Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects«

Abstract

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise (1) God is possible. Second, a demonstration of the “remarkable proposition” (2) If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows: (3) \(g=_{\mathrm{df}} \,\upiota x\square E(x)\). Proposition (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows: (4) \(\diamondsuit E(\upiota x\square E(x)) \rightarrow \, E(\upiota x\square E(x))\). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \(\hbox {F}(y)\) and \(\lnot F(y)\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of (2) turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of (1), i.e. the self-consistency of the concept of a necessary being, remains to be proven.