Wednesday, April 10, 2013

Adams' ontological argument

Robert Adams' modal ontological argument in his piece on Anselm in The Virtue of Faith seems not to get much attention. Adams' modal ontological argument doesn't need S5: it only needs the Brouwer axiom pLMp, namely that if p is true, it not only is possible, but it is a necessary truth that p is possible. Here is a version of Adams' argument. Let G be the proposition that God exists. Then as God is by definition a necessarily existent and essentially divine being, that God exists entails that God necessarily exists:

  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)
And by an application of 1, 6, axiom M (the necessary is true) and modus ponens we can even conclude LG, that necessarily God exists.

This doesn't use S4. So worries about the transitivity of possibility are irrelevant here.

Griffin attributes the Brouwer-based argument to Leibniz.


Michael Birdwell said...

Is there a reason you think this argument receives no attention?

Alexander R Pruss said...

By the way, Plantinga's ontological argument in _The Nature of Necessity_ seems to me to only need Brouwer to conclude that there is a maximally excellent being.

Michael Birdwell said...

Then if Brouwer is more modest, why did he never attempt to formulate it that way? (The question of why people prefer S5 over B has bugged me for a long time, maybe I'm just too ignorant)

Alexander R Pruss said...

Well, S5 has the advantage of elegant simplicity. And S5=S4+B, and S4 is pretty plausible. For we want metaphysical possibility to be the broadest non-arbitrary kind of possibility. But if S4 is false for some possibility concept M, there is a broader non-arbitrary possibility concept that does satisfy S4 (namely, M*, where we say that M*p iff M...Mp for some finite number of Ms).

Unknown said...

Interesting. Thank you for sharing