairs that could conceivably obtain: they represent hypothetically entertainable circumstances and envision conjecturally available situations. Possibilities can be divided into necessary, actual, and merely possible. A possibility is necessarily realized if its obtaining can be established via reductio ad absurdum, that is, if the supposition of their not obtaining entails a (logico-conceptual) contradiction. A possibility is actual if what it claims is indeed the case. Finally, coherent contentions that are neither necessary nor actual are mere possibilities.

We shall employ the propositional variables p, q, r, etc. to range over possibilities at large. To claim p as actually true, we write Tp (or often simply p when this creates no harmful ambiguity).

Since p represents a possibility, we have it that both (∀p)◊p and (∀p)(p → ◊p), where ◊ explicitly betokens the modality of possibility. To affirm that p is necessary, we write □p. It is a matter of logic that □p ↔ ∼◊∼p → p. We accordingly have (∀p)(□p → Tp) and (∀p)(Tp → ◊p).

A proposition is contingent if it is neither necessary nor impossible. With ○ to represent contingency, we have ○p ↔ (∼□p & ∼□∼p) or equivalently ○p ↔ (◊p & ◊∼p).

Possible Worlds

A possible world (abbreviated PW ) will here be represented by the variables w, w1, w2, etc. Such a world embodies a manifold of possibilities which, taken overall, is both consistent (in that never p ∈ w and ∼p ∈ w) and saturated in the sense that for every possibility either it or its negation (but never both) obtains within it.