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Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus

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Abstract

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott’s manuscript about Gödel’s argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours and uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson’s and Bjørdal’s variants of the ontological argument, which are known to be immune to modal collapse.

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Authors and Affiliations

  1. Department of Philosophy, University of Helsinki, P.O. Box 24 (Unioninkatu 40 A) , Helsinki, Finland

    Annika Kanckos

  2. Theory and Logic Group, Vienna University of Technology, Vienna, Austria

    B. Woltzenlogel Paleo

  3. Logic and Computation Group, College of Engineering and Computer Science, Australian National University, Canberra, ACT, Australia

    B. Woltzenlogel Paleo

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  1. Annika Kanckos
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  2. B. Woltzenlogel Paleo
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Correspondence to Annika Kanckos.

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Presented by Heinrich Wansing

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Kanckos, A., Woltzenlogel Paleo, B. Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus. Stud Logica 105, 553–586 (2017). https://doi.org/10.1007/s11225-016-9700-1

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