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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

Let,

@… ≝ 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)

QED

Improving the Formulation of Bonaventure’s OA

The following formulation relies on one less premise than my previous formulation, and avoids the implication that there are not objects which refer to God and which are not completely God, i.e. that there are not objects of thought to which “God” refers (a problem that resulted from the way I formulated P2 in the earlier version).

D1) God is absolutely complete
P1) If no objects to which “God” refers  are objects that truly and completely possess the divine essence, then God is not absolutely complete.
P2) If there is an object to which “God” refers and it truly and completely has the divine essence, then God exists in reality.
C) God exists in reality

Let,

Cx ≝ x is absolutely complete
Dx ≝ x truly and completely has the divine essence
Rxy ≝ x is the entity to which “y” refers
E!x ≝ x exists in reality
g ≝ (ɿx)Cx

1. (∀x)(Rxg → ~Dx) → ~Cg (premise)
2. (∃x)(Rxg ∧ Dx) → E!g (premise)
3. (∀x)(Rxg → ~Dx) (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)(Rxg → ~Dx) (3-9 IP)
11. ~(∀x)(~Rxg ∨ ~Dx)(10 Impl)
12. ~(∀x)~(Rxg ∧ Dx)(11 DeM)
13. (∃x)~~(Rxg ∧ Dx) (12 QN)
14. (∃x)(Rxg ∧ Dx) (13 DN)
15. E!g (2,14 MP)

QED

A Formulation of Bonaventure’s Ontological Argument

franc3a7ois2c_claude_28dit_frc3a8re_luc29_-_saint_bonaventure

Image Source: Wikipedia “Bonaventure

<<Si Deus est Deus, Deus est.>>

Bonaventure writes the following argument:

No one can be ignorant of the fact that this is true: the best is the best; or think that it is false. But the best is a being which is absolutely complete. Now any being which is absolutely complete, for this very reason, is an actual being. Therefore, if the best is the best, the best is. In a similar way, one can argue: If God is God, then God is. Now the antecedent is so true that it cannot be thought not to be. Therefore, it is true without doubt that God exists (Bonaventure, De mysterio trinitatis 1.1 fund. 29 (ed. Quaracchi V 48).

The overly-simplified version of the argument is:

P1) If God is God, then God is.

P2) God is God.

C) God is.

Noone and Houser (2013) write, “…the premise If God is God is not an empty tautology (Seifert 1992, 216–217). It means ‘if the entity to which the term God refers truly possesses the divine essence.’ And the conclusion means that such an entity must exist.”  This inspired me to reconstruct Bonaventure’s argument as best I can.

Informally the argument is:

D1) “God” is the absolutely complete being.
P1) There is an object to which the term “God” refers.
P2) If the object to which the term “God” refers does not truly and completely possess the divine essence, then God is not absolutely complete.
P3) If object to which the term “God” refers truly and completely possesses the divine essence, then God exists in reality.
C) God, the being who truly and completely possesses the divine essence, 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. This definitions is a definite description, 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. 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 stipulate is logically coherent.
Defense of P1: This is to say that the term “God” refers to some imagined, conceived, or real object. The atheist should agree that “God” refers to some object, even if the object is just something in the theist’s fancy.
Defense of P2: Since the antecedent of (P2) specifies a way in which object to which the term “God” refers would be incomplete, it follows of analytic necessity that the object named by “God” is not absolutely complete, i.e. God is not absolutely complete.

Defense of P3: To grant that there is an object which truly and completely possesses the divine essence is semantically equivalent to granting that that which everyone calls “God”, i.e. a perfect being, exists in reality.

Further notes:

  • In other words, it is asking whether the object to which “God” refers is a perfect being. If it is not a perfect being, then “God” means an absolutely complete being and does not refer to an absolutely complete being. There is an “incompleteness” inherent in this relationship, which means that if “God” fails to refer to that which is truly God, then we mean that God, a complete being, is not a complete being. Our sense of “God” would be contradictory in nature.
  • We cannot include in the sense of what “God” is, the notion that “God” refers to something that isn’t completely God.
  • The only consistent alternative is to mean that the object which we name “God” exists in reality, and completely has the divine essence.
  • What Bonaventure is saying is that the sense of “God” must include that it references God, or else the the sense is incoherent. So to grant that there is an object to which the sense of “God” refers is sufficient to prove there is God.

Formally:

Let,

Cx ≝ x absolutely complete
Dx ≝ x truly and completely has the divine essence
Rxy ≝ x is the entity to which “y” refers
E!x ≝ x exists in reality
g ≝ (ɿx)Cx

1. (∃x)Rxg (premise)
2. (∀x)[(Rxg ∧ ~Dx) → ~Cg] (premise)
3. (∃x)(Rxg ∧ Dx) → E!g (premise)
4. Rμg (1 EI)
5. Rμg ∧ ~Dμ (IP)
6. (Rμg ∧ ~Dμ) → ~Cg (2 UI)
7. ~Cg (5,6 MP)
8. (∃x)[Cx ∧ (∀y){[Cy →(y = x)] ∧ ~Cx} (7 theory of descriptions)
9. [Cν ∧ (∀y){[Cy →(y = ν)] ∧ ~Cν (8 EI)
10. [(∀y){[Cy →(y = ν) ∧ Cν] ∧ ~Cν (9 Comm)
11. (∀y){[Cy →(y = ν) ∧ [Cν ∧ ~Cν] (10 Assoc)
12. Cν ∧ ~Cν (11 Simp)
14. ~(Rμg ∧ ~Dμ) (5-13 IP)
15. ~Rμg ∨ ~~Dμ (14 DeM)
16. ~~Rμg (4 DN)
17. ~~Dμ (15,16 DS)
18. Dμ (17 DN)
19. Rμg ∧ Dμ (4,18 Conj)
20 (∃x)(Rxg ∧ Dx) (19 EG)
21. E!g (3,20 MP)

QED

References:

Noone, Tim and Houser, R. E., “Saint Bonaventure”, The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2014/entries/bonaventure/&gt;.

Seifert, Josef, 1992. “‘Si Deus est Deus, Deus est’: Reflections on St. Bonaventure’s Interpretation of St. Anselm’s Ontological Argument,” Franciscan Studies, 52: 215–231.