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.

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