‘Modal’ – Pertaining to the modes of existence (de re) or of propositions (de dicto) as necessary or possible. ‘Necessity’ is a mode of being for a thing or proposition as is ‘Possibility’.
‘Ontological’ – from Greek ontoV for being.
‘Argument’ – designed to logically support a proposition (not to be confused with persuasion which is a psycho-social phenomenon, not a philosophical one).
Throughout this description I shall use standard notation and notation used when the font is restricted to a single typeset as in a text only document for HTTP purposes on the Internet.
The modalities are symbolized as follows:
A square or in typeset  preceding an expression means “It is necessary that…” or “It is necessarily the case that…” or simply “Necessarily…” e.g. as applied to a propositional function.
Ps/Ps – “It is necessarily the case that s is P” where s is a constant referring to some individual and P is a predicate.
A Diamond à or in typeset <> preceding an expression means “It is possibly the case that…” or “It is possible that…” or simply “Possibly…”
Possibility is defined as consistency. àPs/<>Ps reads as “Possibly, s is P” and means that there is no contradiction in attributing P to s. Necessity is defined as “not possibly not the case”. If something cannot not be, then it must be.
Psº~à~Ps or Ps=~<>~Ps
There are many different ways to axiomatize a logic, just as there are different ways to axiomatize geometry. Axioms in some systems will be theorems in others, but since axioms and theorems have the same validity it is only a matter of formal difference. One of the most used systems of modal logic is called S5. There is an interesting theorem in S5 called Brouer’s Theorem.
(PàP)à(àPàP) or (P-->P)-->(<>P-->P)
This theorem is derivable in weaker systems as well.
The modal ontological argument for the existence of God is just a substitution instance for this theorem. There are only two propositions needed.
First comes the definition of God as a being who, IF he exists, does so necessarily, i.e. a Necessary Being. This is only the definition of what God would be like IF he existed. The proposition is formalized as
GàG or G-->G
“If God exists, then he necessarily exists.”
The other proposition is the assertion that it is possible that God exists.
àG or <>G
“Possibly, God exists.”
RULES OF INFERENCE
The only rule of inference needed is Modus Ponens.
PàQ “If P, then Q”
Now we are ready to put the argument together.
(Theorem, sub G for P)
(Def of God)
(1, 2 MP)
(4, 3 MP)
1. (G-->G)-->(<>G-->G) (Theorem, sub G for P)
2. G-->G (Def of God)
3. <>G (premise)
4. <>G-->G (1, 2 MP)
5. G (4, 3 MP)
It is quite a simple argument which makes it hard to understand its fullness. The simple is packed with meaning. As you can see, there is one and only one premise, that it is possible that God exists. If this be granted, then his necessary existence follows. Since all efforts to show that the concept of God is contradictory have failed heretofore I conclude, somewhat reluctantly, that God exists. Kai Neilson tried to argue this in his debate with J.P. Moreland, but didn’t make much progress.
Now I realize that to the average person, this seems like a trick, but the average person is not particularly accustomed to following logical arguments at all, much less highly specialized forms of logical calculi developed by professional philosophers. Most professors at the University level don’t even know modal logic and many have never studied it and some have never heard of it. What do those who know it, but don’t believe in God say? They say that the concept of God is incoherent. I have not yet seen an even slightly plausible argument to that effect. Until I do, the OA will be cogent to me. I might add that I am a convert on this argument. I argued for years that the ontological argument was flawed until someone showed me the modal version. I have always followed Reason wherever it lead and, as usual, it lead to God.
Adams, Robert M., _The Virtue of Faith_, esp. “The Logical Structure of Anselm’s Arguments,” Oxford University Press: 1987.
Moris, Thomas V, _Anselmian Explorations_, esp. “Necessary Beings,” University of Notre Dame Press: 1987.
Plantinga, Alvin, _The Nature of Necessity_, esp. “God and Necessity,” Oxford University Press: 1974, 1992.
Plantinga, Alvin, _The Ontological Argument_, Anchor Books, 1965.
Swinburne, Richard, _The Coherence of Theism_, Oxford University Press: 1977, 1993.
Oddly enough that quotation is linked to a site by an atheist named Adrian Barnett who is attacking my older version of this argument, but he was gracious enough to put this quotation, which I think works against his argument, by a philosopher in the UK.