The BOA with an Actuality Operator “@”

[Note: The following exploration of the Bonavaenturean Ontological Argument (hereafter, the BOA) uses Free Logic and an “actuality” operator.]

Expressed informally

D1) God is the absolutely complete being.
P1) If nothing that satisfies the definite description of God is actually absolutely complete, then God is not absolutely complete.
P2) If something that satisfied the definite description of God is actually absolutely complete, then God exists in reality.
C) God exists in reality

Explanation of D1: Here we stipulate that God is defined as complete in every positive simple attribute, which is to say that by “God”, we mean a perfect being. Given free logic, singular terms that are provided with a definite description do not carry existential import. Maydole (2009, “Ontological Arguments”, Blackwell Companion, 555) explains:

The presupposition is that some referring singular terms and definite descriptions could be free of existential import, and quantifiers should be allowed to range over possibilia (Girle 2003, chap. 4). Otherwise, some referential terms that refer to nonmental things, such as “God” and “the being than which nothing greater can be conceived,” would have to refer to mental things that have existence-in-the-understanding, which makes no sense; or those referential terms would have to have to refer to things that have existence in-reality, which would make the Anselmian ontological argument beg the question.

Maydole’s point with respect to the Anselmian ontological argument applies, mutatis mutandis, to the BOA. This definitions is definite, i.e. it refers to a singular term. Since absolute completeness implies omnipotence, and there can only be one omnipotent being. For, if there were two, one could will contrary to the other, and absurdity would follow, e.g. one wills that at time t1 a surface is entirely red, and another omnipotent being that at time t1 a surface is entirely green.

A stipulation is to be granted, so long as it is coherent, otherwise any conclusion could be deduced from it. As to whether the definition of an absolutely complete being is coherent, it should be noted that perfections, in being both simple and positive, cannot contain any explicit or implicit contradiction, and so the stipulation is logically coherent. For to have a contradiction, one perfection would have to negate the other, either in whole or in part. But for a whole perfection to negate another, the perfection would have to be a negative attribute. And for a part of perfection to negate another perfection, the perfection would have to be complex rather than simple. So perfections are compossible, and the definition coherent. This is based on the Leibnizian argument for the compossibility of perfections.  So here we have a non-question-begging, coherent, definite description.

Defense of P1: The key to defending this premise is to understand how “actually” functions in the argument. In the context of this argument “actually” means that it is the case in our reality. This could be thought in contrast to “imaginably”. For instance, we might say, simply, that Sherlock Holmes is the world’s greatest detective. In one sense, this is true, in that it can be imagined that Sherlock Holmes is the world’s greatest detective. In actuality, though, Sherlock Holmes is not the world’s greatest detective, so it is not completely true that Sherlock Holmes is the world’s greatest detective. That is, “Sherlock Holmes is the world’s greatest detective” is an incomplete expression. The principle behind this premise, then, is the idea that if something is not actually the case, then to say it is the case, simply, is not completely true. Applied, then, to the denial that a thing is actually absolutely complete, and we must infer that it is not completely true that it is absolutely complete. But to deny the complete truth that something is absolutely complete just is to deny that it is absolutely complete.

Defense of P2: This is, of course, not to claim God exists in reality, but is to provide a sufficient condition by which it could be said that God exists in reality. That condition is for an individual to exemplify the perfections of absolute completeness in reality

The Formal Proof


@… ≝ it is actually the case that…
Cx ≝ x is absolutely complete
Dxy ≝ x is the individual by which y is definitionally described
E!x ≝ x exists in reality
g ≝ (ɿx)Cx

1. (∀x)(Dxg → ~@Cx) → ~Cg (premise)
2. (∃x)(Dxg ∧ @Cx) → E!g (premise)
3. (∀x)(Dxg → ~@Cx) (IP)
4. ~Cg (1,3 MP)
5. (∃x)[Cx ∧ (∀y){[Cy →(y = x)] ∧ ~Cx} (4 theory of descriptions)
6. [Cμ ∧ (∀y){[Cy →(y = μ)] ∧ ~Cμ (5 EI)
7. [(∀y){[Cy →(y = μ) ∧ Cμ] ∧ ~Cμ (6 Comm)
8. (∀y){[Cy →(y = μ) ∧ [Cμ ∧ ~Cμ] (7 Assoc)
9. Cμ ∧ ~Cμ (8 Simp)
10. ~(∀x)(Dxg → ~@Cx) (3-9 IP)
11. ~(∀x)(~Dxg ∨ ~@Cx)(10 Impl)
12. ~(∀x)~(Dxg ∧ @Cx)(11 DeM)
13. (∃x)~~(Dxg ∧ @Cx) (12 QN)
14. (∃x)(Dxg ∧ @Cx) (13 DN)
15. E!g (2,14 MP)


Posted on August 2, 2019, in Arguments for God and tagged , , . Bookmark the permalink. 1 Comment.

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