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:
- L(G→LG).
- MG.
- 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.
5 comments:
Is there a reason you think this argument receives no attention?
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.
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)
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).
Interesting. Thank you for sharing
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