Abstract
A simplified variant of Gödel’s ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. The variant presented has been obtained as a side result of a series of theory simplification experiments conducted in interaction with a modern proof assistant system. The starting point for these experiments was the computer encoding of Gödel’s argument, and then automated reasoning techniques were systematically applied to arrive at the simplified variant presented. The presented work thus exemplifies a fruitful human-computer interaction in computational metaphysics. Whether the presented result increases or decreases the attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and theology.
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Notes
- 1.
Some background on modal logic (see also Garson, 2018, and the references therein): The modal operators □ and ◇ are employed, in the given context, to capture the alethic modalities “necessarily holds” and “possibly holds”, and often the modal logic S5 is used for this. However, logic S5 comes with some rather strong reasoning principles, that could, and have been, be taken as basis for criticism on Gödel’s argument. Base modal logic K is comparably uncontroversial, since it only adds the following principles to classical logic: (i) If s is a theorem of K, then so is □s, and (ii) the distribution axiom □(s → t) → (□s → □ t) (if s implies t holds necessarily, then the necessity of s implies the necessity of t). Modal logic KT additionally provides the T axiom: □s → s (if s holds necessarily, then s), respectively its dual s → ◇ s (if s, then s is possible).
Modal logics can be given a possible world semantics, so that □s can be read as: for all possible worlds v, which are reachable from a given current world w, we have that s holds in v. And its dual, ◇s, thus means: there exists a possible world v, reachable from the current world w, so that s holds in v.
- 2.
¬ϕ is shorthand for λx ¬ ϕ(x).
- 3.
Alternatively, we may postulate A3’: The conjunction of any collection of positive properties is positive. Formally, \( \forall \mathcal{Z}.\left(\mathsf{P} os\mathcal{Z}\to \forall X\left(X\sqcap \mathcal{Z}\to \mathsf{P}X\right)\right) \), where \( \mathsf{P} os\ \mathcal{Z} \) stands for \( \forall X\left(\mathcal{Z}X\to \mathsf{P}X\right) \) and \( X\sqcap \mathcal{Z} \) is shorthand for \( \square \forall u.\left( Xu\leftrightarrow \left(\forall Y.\mathcal{Z}Y\to Yu\right)\right) \).
- 4.
Monotheism results are of course dependent on the assumed notion of identity. This aspect should be further explored in future work.
- 5.
In this countermodel, the two possible worlds i1 and i2 are both reachable from i2, but only world i1 can be reached from i1. Moreover, there is a property ϕ, which holds for the (single) God-like entity e in world i2 but not in i1. Apparently, in world i2, modal collapse ∀s (s → □ s) is not validated, since (ϕ e) holds in i2 but not in i1, which is reachable from i2. The positive properties in this countermodel are as follows : ϕ is positive only in world i2, while property ϕ′, which holds for e in i1 but not in i2, is positive only in i1. λx ⊤ is a positive property in both possible worlds, and λx ⊥ in none of them.
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I thank Andrea Vestrucci and the anonymous reviewers for valuable comments that helped improve this chapter.
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Appendix: Sources of Conducted Experiments
Appendix: Sources of Conducted Experiments
The universal meta-logical reasoning approach at work: exemplary shallow semantic embedding of modal higher-order logic K in classical higher-order logic
Verification of Scott’s variant of Gödel’s ontological argument in modal higher-order logic KB, using first-order and higher-order possibilistic quantifiers; the theory HOML from Fig. 1 is imported
Simplified ontological argument in modal logic K, respectively KT, using possibilist first-order and higher-order quantifiers
Simplified ontological argument in modal logic K, respectively KT, using actualist first-order quantifiers and possibilist higher-order quantifier
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Benzmüller, C. (2023). A Simplified Variant of Gödel’s Ontological Argument. In: Vestrucci, A. (eds) Beyond Babel: Religion and Linguistic Pluralism. Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-031-42127-3_19
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