Things I Know But Cannot Prove

I will argue in what follows that much of what we know we cannot prove to ourselves.

The argument of this post is of great significance to the issue of the rationality of religious belief. When the argument is complete, I will turn to that issue.

I will begin with a set of definitions. The last will be a definition of 'proof'. (No doubt these definitions are a bit rough, but they will do.)

1. An argument is an ordered set of propositions. One is the conclusion. The rest are the premises. The premises are offered in support of the conclusion. (They need not actually give it any support. We do no wish to define 'argument' in such a way that all arguments are good arguments. Some arguments are evidently quite bad and do not lend any support to their conclusions.)

2. Arguments must have at least these two virtues if they are any good at all. (a) They must have all true premises. (b) The premises must be such that, if true, they do lend support to the conclusion.

These two properties of good arguments are independent of one another. An argument can have all true premises and yet not support its conclusion at all. Here's an example:

3 is odd.
My son is 3.
Thus my son is odd.

Both premises are true, but the conclusion (no matter how we understand the word 'odd' within it) quite obviously does not follow from them.

Moreover, an argument some (or all) of whose premises are false can be such that, had the premises been true, they would have lent support to the conclusion. Example:

5 is even and greater than 2.
No prime greater than two is even.
Thus 5 is not prime.

Clearly if the premises had been true, that truth would have been 'transmitted' to the conclusion. But of course one of the premises is false and thus the argument is no good.

Call an argument valid just if it is such that had its premises been true, they would shown the conclusion true, or at least likely true.

Call an argument sound just if it is valid and has all true premises.

3. Not all sound arguments prove that their conclusions are true. By definition the conclusions of sound arguments do follow from true premises, but this does not imply that they have been proven. Example:

2 is prime.
Thus 2 is prime.

The one premise is true, and the relation of conclusion to premise is such that the premise cannot be true and yet the conclusion false. Thus the argument is both valid and sound. But it does not prove that its conclusion is true. Rather it assumes the truth of its conclusion within the premises and thus cannot possible prove its conclusion. No proposition can be both assumed true and yet, at the same time, proven true. Merely assumed and proven are inconsistent.

We might attempt a definition of 'proof' based upon this insight. Call an argument 'circular' just if its conclusion is assumed true within the premises. What then do we say of this attempt at a definition of 'proof'?

A proof is a sound, non-circular argument.

Unfortunately this definition will not do. Some sound, non-circular arguments do not prove their conclusions. Example:

The only prime that lies between 7902 and 7908 is 7907.
Either it is not the case that the only prime that lies between 7902 and 7908 is 7907 or 2 is prime.
Thus 2 is prime.

A quick google of 'prime list' will verify the first premise. The first of the parts of the second premise ('it is not the case that the only prime that lies between 7902 and 7908 is 7907') is false. But the second part ('2 is prime') is true. 2 is the first prime. Thus the whole of the second premise is true, for any propisiton of the form:

Either p or q.

is true if one of its constituent propositions is true.

Moreover, the conclusion of the argument does follow from the premises. The form of the argument is this:

p
Either not-p or q.
Thus q.

If the premises of an argument of this form are true, the conclusion must be true as well.

So then our argument is valid and has all true premises. This of course means that it is sound. Moreover, it is not circular. The conclusion is assumed nowhere in the premises. (It is part of the 'Either-or' we find in the second premise, but that it is does not imply that it is assumed. Rather what is assumed is the whole of the 'Either or ', and that could be true even if the second of its parts was false.) But is our argument a proof of its conclusion? It cannot be. Before we considered this argument, we were more certain of the truth of its conclusion than we were of the truth of either of its premises.

4. This will lead us to the proper definition of 'proof'. A proof begins with what is more certain and leads us to that which is less certain. Of course if the argument is successful, we will thereby become more certain of the conclusion than we were before. (How much more? That will depend upon the strength of the argument. We cannot become more certain of the conclusion that we were of the least certain premise. Moreover, not all arguments give the same degree of support to their conclusions. The degree of certitude of the conclusion will depend both upon the degree of certitude of the premises and upon the degree to which the premises lend support to the conclusion.) But before we examine the argument, its conclusion must be less certain to us than its premises if it is to constitute a proof of its conclusion.

So, then, let us say this:

A proof is a sound argument the conclusion of which was less certain to us than any of its premises before we began our examination of it.

Let me make a few comments about this definition. (i) It entails that proofs are not circular, for in a circular argument the conclusion is just as certain as the premises, for in a circular argument the conclusion is assumed within the premises. So then we do not need to add to our definition the condition that proofs must be non-circular. (ii) The definition quite explicitly relativizes the notion of proof to the degree of certitude that a certain person has of its premises and its conclusion. Thus what constitutes a proof for me will not in all cases constitute a proof for you. This on reflection should be obvious. An innocent man needs no proof of his innocence. But if his wife and children were brutally murdered and the police have as yet no reason to rule the husband out as a supsect, the police do need proof of his innocence. If the police find that proof, it will be a proof for them but not for the husband. He knew all along that he did not kill his family and cannot in any sense have it proven to him that he did not. In what follows, I will not simply speak of proofs but of proofs for this or that person. (iii) The definition just as explicity relativizes the notion of proof to a certain time, viz. the time at which the argument is considered. This leaves open the possibility that what was for a me a proof at one time will not be a proof at a later time. (iv) Might we need to add some other condition to the argument? Perhaps we do. But even if so, we have identified a property that must be had by all proofs. This property is all that I need for the argument to follow.

5. I know that I love my wife and children. I know that I now sit in my study and write about the nature of knowledge. I know that I like red wines much better than whites.

But I cannot prove any of these things to myself. I can, of course, produce sound non-circular arguments for each of these. But any such argument will contain at least one premise that is antecedently no more certain for me than the conclusion, for I am as sure of these things as I am sure of anything.

Perhaps I ought to attempt a proof of the first to drive home my point. So, then, how might I prove to myself that I am a father of three children? Here's one way:

I recall the birth of each of my three children.
None have died.
Thus I am a father of three.

The argument is sound and non-circular. But it doesn't really prove anything to me, for I was just as certain of its conclusion as I was of its premises before I constructed the argument. The argument did not proceed from the more certain and upon that as basis proceed to the less certain. Rather, the three propositions within the argument were equally certain to me before I constructed the argument. The argument fails as a proof.

Moreover, no amount of ingenuity will produce an argument that for me constitutes a proof of the proposition that I am a father of three. For I am as certain that I am a father of three as I am of anything. Thus no set of propositions could possible prove to me that I am a father of three (though of course many sets of true propositions non-circularly entail that I am a father of three).

6. Quite obviously, then, much of what we know we cannot possibly prove to ourselves - my three examples could be added to quite easily. The reason for this is simple: many of us believe a large set of propositions that for us are at least as certain as anything else we believe.

Might someoneone object here that if I cannot prove, for example, that I have three children, I do not really know it? No doubt some will. My response is two-part. (i) Surely reflection on the nature of knowledge and of proof should never lead us to doubt those simple truths that we believed before reflection began. I do know that I have three children, and no amount of philosophizing can ever cast even an iota of doubt on it. But can I prove that I have three children? I can produce nothing like a proof of this belief. (I can produce sound, non-circular arguments ad infinitum for this belief. But none are proofs; none begin in the more certain and proceed to the less certain.) (ii) The objector seems to assent to the proposition that if a thing cannot be proven, it cannot be known. Let us then ask for the proof of this proposition. (The proposition is meant to apply universally and thus must apply to itself.) I for one know of no argument with any prima facie plausability that looks anything like a proof of this proposition. Indeed I think it likely that no such argument exists. Thus the proposition that knoweldge requires proof is, if true, false. But no proposition can be both true and false, and thus it is false. There can be knowledge without the possibility of proof.

7. What relevance has this to the issue of the extent of religious knowledge? Often the believer is asked by the skeptic to produce proofs of her beliefs, and if she is unable to do so the skeptic often assumes that the believer does not really know what she claims to know. Lack of proof is often assumed by the skeptic to undermine the rationality of belief. But as should be clear, mere lack of proof by itself does nothing to undermine the rationality of what we believe, for much of what we know we cannot prove.

For the skeptic to make her case, she must prove that religious belief is among the class of beliefs that do require proof if they are to be known. (Surely this class is not empty. Much of what we know we know only because we can prove it, or at least by argument show it quite likely. Much of science proceeds in this way.)

Let me here say that I for one know of no way to prove that no religious belief can be known if it is not proven. Indeed I suspect that some religious beliefs have a prima facie right to claim of themselves that they are in the class of known but unprovable propositions. If the skeptic believes otherwise, it is incumbent upon her to show otherwise. The ball is in the skeptic's court.

8. Of course there's much that remains to be said on the subject. (It seems to me that what remains is much more difficult than what has been said.) If we were to continue to philosophize, the first question that we must answer would of course be this: how do we distinguish a belief that requires proof from one that does not? Others would follow soon thereafter. It seems that those that do not need proof are know directly, or immediately. But what does this mean? Moreover, what are those cognitive faculties that directly, or immediately give rise to knowledge? I have little insight into how to answer these. But that in no way undermines the conclusion of this post: much of what we know we cannot prove.