A Combinatorial Theory of Compossibility in Leibniz's Metaphysics

Many contemporary metaphysicians think that for any two distinct things, it is always possible for them to coexist with one another. Leibniz gives a somewhat different answer: two distinct things are able to coexist with one another only when they are compossible. God cannot create all possible substances together because not all of them are compossible. But what is the basis within Leibniz’s philosophy for the incompossibility of substances? This has been one of the most hotly contested issues in the recent secondary literature. Four kinds of interpretations have been presented. Logical interpretations maintain that compossibility is ultimately nothing but logical consistency. Advocates of logical interpretations argue that two possible substances are compossible just in case their complete concepts are logically consistent. In contrast, lawful, cosmological, and packing interpretations assume that possible substances are logically independent of one another. They maintain that any two possible substances are per se compossible. However, God is precluded from actualizing all possible substances by some non-logical constraints. The literature has long been dominated by variations of those four approaches. In this dissertation, however, I show that there is one important issue that has been largely overlooked: the compossibility relation is intransitive. Intransitivity will be problematic for all the above interpretations since they all seem to agree that the compossibility relation is transitive. According to logical interpretations, each possible substance is compossible with and only with its world-mates; thus, compossibility is an equivalence relation (reflexive, symmetric, and transitive). According to lawful, cosmological, and packing interpretations, the compossibility relation is trivially transitive since any two possible substances are per se compossible. However, there are passages where Leibniz suggests that the compossibility relation is intransitive. If compossibility is intransitive for him, then none of those four approaches is on the right track. This indicates that we need a new approach to the puzzle of impossibility. In my dissertation, I present a novel interpretation of compossibility. My alternative has the following features: (1) It uses combinatorial principles to solve the problem of compossibility; God calculates all the possible combinations and the sum of the perfection of each combination by simple mathematical principles. But (2) the combinatorial principles I am relying on are non-Humean. Thus, I deny that everything can be combined with everything else. The intransitivity of compossibility is in fact a natural consequence of non-Humean combinatorialism. Moreover, (3) my view can provide solutions to important puzzles of compossibility. More specifically, it can explain both that (i) Spinozistic necessitarianism is logically impossible and that (ii) the World-Apart scenario is logically possible for Leibniz. Thus, my work will reveal in a rigorous manner what Leibniz has in mind when he says that his metaphysics is nothing but the “Divine Mathematics.”