Why Axiom S5 of Modal Logic is not Decisive for Accepting the Ontological Argument

In this brief paper, I want to share an ontological argument made by Robert Adams without presupposing the validity of axiom S5 of modal logic. Often I hear, to be anecdotal, that Immanuel Kant was the philosopher who showed how the ontological argument failed as a natural theological argument. While I am not a logician (much less a modal logician), this might be of use to those who deny the ontological argument for the existence of God along Kantian lines (or to those who encounter people who argue along those lines). While I may not even understand the argument (as I am sure is probably the case), I will nonetheless provide a dilemma for the person who denies the ontological argument in virtue of their denying axiom S5 of modal logic. So, I will begin by outlining (extraordinarily briefly) what the ontological argument is (beginning with St. Anselm), reiterate the philosopher Immanuel Kant’s objection, and then conclude with Adams’ ontological argument which plausibly avoids his objection.

The ontological argument, simply, is a natural theological argument for the existence of God which begins with

(1) God is that than which nothing greater can be conceived (thought) (St. Anselm)

and concludes with

(2) Therefore, God exists.

Inherent in the concept of God, then, is the property of existence. While this was problematic for even St. Anselm’s contemporary Gaunilo, the philosopher Immanuel Kant’s objection was based on his denial of existence as a predicate. So, as some argue, one need not take a standpoint in the metaphysics of modality which includes the acceptance of S5.

To this, I have very little to say asides that Robert Adams, in his The Virtue of Faith, has an ontological argument without S5 that, as Alex Pruss notes, “only needs the Brouwer axiom p→LMp, namely that if p is true, it not only is possible, but it is a necessary truth that p is possible.” Pruss schematizes a version of the argument as follows:

  1. L(GLG).

Add that possibly God exists:

  1. MG.

The proof is simple:

  1. MLG. (By 1 and 2 and K)
  2. ~GLM~G. (Brouwer)
  3. MLGG. (Contraposition on 4)
  4. G. (Modus ponens on 3 and 5) [1]

So, while some may have avoided the ontological argument axiomatically from their views in the metaphysics of modality i.e., denying S5, it is open for discussion whether or not Adams has solved this problem.

[1] This was retrieved (April 10, 2016) from Alex Pruss’ philosophical blog: http://alexanderpruss.blogspot.ca/2013/04/adams-ontological-argument.html.

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