The Ontological Argument
Peter Suber, Philosophy Department, Earlham College

I've seen this argument discussed in several places around the web, sometimes labelled "Suber's version of the ontological argument" or "Suber's argument" for God's existence. Those labels are unfair to Charles Hartshorne, who formulated the argument. My refinement of his formulation is trivial. The labels are also unfair to me, since I do not endorse the argument or believe its conclusion. I consider the argument valid but unsound.

Note to students. This web page is not an assignment. While we will study the ontological argument in class, we will not study the logical formalization of it presented here. This is strictly FYI for those of you willing to work through the formalism.

The ontological argument for the existence of God was first framed in the Proslogion of Saint Anselm of Canterbury (1033-1109). Important variations occur in Descartes, Leibniz, Spinoza, and Hegel, and arguably in every rationalist theory of an absolute being. The gist of the argument is to prove that God exists from the mere possibility, or mere idea, of God's existence. It asserts that God's essence implies God's existence.

Kant was the first to name the argument "ontological". Charles Hartshorne thinks it should be called the "modal" argument, since it relies on the modal categories of possibility, actuality, and necessity. Here is my restatement of Hartshorne's restatement of the argument in the formalism of contemporary modal logic. (See Hartshorne's The Logic of Perfection, Open Court, 1962, at p. 51.) We use the standard notation in which and are the modal operators.

p = p is necessarily true

p = p is possibly true

p = p is (actually) true

The proof will use rules of inference from standard non-modal logic, and a few from modal logic. (NB: there is no standard modal logic.)

  1. One rule from the former that is not taught in our Symbolic Logic course (but which is perfectly valid) is p q, q r, p r. For our purposes here, let us call this the rule of substitution.

    Here are three rules from modal logic that you might not already know.

  2. If p is necessarily true, then p is actually true, that is, p p.

  3. Becker's postulate holds that modal status (except for actuality) is always necessary. Hence it asserts p p and p p. Becker's postulate does not assert that p p; if it did, it would directly assert one of Anselm's critical premises, when p is the assertion of God's existence. (More on this in the notes following the proof.)

  4. The modal version of modus tollens holds that if the consequent of a conditional is necessarily false, then its antecedent is necessarily false; or (p q) (~q ~p).

Let  p  abbreviate (x)Px, "something perfect exists". Let "" signify strict implication, that is, (p q) ~(p · ~q). Here is the "ontological" or modal proof of p:

1.   p p Anselm: perfection cannot exist contingently
2.   ~~p Anselm: perfection is not impossible, or p
3.   p p modal axiom (rule B above)
4.   p ~p principle of excluded middle
5.   ~p ~p Becker's postulate (rule C above) applied to ~p
6.   p ~p 4, 5, substitution (rule A above)
7.   ~p ~p 1, modal modus tollens (rule D above)
8.   p ~p 6, 7, substitution (rule A above)
9.   p 8, 2, disjunctive syllogism
10.   p 9, 3, modus ponens

Of these 1, 2, and 5 are postulates that one might deny. Together 1 and 2 say that God is possible, and if God is actual, then God necessarily exists. Together with the conclusion, it is almost as if Anselm had argued that possibility implies actuality, and actuality implies necessity. But while such an argument would be invalid, the argument above is valid. (You decide whether it is sound.)

Anselm needs only two premises (1-2 above): that perfection cannot exist contingently (that is, if something is perfect, it is necessarily perfect), and that perfection is not impossible. Premises 3-5 are explicit statements of logical principles.

Notice that if we apply Becker's postulate to the modal category of actuality, we would get p p. This is one of Anselm's two unproved premises. This should show that Becker's postulate is not innocuous. Also notice that we take as a modal axiom, p p. Together with the proposition, p p (from Anselm or Becker), we may infer that p p. This is not a reductio ad absurdum; it is a window onto the modal logic at work here —or (for Anselm) onto the nature of perfection.

One might well ask whether this reasoning could prove that any proposition that is possibly true is actually true. The answer is —it depends. For Anselm the answer is no. The first premise (from Anselm) asserts that perfection cannot exist contingently. This is essential to the proof. If it can be said of any other proposition, q, that q q, then it would indeed follow that we could prove q from its mere possibility (and this modal logic apparatus). But for Anselm only perfection has this property. On this reading it is essential that the argument is about perfection, or about God only as perfect, rather than about God under some other description or about devils, islands, or shoes. But the answer changes if we take p p not from Anselm's theology but from Becker's logic (applied to the modal category of actuality), and assume that Becker's logic applies to all propositions. Then Becker's logic alone could give us for any proposition, q, the crucial premise, q q. Then any proposition could be proved by an ontological argument. For this reason, the modal logician, Arthur Prior, has said: "Modal logic is haunted by the myth that whatever exists exists necessarily."

Becker's logic is not arbitrary, even if it is optional. The key postulate here that modal status is always necessary is affirmed even by thinkers with no interest in formal logic who reject the ontological argument (see e.g. Kierkegaard's "Interlude" to the Philosophical Fragments).

This file is an electronic hand-out for the course, Rationalism & Empiricism.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

Ribbon] Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A. Copyright © 1998, Peter Suber.