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 p→LMp, 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:
- MLG. (By 1 and 2 and K)
- ~G→LM~G. (Brouwer)
- MLG→G. (Contraposition on 4)
- G. (Modus ponens on 3 and 5)
This doesn't use S4. So worries about the transitivity of possibility are irrelevant here.
Griffin attributes the Brouwer-based argument to Leibniz.