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Formalization, Mechanization and Automation of G odels Proof of Gods Existence

Christoph Benzm uller1 and Bruno Woltzenlogel Paleo2


arXiv:1308.4526v4 [cs.LO] 10 Sep 2013

Dahlem Center for Intelligent Systems, Freie Universit at Berlin, Germany 2 Theory and Logic Group, Vienna University of Technology, Austria

Attempts to prove the existence (or non-existence) of God by means of abstract ontological arguments are an old tradition in philosophy and theology. G odels proof [12,13] is a modern culmination of this tradition, following particularly the footsteps of Leibniz. G odel denes God as a being who possesses all positive properties. He does not extensively discuss what positive properties are, but instead he states a few reasonable (but debatable) axioms that they should satisfy. Various slightly dierent versions of axioms and denitions have been considered by G odel and by several philosophers who commented on his proof (cf. [19,2,11,1,10]). Dana Scotts version of G odels proof [18] employs the following axioms (A), denitions (D), corollaries (C) and theorems (T), and it proceeds in the following order:3 A1 Either a property or its negation is positive, but not both: [P () P ()] A2 A property necessarily implied by a positive property is positive: [(P () x[(x) (x)]) P ( )] T1 Positive properties are possibly exemplied: [P () x(x)] D1 A God-like being possesses all positive properties: G(x) [P () (x)] A3 The property of being God-like is positive: P (G) C Possibly, God exists: xG(x) A4 Positive properties are necessarily positive: [P () P ()] D2 An essence of an individual is a property possessed by it and necessarily implying any of its properties: ess . x (x) ( (x) y ((y ) (y ))) T2 Being God-like is an essence of any God-like being: x[G(x) G ess . x] D3 Necessary existence of an individual is the necessary exemplication of all its essences: NE (x) [ ess . x y(y )] A5 Necessary existence is a positive property: P (NE ) T3 Necessarily, God exists: xG(x) Scotts version of G odels proof has now been analysed for the rst-time with an unprecedent degree of detail and formality with the help of theorem provers; cf. [17]. The following has been done (and in this order): A detailed natural deduction proof. A formalization of the axioms, denitions and theorems in the TPTP THF syntax [20]. Automatic verication of the consistency of the axioms and denitions with Nitpick [8]. Automatic demonstration of the theorems with the provers LEO-II [5] and Satallax [9]. A step-by-step formalization using the Coq proof assistant [6]. A formalization using the Isabelle proof assistant [16], where the theorems (and some additional lemmata) have been automated with Sledgehammer [7] and Metis [15].

This work has been supported by the German Research Foundation under grant BE2501/9-1. A1, A2, A5, D1, D3 are logically equivalent to, respectively, axioms 2, 5 and 4 and denitions 1 and 3 in G odels notes [12,13]. A3 was introduced by Scott [18] and could be derived from G odels axiom 1 and D1 in a logic with innitary conjunction. A4 is a weaker form of G odels axiom 3. D2 has an extra conjunct (x) lacking in G odels denition 2; this is believed to have been an oversight by G odel [14].

G odels proof is challenging to formalize and verify because it requires an expressive logical language with modal operators (possibly and necessarily ) and with quantiers for individuals and properties. Our computer-assisted formalizations rely on an embedding of the modal logic into classical higher-order logic with Henkin semantics [4,3]. The formalization is thus essentially done in classical higher-order logic where quantied modal logic is emulated. In our ongoing computer-assisted study of G odels proof we have obtained the following results: The basic modal logic K is sucient for proving T1, C and T2. Modal logic S5 is not needed for proving T3; the logic KB is sucient. Without the rst conjunct (x) in D2 the set of axioms and denitions would be inconsistent. For proving theorem T1, only the left to right direction of axiom A1 is needed. However, the backward direction of A1 is required for proving T2.

This work attests the maturity of contemporary interactive and automated deduction tools for classical higher-order logic and demonstrates the elegance and practical relevance of the embeddings-based approach. Most importantly, our work opens new perspectives for a computerassisted theoretical philosophy. The critical discussion of the underlying concepts, denitions and axioms remains a human responsibility, but the computer can assist in building and checking rigorously correct logical arguments. In case of logico-philosophical disputes, the computer can check the disputing arguments and partially fulll Leibniz dictum: Calculemus Let us calculate!

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