Abstract
We propose a modification of Gödel’s ontological argument for God’s existence from his ‘Ontologischer Beweis’ manuscript (1970). We follow a Leibnizian onto-theology, especially two of Leibniz’s letters from 1676 and 1677, to which Gödel could relate. We consider two differences between Gödel and Leibniz. We argue for the superiority of Leibniz’s ideas, while preserving the main structure of the Gödelian argument. Our first aim is to bring Gödel’s concept of positiveness closer to the idea of a Leibnizian perfectio which should not be understood via negations. Our second goal is to analyze the concept of being necessary in terms of a Leibnizian demonstrability. To this end, we formulate an S4 version of Gödel’s argument without using negative predicate terms. We sketch a model for our theory that allows us to express a few specific properties of the Leibnizian God.
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Notes
- 1.
A survey of the logics used in various formalizations of OB is given by Świetorzecka (2016, 25–29). Modal frames were investigated earlier by Kovač (2003). In addition to S5, the mostly used modal logics which do not require many changes in the original structure of the argument in OB are K5, KD45, and KB.
- 2.
Interestingly, the formula (4)
follows from the law about the rationality of necessary statements, which is also attributed to Leibniz by Perzanowski. If we assume that contingency is described by \(\overline {K} A=: \Diamond A \land \Diamond \neg A\), then we can express the statement “Everything that is necessary, is not contingent” as: 
. Using the standard definition: 
, we obtain precisely 
(cf. Perzanowski 1989, 99). - 3.
The rule ⊢ A → B⇒ ⊢◊A →◊B is derivable from
. The theory GO
o is 
transparent in the sense that for every A which is GO
o thesis we get 
only with modus ponens and rules for ∀ (Czermak 2002, 317). - 4.
We need
closure of general closure of G
↔ because we are not in S5. - 5.
The inference from \(\neg \mathsf {P}(\overline {I})\), where: Ix ↔ x = x, and A2 to Th1 was elaborated by Czermak (2002, 316). Christian has shown that the Scott theory is deductively equivalent to the theory which comes from it by replacing A1 with \(\neg \mathsf {P}(\overline {I})\) (Christian 1989, 6–7). Our observations 1 and 4 use the formula ∃φ¬P(φ) which is weaker than \(\neg \mathsf {P}(\overline {I})\).
- 6.
In some of his philosophical notes, Gödel considered positiveness in the sense of “purely good” or “assertions” and suggested a simplification of the ontological argument without A1 (Gödel 1995, 435).
- 7.
Kovač stressed this connection with respect to the passage quoted above. He formalized it in a fragment of second order logic with a ◊ version of axiom 4: ◊◊A →◊A (Kovač 2017). According to this approach, we begin with the assumption
(a1)
then (A is provable from other propositions or true per se).Compossibility of two properties is defined as follows:
(Comp) Comp(λx.(Xx ∧ Y x)) ⇔◊∃x(λx.(Xx ∧ Y x))(x).
We consider two perfections X and Y . We assume indirectly that (1) ¬Comp(λx.(Xx ∧ Y x)). Because (1) is not provable (X and Y are not analyzable) and not true per se, by (a1) we have (2)
. By 
and 2 we obtain (3) ◊Comp(λx.(Xx ∧ Y x)) and so by ◊ version of 4 we obtain (4) ◊∃x(λx.(Xx ∧ Y x))(x). - 8.
The Leibnizian God is neither finite nor infinite in time and space (Futch 2008, 171–194).
follows from the law about the rationality of necessary statements, which is also attributed to Leibniz by Perzanowski. If we assume that contingency is described by 
. Using the standard definition: 
, we obtain precisely 
(cf. Perzanowski 
. The theory 
transparent in the sense that for every A which is 
only with modus ponens and rules for ∀ (Czermak 
closure of general closure of G
↔ because we are not in 
then (A is provable from other propositions or true per se).
. By 
and 2 we obtain (3) ◊Comp(λx.(Xx ∧ Y x)) and so by ◊ version of 4 we obtain (4) ◊∃x(λx.(Xx ∧ Y x))(x).References
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Świetorzecka, K., Łyczak, M. (2020). An Even More Leibnizian Version of Gödel’s Ontological Argument. In: Silvestre, R.S., Göcke, B.P., Béziau, JY., Bilimoria, P. (eds) Beyond Faith and Rationality. Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-43535-6_6
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