Kurt Gödel is best known to mathematicians and the general public for his celebrated incompleteness theorems. Physicists also know his famous cosmological model in which time-like lines close back on themselves so that the distance past and the distant future are one and the same. What is less well known is the fact that Gödel has sketched a revised version of Anselm's traditional ontological argument for the existence of God.
How does a mathematician get mixed up in the God-business? Gödel was a mystic, whose mathematical research exemplified a philosophical stance akin to the Neo-Platonics. In this respect, Gödel had as much in common with the medieval theologians and philosophers as the twentieth-century mathematicians who pioneered the theory of computation and modern computer science. However, a deeper reason for Gödel's contribution to the ontological argument is that the most sophisticated versions of the ontological argument are nowadays written in terms of modal logic, a branch of logic that was familiar to the medieval scholastics, and axiomatized by C. I. Lewis (not to be confused with C. S. Lewis, or C. Day Lewis for that matter). It turns out that modal logic is not only a useful language in which to discuss God, it is also a useful language for proof theory, the study of what can and cannot be proved in mathematical systems of deduction. Issues of completeness of mathematical systems, the independence of axioms from other axioms, and issue of the consistency of formal mathematical systems are all part of proof theory.
Talking about proof theory often feels like discourse about God:
Some of the pioneering work of Kurt Gödel showed that the modal logic of philosophers which was used to analyse the ontological argument for the existence of God was also very useful in proof theory and metamathematics.
Kurt Gödel was born in what is now Brno in the Czech Republic in 1906. (At the time it was part of the Austrian-Hungarian empire.) In 1923, he attended the University of Vienna and received his doctorate in 1929 under the supervision of Hans Hahn (1879-1934), best known to mathematicians as one half of the Hahn-Banach Theorem. Gödel joined the faculty of the University of Vienna, and became a member of the famous group of positivist philosophers until 1938. While he was a part of this philosophical circle, his thinking was also much influenced by the work of Leibniz. This was to influence his thinking about the ontological argument.
With the outbreak of World War II, Kurt Gödel decided to leave Vienna. He emigrated to the United States in 1940, and joined the Institute for Advanced Study in Princeton in 1953 until his death in 1978.
As I mentioned above, the most common setting for the discussion of ontological arguments for the existence of God is the framework of modal logic. What is modal logic and why do we need it?
Consider the following "proof" for the existence of God. Let us call it the argument from omniscience.
Now, not for one minute do I entertain the idea that this argument will convince anyone about God's existence. Rather, I wish to argue that it is modally naive. What have we managed to prove by the argument above, if not the existence of God?
The argument defines God to be an omniscient and rational individual. Now mathematicians tend to broadly accept the idea that you can define terms as you like. There is no claim that this is in particular the Judeo-Christian God, or the God of any other religious group. We would all accept, I think, that whether this being should be called God or not, a proof of the existence of an omniscient rational being is no small accomplishment. So that is not the problem with the argument.
We could argue black and blue about the possibility that a rational individual might not believe in his own existence. Descartes claimed Cogito ergo sum, and most of us accept that to doubt your own existence would be a pretty strange state of mind.
The real problem is that the argument makes an assumption that is not brought out explicitly. It assumes that it is possible for an omniscient rational individual to exist, where omniscience includes knowledge about one's own existence. So what the argument really seems to show is that (for God as defined):
We can write this in the symbolism of modal logic as
where g is the statement that a rational omniscient being exists. The symbol that looks like a magnet on its side represents material implication. The statement ab is true for material implication if it is not the case that a is true and b is false.
We can also write the conclusion above as
where denotes weak disjunction -- equivalent to "and/or" in ordinary parlance, and ~ denotes negation. Other operators include ab, "a is equivalent to b" and a&b, "a and b".
The symbols and are called modal operators, and denote the concept of necessary (as opposed to contingent or accidental) truth, and possible truth. For example, the statement
is true because the statement is provable, and hence necessarily true. However
is false, as I understand trees and Brooklyn. That a tree grows in Brooklyn is a contingent or accidental truth, best formalized as
where t is the statement that "a tree grows in Brooklyn." We don't really need two modal operators, because it is possible to write in terms of and vice versa. Thus
is a tautology.
We must not presume that the only necessary truths are those which we can prove. Gödel showed the weakness of that presumption with his first and second incompleteness theorems. Nor should we presume that mathematical and logical truth encompass all necessary truths. There may well be many others. Plato felt that necessary truths could be found in aesthetics and ethics, also.
What are the axioms or postulates of modal logic? In addition to the usual postulates of propositional logic (which are equivalent to verification by a truth table), modal logic also requires postulates such as
for all propositions a. The contrapositive of this family of postulates is
Another principle which is commonly used is the postulate
which is modal modus ponens.
Also useful is the necessitation postulate, that
i.e., a is itself a postulate or a theorem.
These postulates represent the common core of modal logic. Two additional postulates are often added to these to give it additional strength. The first of these is
and the second is that
Together, these two postulates state that the modal status of a proposition is a necessary truth. The principle that the modal status of a proposition is a necessary truth is call Becker's postulate.
We shall return in greater detail to these rules of inference on the next page.
John Spriko's beautiful portrayal of the Penrose Triangle gives us a window into the first of our four modal worlds: the world of propositions which are necessarily false. What we see in this picture is geometrically impossible. You may wish to check out John Spriko pages to see what he thinks about the Penrose Triangle. Is our belief in the impossible just a limitation in ourselves, or something fundamental to the nature of Platonic reality?
Parallel universes, alternate histories and fantastic visions are all food for thought in the second of our four modal worlds: the world of propositions which are contingent but false. Such worlds are visions of what is possible, even if we know that some of these visions will never come true. In this picture, we see Slawek Wojtowicz's eerie portrayal of the suspended bodies of "Baby Boomers" floating in the upper atmosphere of their planet.
The world of contingent truth is the appropriate setting for the sciences. While science fiction writers can indulge in faster-than-light travel or the grandfather paradoxes of time travel, the scientist is constrained by what is possible as well as what is known. That leaves plenty of room for adventure.
I have chosen this striking fractal picture of a mandelbrot bud within the mandelbrot set as the window into the world of necessarily true propositions. The infinite complexities of the mandelbrot set are a reminder of both the elegance of mathematical truth and its subtlety. While some argue that the world of mathematics is a product of the human mind, one can only conclude in looking at the mandelbrot set that perhaps the creation has outgrown its creator. What wonders remain to be discovered in the infinite depths of this set?
While modal logic may look a bit strange, it is in many respects more consistent with the logic of ordinary discourse than is the propositional logic that mathematicians use. One area where modal logic is useful is in discussing counterfactual propositions. For example, the statement
can be meaningfully debated in modal logic. However, in propositional logic, this statement would be trivially true, because any statement of the form ab is true in propositional logic if the statement a is false. Other trivially true statements in propositional logic include
Unfortunately, this statement is trivially true in propositional logic, as well as
There seems little point in having a debate. Suppose we let r stand for the proposition that the Roman empire did not fall to barbarians, and c stand for the proposition that computers use Roman numerals. Then in propositional logic the statements rc and r~c are both true because r is false. On the other hand, in modal logic (reflecting more appropriately ordinary discourse) it is reasonable that (rc) is true and (r~c) is false, or vice versa. Modal logicians call this form of implication entailment to distinguish it from material implication as defined by the logical connective .
In order to use counterfactuals in argumentation, it is important to distinguish different types of modal necessity or modal possibility. For example, an argument based upon the possible consequences of Napoleon winning at Waterloo, should not be able to assume that the laws of physics are also changed. In a certain sense, the laws of physics would appear to be more necessary than the fact that Napoleon lost the battle of Waterloo. While this may be a bit presumptuous from a philosophical standpoint, it is certainly the basis for ordinary discourse. In his book The Nature of Necessity, Alvin Plantinga distinguishes between natural necessity on one hand, and broad logical necessity on the other. The statement
is impossible in the modal sense of natural necessity, while it is possible in the modal sense of broad logical necessity.
To my way of thinking, it is doubtful whether we can restrict ourselves merely to these two modal concepts of necessity (or possibility). Language and the world abound in various restrictions on possibility. Nor can science ignore modal logic and relegate it to the realms of metaphysical speculation. For example, in order to understand causal relationships -- to ask how one event can be a cause of another -- it would appear that we have to work with counterfactuals in science. If I say that a certain drug causes a certain response in a subject, then I would appear to be saying that if the drug is administered then the response will follow, whereas if the drug is not administered the response will not follow. One of these two statements is a counterfactual argumentation.
"Well, I stayed up late the other night and saw a man in my living room dressed in a red tunic. He had a large sack over his shoulder, was rather fat, and had a full white beard. Upon my asking who he was, he replied `Just call me Santa, buddy,' and leaped out a window."
What should we think of this? We can examine the logic of the argument for Santa's existence. It goes something like this:
1. Santa Claus is [characterized as]
a fat man who wears a red suit and has a white beard.
2. A fat man wearing a red suit and having a white beard was in my living room the other night.
Therefore by 1 and 2:
3. Santa Claus was in my living room last night.
4. If Santa Claus does not exist, then he would not have been in my living room the other night.
Therefore by 3 and 4:
5. Santa Claus exists.
Despite strong evidence from millions of children around the world, the reader is to be forgiven for failing to find this argument convincing. Next, let us suppose that someone else also believed in Santa Claus and provided the following testimonial:
"I was sitting in my living room last Christmas Eve around midnight, and was astonished to see a man appear from the fireplace and enter the room carrying a large bag. I could have sworn that the chimney was not wide enough to permit him ingress, but he managed to do it anyway. Putting his right forefinger to his lips and making a shushing noise, he quietly removed a lego set, a toy tractor and a full sized washing machine from his bag. How he managed to get the washing machine down the chimney in that small bag I do not know. He wasn't dressed like Santa Claus: his suit was brown, not red, and his beard was red, not white. However, he had a very merry twinkle in his eye, and looked quite the jolly fellow. He magically disappeared up the chimney in a mysterious fashion, taking some cookies with him. I'm sure it was Santa Claus."
We can identify a certain logical structure to this argument as well:
1. Santa Claus is [characterized as]
a magical merry fellow who brings presents on Christmas Eve.
2. A magical merry fellow was in my living room delivering presents on Christmas Eve.
Therefore by 1 and 2:
3. Santa Claus was in my living room on Christmas Eve.
4. If Santa Claus does not exist, then he would not have been in my living room on Christmas Eve.
Therefore by 3 and 4:
5. Santa Claus exists.
Now we might be tempted to question the reliability of this individual's account. But let us suppose for the moment that both accounts are accurate descriptions of what actually occurred. It is obvious that the second account gives us much greater reason for believing in Santa Claus than the first. Why is this?
The basic difference between the two logical arguments is in the definition of Santa Claus. If Santa Claus were defined to be an individual who is fat, has a white beard and wears a red suit, then clearly any person who matched that description would, by definition, be Santa Claus. Clearly there are countless people who can (and do) match that description in shopping malls everywhere. So Santa is not uniquely defined by that description. There is another problem as well. If Santa is defined to be someone who wears a red suit, then does he cease to be Santa when he takes it off? Would he cease to be Santa if he shaved? Would he become someone else if he went on a diet and exercise program and lost a little weight?
The question is of practical importance. We all know or know of individuals who have lost (or gained) so much weight that their friends can hardly recognize them. Police agencies everywhere hunt for suspects matching certain descriptions and know full well that these descriptions are not entirely satisfactory as identifying characteristics. What are the defining characteristics of any individual? In formal philosophical terms, how do we determine whether an individual is truly to be identified as "Elvis Presley" or is just someone who matches his description?
Modal logicians define a property of an individual to be essential if it is necessary that that individual has that property. So, if x is the individual known as "Santa Claus," and F is the property of being magical, then F is an essential property if Fx is a true proposition. Otherwise a property is said to be contingent. Thus being magical would appear to be an essential property of Santa Claus while being fat, for all we know, is contingent. Whether Santa exists is irrelevant here. The fact that Santa might or might not exist can only be determined by having a clear enough knowledge of Santa's essential properties that we can identify him if we were to bump into him.
An additional concept that we will need is that of the essence of an individual or object. The essence of an individual is a strengthening of the notion of an essential property, to make it maximal. As mathematicians would say, an essence is a saturated essential property.
Definition A property F is said to be an essence of an individual x if for every property G, the proposition
Note that, in particular, the proposition FxFx is a theorem of propositional logic. So, by the necessitation axiom, we have (FxFx). So if F is an essence of x, we can deduce that Fx, i.e., that F is an essential property of x. Of course, not every essential property need be an essence.
It is now time to consider the original ontological argument first proposed by Anselm.