A logical God?

Probably the least popular of all the familiar arguments that are from time to time offered to prove the existence of God is the Ontological Proof.  Here is a one-paragraph synopsis of Saint Anselm’s version of the Ontological Proof, taken from the Stanford Encyclopedia of Philosophy:

The first, and best-known, ontological argument was proposed by St. Anselm of Canterbury in the 11th. century C.E. In his Proslogion, St. Anselm claims to derive the existence of God from the concept of a being than which no greater can be conceived. St. Anselm reasoned that, if such a being fails to exist, then a greater being—namely, a being than which no greater can be conceived, and which exists—can be conceived. But this would be absurd: nothing can be greater than a being than which no greater can be conceived. So a being than which no greater can be conceived—i.e., God—exists.

Even believers tend to react to the Ontological Proof with distaste and irritation.  So it was rather interesting when, in 2013, German logicians Christoph Benzmüller and Bruno Woltzenlogel Paleo proved that Kurt Gödel’s demonstration that the basic axioms of Logic K, a form of modal logic developed by Saul Kripke (the “K” in “Logic K” stands for “Kripke,”) imply that the Ontological Proof is sound.

Logic K is not the only possible system of logic, so this implication does not by itself prove that God exists.  What makes Professors Benzmüller and Woltzenlogel Paleo’s work so interesting is that Logic K is an extremely simple system, especially as compared with a system like arithmetic, which as Gödel himself showed is infinitely complex in its basic axioms.  The reasoning we use in practical life adds manifold layers of complexity to propositional frameworks such as those of formal logic or mathematics.  If something as specific as monotheism can come springing out of something as spare as the basic axioms of Logic K, then the idea that any form of rigorous intellectual activity can be neutral regarding the kinds of questions monotheism is supposed to answer becomes tenuous.

That is not to say that our cultural formation precedes our intellectual activity, and so that all of our systematic reasoning is infused with the particular circumstances of the society in which we were raised, often in ways of which we are unaware.  It would no doubt be true to say this; however, it is a statement that rests on the findings of the social sciences, expressed in language that has grown up in the development of those sciences.  And the social sciences themselves derive their authority from their status as products of rigorous intellectual activity.  If all such activity is already implicated in theology, then an attempt to confine the implications of Professors Benzmüller and Woltzenlogel Paleo’s work to areas already explored by the social sciences is an attempt to minimize the scope of the problem.

A God who holds the world record for eating the most skateboards is greater than a God who does not hold that record

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Nor is it even to say that as we develop a system of reasoning we are condemned to stack the deck, consciously or unconsciously, in favor of our own religious commitments.  Aristotle grew up in a society in which monotheism was an alien phenomenon which, on those rare occasions when it would be mentioned, was regarded with undisguised contempt. Yet, as such Muslim and Christian commentators on Aristotle as Ibn Sina, Ibn Rushd, and Thomas Aquinas showed many centuries ago, Aristotle’s logic works best when it is applied to a monotheistic universe.  Aristotle himself would no doubt have regarded this as a reductio ad absurdum of his work, and would have gone back to the drawing board to produce a new system of logic, one that fit with what he regarded as the real world of multiple gods and other beings whom it was obligatory to worship.  Perhaps he would have succeeded in creating such a system; he was Aristotle, after all, and was as well equipped as anyone has ever been to accomplish such a thing.  But as it happens, he never had occasion to try, and for two thousand years Aristotle’s logic was the prevailing system in the world from India to Ireland.

When Aristotle’s system of logic was in favor, the work of men like Ibn Sina, Ibn Rushd, and Thomas Aquinas gave compelling grounds for accepting monotheism.  That Aristotle, as a polytheist from a resolutely polytheistic culture, could not be accused of stacking the deck to produce a system that supported monotheism, certainly added to the force of these grounds.  Nowadays, Aristotle’s logic is obsolete, and so one could hardly expect logicians to become monotheists simply because the Medieval Scholastics found in it support for monotheism.

Still, that it is monotheism that jumps out, not only from a logical system constructed by a rabbi’s son like Saul Kripke on the basis of a metaphysics constructed by vaguely Christian thinker like Leibniz, but also from a system constructed by the thoroughly pagan Aristotle, does make it difficult to claim that the relationship between monotheism and systematic reasoning is entirely an illusion resulting from indoctrination in monotheism.  It is likely that the idea of a single deity who is the supreme creator, ruler, and judge of the world is a sort of default position built into the whole project of codifying the rules of logic.

Just as it does not follow from the fact that Logic K rests on axioms which, taken together, imply the existence of God, that God in fact exists, so it would not follow from God’s status as a default hypothesis of formal logic that God in fact exists.  Like all other human activities, formal logic is a byproduct of any number of particular and contingent circumstances, starting with the biological adaptations that enabled our ancestors to survive, continuing through the particularities of our cultural backgrounds, and continuing through the countless vicissitudes that make it possible to distinguish the life of one individual from that of another.  It may well be that formal logic, mathematics, and the sciences, pursuits in which only a small minority of the people in the world today and only a minuscule percentage of all the people who have ever lived take an interest, will ultimately prove to be trivial matters sharply limited in their ability to cast light on the weightiest matters.  Perhaps the sorts of things most people find more interesting and which a majority has always found to be more interesting will prove to be more powerful aids to understanding, or perhaps systematized reasoning in the forms we now know will ultimately turn out to be relatively trivial preparations for some new form of understanding that awaits us in the future.  Perhaps neither of those things will happen, but we will simply come to accept a tendency to monotheism as a not-very-interesting shortcoming inherent in projects to codify the rules of correct reasoning.

Of course, monotheism is also a minority pursuit in the overall picture of humanity.  At no point in the history of the world has a majority of the human race been monotheistic in its views.  Today Christians, Muslims, Jews, and members of other monotheistic groups are probably more numerous than ever before, yet they still comprise well under half the world’s people.  What is more, monotheism seems to have been invented only once, in Babylon during the Captivity, while polytheism, animism, ancestor-worship, and other religious orientations all likely arose independently in many times and places.  In that context, monotheism looks like a freak occurrence.

It is that very freakishness that makes the recurrence of monotheism at the roots of logical systems a matter of interest.  If something so particular can keep cropping up wherever people make their most intense attempts to be general, what oddities might come out of the far more complicated sets of axioms that underlie applied reasoning?  In the light of what Professors Benzmüller and Woltzenlogel Paleo have shown about Logic K, we could hardly be surprised if hidden somewhere in the axioms of trigonometry were a recipe for kosher chicken soup, or for that matter if a description of the Loch Ness Monster were encoded somewhere in Newton’s Laws of Motion.

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