Leibniz's ontological proof of the existence of God and the problem of ``impossible objects``
This article examines one of Leibniz's ontological proofs for the existence of God: \textit{Probatio existentiae Dei ex ejus essentia} (1678, now published in AII,1). Leibniz's proof builds on a criticism of the classical ontological proof (hereinafter OA) endorsed, among others, by Descartes: {\parindent=0.7cm\begin{itemize}\item[--] God is, by definition, the most perfect being. \item[--] The most perfect being includes all perfections. \item[--] Existence is a perfection. \item[--] The most perfect being includes existence. \item[--] Existence follows from the idea of God. \item[--] Therefore God exists. \end{itemize}} According to an objection which goes back to Gaunilo of Mormoutiers, this argument is fallacious because it confuses the existence of an idea with the existence of a thing denoted by this idea. Hence, we could, using the formal proof-structure of OA, prove the existence of all sort of non-existing things. Leibniz holds that OA merely proves the conditional: ``if God is possible, then God exists'' and it is incomplete, because it fails to prove that God is possible. A complete proof by OA would follow either of the schemes below, depending on two different conceptions of God as, respectively, ``the most perfect being'' or ``the necessary being'': OA1 {\parindent=0.7cm\begin{itemize}\item[(i)] The most perfect being is possible. \item[(ii)] If most perfect being is possible, then God exists. \end{itemize}} or: OA2 {\parindent=0.7cm\begin{itemize}\item[(i)] The necessary being is possible. \item[(ii)] If the necessary being is possible, then God exists. \end{itemize}} The author's paper considers Leibniz's attempt to prove OA2, analysing it using firstly the formalism of modern modal logic and then using a logic of concepts developed by Leibniz himself. The interpretation of Leibniz's proof through the lens of modal logic brings to light certain ambiguities in Leibniz's own reasoning. In particular, the impossibility of an entity x is understood, in the course of the same proof, in two ways: i) Firstly, in a propositional sense: ``it is impossible that x exists'' or ii) in the sense that x is impossible if contradictory propositions are true about x. The equivalence between i) and ii), which appears to be crucial to carry out the proof, has not been proven by Leibniz. Moreover, the meaning of ``impossibility'' given in ii) is utterly problematic in the light of Leibniz's logical system. Leibniz argued that a definition of an ``impossible object'' as an object which involves a contradiction can entail the existence of impossible objects such as a ``square circle''. According to Leibniz, such an impossible entity exists because the proposition: ``a square circle is not a circle'' is true: {\parindent=0.7cm\begin{itemize}\item[(i)] A square is not a circle (T). \item[(ii)] A square circle is a square (T). \item[(iii)] A square circle is not a circle. \end{itemize}} At the same time, the proposition: ``a square circle is not a circle'' involves a contradiction, hence the square circle is an impossible object. The author points out a flaw in the argument, as the premiss (ii) can be translated using first-order predicate logic as follows: ``there exists a x, such that x is a Circle and x is a Square'', which is a false proposition. Moreover, it negates premiss (i) (in the author's translation: there exists a x, such that if x is a square, then x is not a circle). Even if Leibniz's example is flawed, existing impossible objects may be attested on the basis of mathematical practice. For example, let us consider the case of imaginary numbers: {\parindent=0.7cm\begin{itemize}\item[(i)] According to the Euclidean notion of number, the square of every known number is \(>\) than 0. \item[(ii)] There is a entity which is a number and whose square is \(<0\) (this is the imaginary number ``i''). \item[(iii)] Conclusion: there is a number whose square is both more than and less than 0. \end{itemize}} The premisses are mutually inconsistent but both true, while the conclusion states two contradictory propositions about a certain object -- the number we denote with the symbol ``i''. On the other hand, the existence of imaginary numbers may be ascertained not on logical grounds, but on the grounds of their reliability as it was customarily done in XVI and XVII mathematics (as, for instance, in Bombelli, which introduced imaginary numbers are computational devices). The case of imaginary numbers was an example certainly known to Leibniz. Although no mention of them can be found in Leibniz's \textit{Probatio}, it seems worth investigating with more detail whether the appeal to imaginary numbers supports Leibniz's thesis according to which some impossible objects really exist. In the final section of the paper, the author takes a different perspective and examines Leibniz's argument through the framework of Leibniz's own logical calculus or ``algebra of concepts''. The conclusion is that in this framework we can prove the following: ``If God is possible, then God exists'', but we cannot prove the premise: ``God is possible''. Eventually, Leibniz's proof of God's existence fails to prove what it set out to prove, namely the possibility of God.
scientific article; zbMATH DE number 6706574