Modal arguments

Arguments for the Existence of God
by Metacrock - edited by JMT
Used with Permission

Modal Arguments: XII and XIII

This first argument is based upon the ontological argument. If you have not already read the Mode's of Being page, please do so, as it contains crucial concepts necessary for understanding the next several arguments. This argument assumes that the reader has already looked at that page. These arguments turn on modal logic which is a special form of logic that analyzes modes of existence, necessity and contingency.

The Late Charles Hartshorne

Hartshorne (1897-2000) Lived to be 103, at the time of his death in the Fall of 2000, he was known as "the greatest living Metaphysician." Hartshorne was one of the major forces in the "back to God" movement in Philosophy (a term coined by Christianity Today in a 1979 article. His first and greatest claim to fame is as the second most influential voice in process philosophy, along with Alfred North blackhead, but he is also credited as the man who brought the Ontological argument back from ignominious defeat by Kant almost two centuries earlier. Hartshorne was also a recognized authority on birdsong, and an authority on bicycles, having never driven a car a single time in his centogenerian lifespan.

XII. Hartshorne's Modal Argument.

A. The logic of the argument:

This argument is analytical, it proceeds from the basis in logic to argue that the concept of God is such that if we understood the meaning of the terms we would have to conclude that God must exist. Naturally that is a very controversial position. Many Christians and other theists reject the ontological argument on the grounds knowledge must be somewhat empirical. Nevertheless the argument has been used for a long time, and despite its many apparent deaths, it keeps returning in one form or another. Perhaps the best book on the subject is The Many Faced Argument by John Hick. Somehow the ontological argument just wont die. I feel that this is not so much because the argument itself is true as a proof, but because it gets at something deeper than proof, something to do with the way to think about God, and it strikes a deep cord in our consciousness, even though as a proof it may fail. For this reason alone it is important to know, if only to know the concept itself.

1) God can be analytically conceived of without contradiction.
2) Therefore God is not impossible.
3) By definition God cannot be contingent.
4) Therefore God is either necessary or impossible.
5) God is not impossible (from 2) therefore, God is necessary.
6) Whatever is necessary by the force of Becker's modal theorem must necessarily exist.

(This is actually my re-statement of what Hartshorne is saying).

Hartshorne's actual modal logic looks like this:

The OA: an assessment:

by Ed Soybean Hartshorne's ontological argument is based on Anselm's second argument and claims that God's existence is logically necessary. Hartshorne's argument is given here, where "N(A)" means "it is logically necessary that A," "~A" means "it is not the case that A," "-->" is strict implication, "v" means "or," and "g" means "God exists":

g --> N(g)
N(g) v ~N(g)
~N(g) --> N(~N(g))
N(g) v N(~N(g))
N(~N(g)) --> N(~g)
N(g) v N(~g)
N(g) --> g

This argument is valid. Furthermore, given an Angelina conception of God, premises one and five are sound. Premise two is just the law of the excluded middle, and premise three is a law of the modal logic S5. Premise nine is obviously sound, so this leaves premise seven as the only premise to question. Premise seven says that it is logically possible that God exists.

Yes, those funny lines, "g-->N(g)" are the argument, those are the formal symbols used in modal logic.

B. God's Possibility vs. Impossibility.

The argument turns on the distinction between necessity and contingency, and upon the distinction between mere possibility and the nature of necessary being as not mere possible. In other words, God is either necessary or impossible. If God exists than he is ontologically necessary, because he is logically necessary by definition. But if he does not exist than it is ontologically impossible that he exists, or could come to exist. This is because God cannot be contingent, by definition. A contingency is just not God. So if God is possible, he can't be "merely possible" and thus is not impossible, which means he must be necessary.

God is conceivable in analytic terms without contradiction:
The universe without God is not conceivable in analytical terms; it is dependent upon principles which are themselves contingent. Nothing can come from a possibility of total nothingness; the existence of singularities and density of matter depend upon empirical observations and extrapolation form it. By definition these things are not analytical and do depend upon causes higher up the chain than their being (note that the skeptic at this point probably denies the validity of analytic proofs but to reverse the argument must accept such proof).

Since the concept is coherent and not contradictory and is derived from analytic terms, to reverse the argument the atheist must show that God is impossible since the burden of proof is now on the one arguing that a contingent state of affairs could produce a universe in which being has to be.

D. Answering Objections:

1) The argument can be reversed

Atheists have tried to reverse the argument merely by saying:

1) either God exists or he doesn't
2) God is either necessary or impossible. Necessary if he exists, impossible if he does not
3) God is impossible
4) Therefore God does not exist.

But of course this is merely stipulation. They assume that what the argument is doing is just stipulating everything that has been said about God, but on the "Modes of Being" page I show that each of these modalities of existence are logical deductions. Either a thing exists or it does not. One can equivocate about the meaning the term "existence," but here I clearly mean concrete actual existence in the "real" world. If a thing does not exist it is either that it could, but just doesn't happen to exist, or that it cannot exist because it is a conceptual contradiction, such as square circles, or round triangles and so on. Therefore, if it does exist, it is either that it exists contingently or that it is not contingent but exists necessarily (that is it could not fail to exist without contradiction). These are the four most basic modes of being and cannot be denied. They could be subdivided, for example fictional contingency, such as Superman or Dick Tracy, that which would be contingent if it had real concrete actuality, but is merely a fictional concept. But the four modes are the basic logical deductions about the nature of existence.

The idea that the argument can be reversed just by switching the lines and declaring God impossible merely begs the question. Is God really impossible just because we can utter those words? Is God logically necessary just because we can utter those words?. No, but that's not what is being said. God is logically necessary as a concept. That is the nature of the God-concept, that's the idea of God. To deny that would be like saying "how do you know that tables are things to put things on?" Or "how do you know that triangles have three sides?" The question is one of actuality, so if it is possible that God exists than God is ontologically necessary and thus has real concrete existence because since God is not contingent it cannot be that God is "merely possible." If it is at all possible that God exists, than it's not impossible. To show that the argument can truly be reversed the atheist must show why God is impossible, and to do that he/she must show that God cannot be understood analytically without contradiction.

Another attempt at reversing the argument, which is always used on message boards when I make this argument: just to put not in front of each line. "It is possible that god does not exist." The premise is they don't have to prove God is impossible, but just that the possibility of God's not existing reverses the argument.

The problem is, the premise is false. If god is not analytically impossible (contradictory) then God must exist. Thus it is not true that it is possible that God does not exist. The logic works like this:

(1) If God is indeed possible, the God cannot be impossible.

(2) to say God is not possible is the same as saying god is impossible.

(3) if something is possible, it can't be impossible.

(4) you must show why God is impossible.

(5) I have shown why God is possible, because God is conceivable without contradiction.

(6) anticipating answer on entity and consciousness, consciousness is not a primary quality of God. Other things are conscious, that is not something uniquely establishes God as God, logical necessity is such a thing.

(7) If God is possible, and can't be impossible, and can't be contingent, then to be possible for God is to be logically necessary. Thus it does not work to say God is not possible because it isn't true, thus it's a false premise.

To make good on any reversal they must show a contradiction in the concept of God. To this they always retort "well you can't prove that God is not contradictory." But I don't have to prove that. One can assume that if there is no contradiction it is not contradictory. They are the one's seeking to make the reversal, so it's their burden of proof. But to prove that God is possible all one need do is conceive god analytically without contradiction. what else could one do to prove a possibility?

2) The assumption that we are merely loading the concept with terms that make it necessary, or that the definition of God as necessary is arbitrary.

This is really the same argument one must make to reverse the argument of necessary being. This is what atheists always argue. The first thing they say bout it is that we are just arbitrarily sticking on the term "necessary" and playing word games. Some go so far as to try and demonstrate this by sticking the term necessary on other things, such as "purple cow" or anything they think of, and that's supposed to show what we are doing. I regard this move as nothing more than a demonstration that they do not understand the concepts The necessity of necessity and why it must be applied to God is demonstrated on the "modes of being" page. Moreover, this move is nothing more than the perfect Island argument. It can't work because it merely enthrones contingencies. Our reason for saying that God is necessary is much more logical and organic and is much more than a mere word game.

While it is true that God as being itself is a pre-given postulate and is independent of proof because it is part of the definition of God, the realization that being has t be means that this must be the case.

3) The assumption that we are lending existence to a fictional being.

This is merely an assumption. The necessary existence of God is implied in the possibility of God's existence and the realization that the only alternative is impossibility. God is possible and thus necessary. Some have tried to argue that they are breaking up the four categories with a 5th not seen, that of "fictional" but that applies to the category 4 that of non-existing contingency.

4) Equivocating between types of necessity.

The argument says that to say God is necessary as a postulate of definition is speaking of ontological necessity, than to assert the actuality of it is moving from logical to ontological necessity.

To say that a thing is logically possible is to say that it might have existed in the past or may exist in the future. But for God to exist he must always have existed; in the past, in the future, or all time. Given logical necessity the logical possibility of God 's non existence is impossible. Therefore, ontological necessity implies logical necessity. One implies the other and it is a rational move from one to the other.

This argument may seem like merely a trick of words, and modal logic may be controversial, but it turns on very basic logic, such as modus tolens or modus ponens which is accepted by all logicians. On Argument 1 I document Anthony Flew saying that the logical categories of "Necessary" and "contingent" truth are accepted by all logicians.

Concise introduction to the Modal Ontological Argument for The Existence of God.


‘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 "onto" 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.”

The only rule of inference needed is Modus Ponens.
PàQ “If P, then Q”
Therefore Q
Now we are ready to put the argument together.

1. (GàG)à(àGàG)
2. GàG
3. àG
4. àGàG
5. G
(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.

XIII.A non-modal version of Hartshorne's ontological argument from perfection.

This section (non-Modal version of Hartshorne's Perfection argument) was written by a friend of mine screen named URBILD. He's a graduate student in theology.

The following argument is from Hartshorne's Man's Vision of God and the Logic of Theism. Further discussion of the OA can be found in The Logic of Perfection and Anselm's Discovery.
A). Deduced from Divine Perfection.

The Ontological Argument is deduced from the divine perfection, i.e. the relation between the possibility and actuality of God. With finite ideas, the task of knowledge is to decide among three cases:

(1) the type of thing conceived is impossible and hence non-existent;
(2) the type of thing is possible, but there is no actual example;
(3) the thing is possible and there is an example.

The OA holds that only (a) and (b) need to be considered since (b) is meaningless or nonsensical. If it can be shown that the idea of God is not nonsensical, then it follows that it has an actual object, since a "merely possible" God is inconceivable (p.299). Where impossibility and non-actualized possibility are both excluded, nothing remains but actuality (p.300).

B) God cannot be a mere possibility.

The idea of God is the idea of a being everlasting in duration and independent in terms of his individual essence. In other words, God is the only conceivable object which must be un-produced. Gaunilo's objection that if a perfect being must exist, then a perfect island must exist is unfounded for clearly, an island is not in essence un-produced or self-sufficient (p.303).

C) Famous objection answered: Existence not Predicate.

Another objection to the OA is that existence is not a predicate, and hence cannot be implied by the predicate "perfection". But the mode of a thing's existence is included in every predicate. Contingent existence is implied by predicates describing finite beings. Necessary existence is a predicate that uniquely belongs to God. The necessary being is that individual in which existence implies not simply existence, but that there is no separation between possibility and actuality (p.306).(this was the objection of Kant and Bertrand Russell).

D) Unique relation of God and existence.

The OA argues from the unique relation between God and existence. By definition, God's relation to existence is unique. God is the only unsurpassed and unsurpassable being. To object to this idea is to object to the idea of God, not merely the affirmation that "There is a being corresponding to the idea" (p.310).

Now, there is no good reason to object to the exceptional status of God's existence. What is it that gives us the aspect of identity by which we define existence as such? We may try to define existence as such through our own personal identity. But this definition would be solipsistic. Hence, there must be some further aspect of identity like ourselves in being concrete existent, but unlike ourselves in being able to constitute the all -embracing register of existence -- that is what God is.

E. Answer on Perfect Island/purple cow.

An atheist friend of mine on the CARM board likes to use a purple cow rather than an island but it's the same argument. Guanilo argued against Anselm that through his argument anyone could prove anything; he could prove the existence of a perfect island. Usually the atheist will copy the same wording of the original argument the theist has given but using the word "island" or whatever rather than God.

But if this argument was sound it would have killed the OA long ago and yet it keeps coming back. Here is the answer of one philosopher as to why Guanilo's argument fails:

Guanilo says,

"If a man should try to prove to me by such reasoning that this island truly exists, and that its existence should no longer be doubted, either I should believe that he was jesting, or I know not which I ought to regard as the greater fool: myself...; or him, if he should suppose that he had established with certainty the existence of the island" (72b).

Taking stock of Gaunilo's criticism, I want to make two points:

* First, Gaunilo's argument, even if true, doesn't really tell us where Anselm's argument goes Wrong. It doesn't really demonstrate that Anselm's argument doesn't' work.

* Second, this argument only works for contingent things. IT proves that you can't argue from a contingency to a necessity, but it doesn't prove the same of necessary things, that which cannot fail to exist and could not have been otherwise.

* Since the whole point of Anselm's argument is that the unique nature of the case of God proves to be necessary being and is proven, thus the argument of Guanilo doesn't even apply.

F. Modal version of Hartshorne's Perfection Argument.

"q" for "( ]x)Px," there is a perfect being or perfection exists.
"N" for "it is necessary (logically true) that"
"~" for "it is not true that"
"v" for "or"
"p-->q" for "p strictly implies q" or "N~(p&~q)".
1. q-->Nq "Anselm's Principle:" perfection could not exist contingently
2. Nq v ~Nq Excluded Middle
3. ~Nq-->N~Nq Form of Becker's Postulate: modal status is always necessary
4. Nq v N~Nq Inference from (2,3)
5. N~Nq-->N~q Inference from (1): the necessary falsity of the consequent
implies that of the antecedent (Modal form of modus tollens)
6. Nq v N~q Inference from (4,5)
7. ~N~q Intuitive postulate (or conclusion from other theistic
arguments): perfection is not impossible
8. Nq Inference from (6,7)
9. Nq-->q Modal axiom
10. q Inference from (8,9)

[from Baird's Handout] (from Charles Hartshorne, The Logic of Perfection (LaSalle, Ill.: Open Court, 1962), pp. 50-51, using some of the modifications by C. Stephen Evans, in Philosophy of Religion (Downers Grove, IL: Inter-Varsity Press, 1985), p. 48.)

By Metacrock. Used with Permission.
For more articles by the same author, see Doxa.