# Logic Expressions and Notation

I am brand new to logic so am trying to teach myself how to write logic expressions. Let's say I take the following Modal ontological argument:

-Premise 1: It is possible that God exists.
-Premise 2: If it is possible that God exits, then God exists in some possible worlds.
-Premise 3:If God exists in some possible worlds, then God exists in All possible worlds.

-Premise 4: If God exists in all Possible Worlds,then God exists in the actual world.

-Premise 5: If God exists in the actual world, then God exists

or the summary of that argument....

-Premise 1: It is at least possible (◊) for God to exist

-Premise 2: If God’s existence is possible, then necessarily (□), God does exist

-Premise 3: Therefore it follows (⊨), necessarily (□), God exists (∃)

In terms of notation what is the best and simplest way to express the premises and argument where P denotes God?

◊P → □P ⊨ ∃P

I know this is most likely wrong so any help much appreciated

• Maybe: ◊∃P, ◊P → □P ⊨ ∃P... But syntax of Modal Logic is slightly different: we have to quantify individuals and not propositions. Thus, ◊∃xGod(x) to mean "It is possible that God exists**. Commented Nov 30, 2022 at 16:23
• Thanks Mauro ALLEGRANZA that this VERY helpful. I hadn't even appreciated why modal logic was required in this example, but now I think I do, as it is attempting to quantify the truth of the judgement in terms of possibility. necessity or impossibility Commented Nov 30, 2022 at 18:33
• @haxor789 I wasn't actually concerned with the argument itself I was using it as an example since it was recently shared on here. For what it's worth, I don't understand Premise 3 either, as it does not seem to necessarily follow. I can only think that it means if God is necessary and exists in one possible world then be necessity he would exit in ALL possible worlds. Commented Nov 30, 2022 at 18:53
• As a side comment, 2, 4, and 5, look like they should be inferable from the logic and so should not be called premises. It depends, of course, on what logic you are using, but just from general principles, only 1 and 3 look like proper premises. Commented Nov 30, 2022 at 22:41
• @haxor789, you might find the SEP article interesting: plato.stanford.edu/entries/god-necessary-being Commented Dec 2, 2022 at 0:36