Thursday, March 16, 2006

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.

9 comments:

Joe T said...

Hi Franklin,

I'm having trouble with part 3 of this post. Specifically with your statement that "[s]ome sound, non-circular arguments do not prove their conclusions."

Here's your example of such an argument:

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.


I don't see how this cannot be called circular, since it has as an implied premise the definition of what is and what is not a prime number.

Further along in section 3, you boil the above argument down to an abstraction:

p
Either not-p or q.
Thus q.


This is clearly non-circular, but it also just as clearly does prove its conclusion.

I have not been able to come up with an example of a sound, non-circular argument that does not prove its conclusions. Am I missing something?

Franklin Mason said...

Good to talk to you again, Joe.

The argument about primes does presuppose that we know what a prime is. But I do not think it plausible to say that the definition of 'prime' is a supressed premise. If it were, then the defintion of every term within the argument would be a supressed premise; and this seems a mistake. Moreover, even if we are to say that the definition of 'prime' were a surpressed premise, the argument would be circular only if that same definition were asserted in the conclusion. However it is not. The conclusion asserts only that 2 is prime.

Later, you say of the argument 'abstraction' (an 'argument-form' I would call it) that it is clearly non-circular. But the argument to do with primes has the very form displayed in the abstraction. Thus one is circular just if the other is circular. One cannot be circular and yet the other not.

I admit that, in a sense of 'prove', sound non-circular arguments prove their conclusions. They begin with a premise-set that does not include their conclusions and validy reach their conclusions. But not all such arguments amount to anything like an extension of knowledge. They might move from something we already know to something we already know. In the sense of 'proof' that I had in mind, a proof moves from what is known to what is not. Proofs extend knowledge.

Hope this makes at least a bit of sense.

Franklin

Joe T said...

Hi Franklin,

Thanks for your response. It's good to talk to you again, too.

A little while after writing my comment I knew it had problems. I concede that knowing what a prime is is not necessarily, for the sake of this argument, knowing that any given natural number, even 2, is a prime, so your argument in 3 is not circular.

I still have a problem with it, though, and it has taken me a while to figure out how to get it in writing cogently. I don't intend to show my work below, because it would clutter up your blog, but I can elaborate if needed.

Using just the argument form for now, you wrote:

p
Either not-p or q.
Thus q.


I'm not sure from context if your "either-or" construct is an "exclusive or" or not, but if it is, then second statement of the argument form can be shown to be logically equivalent to "p if and only if q", and if it is not (i.e. if it is the "garden variety" logical or), the second statement can be shown to be equivalent to "p only if q". I'll assume the "p only if q" equivalence from now on, since it is itself implied by "p if and only if q", and what I write below will pertain in either case.

Substituting the English back into the "p only if q" form of the statement, we get "7907 is the only prime that lies between 7902 and 7908 only if 2 is prime."

If this is true, we can also assume the contrapositive is, i.e.: "2 is not prime only if 7907 is not the only prime between 7902 and 7908".

Neither of these is self-evidently true to me. Which numbers are prime and which aren't prime depend on the factors they have or don't have, and not on the primeness of other numbers.

That's all I have for now. I look forward to your response.

Joe

Franklin Mason said...

Joe,

I should have commented upon a certain assumption that I made within the context of the argument. The assumption is that - as the logicians say - the logical operators 'either p or q' and 'p only if q' are truth-functional. This means simply that the truth-value of the whole depends solely upon the truth-values of p and q; it does not depend upon any relation that might obtain, or fail to obtain, between p and q.

For any p and q, the truth-value of p only if q is uniquely determined thus ('T' stands for true and 'F' for false):

p q p only if q
T T T
T F F
F T T
F F F

If we understand 'p only if q' in this way, the argument to do with primes does come out valid. Moreover, it does have all true premises. But, as I said before, it does not seem to extend our knowledge in any way.

I grant that the logicians 'only if' is not the 'only if' found in ordinary langauge. That 'only if' does assert a relation between its first and second part. The logicians 'only if' contains, as it were, a certain residue of the ordinary language 'only if' that makes it quite useful in the study of validity.

Joe T said...

Franklin,

Although I was unfamiliar with the term "truth-functional" before I read your comment, I am familiar with the concept, and "truth-functional" definitions of "p only if q' (etc.) were what I had in mind when I wrote my comment, though admittedly my language was imprecise and I did conflate them with the "ordinary language" definitions, which was confusing and unnecessary.

I'll have more to add later.

Franklin Mason said...

Error correction:

The last line in the truth-table for 'p only if q' should end not with a 'F' but with a 'T'.

Joe T said...

Hi Franklin,

Just wanted to get back to this before I forget about it again.

I'll try to be brief. You have said that "2 is prime" (aka "q" from the argument form) is not a premise of the argument about primes we've been discussing.

If that is true, then how can the statement "Either not-p or q" be true, as you say it is, given that "p" is a premise of the argument?

Now, clearly it can be shown to be true trivially from the definition of primes, but in the context of the argument as stated (and this is how I am used to evaluating these things), with the premises available to me, it seems that "either not-p or q" cannot be said to be either true or false.

To say it is true seems to me to imply that "q" is a premise, and therefore the argument is circular.

Now, if you're saying it's ok to show that q is true outside the scope of the current argument, at least for the purpose of evaluating the soundness of that argument, then I think I understand what the disconnect has been here.

Doctor Logic said...

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.

I think that we may not be applying the definition of knowledge with consistency, here.

As you stated in one of your previous posts on Christianity, one can hardly believe that a statement in an unknown language is true. I agree. We cannot assert a proposition that has no meaning. I think that a meaningful proposition is one that has logical consequences for other independent propositions.

Your claim that you are the father of three carries a multitude of implications, any of which can be (and probably have been) tested. You might say that, to the extent you know what the proposition means, it has been empirically verified.

In contrast, religious knowledge is frequently an oxymoron. Religious propositions are, by chance or design, meaningless. There are no other verifiable propositions that follow from "God exists" or "God is good." Nothing we have experienced or will experience can lend credence to such claims. The acid test is to ask the religionist whether there is any experience that could convince them that their claims were false. If they say no, then the meaning they attribute to such claims is empty. Those claims are not about the world of experience. At best, those claims restate the totality of experience.

When we claim to know a proposition is true, we mean that we experience high confidence in that proposition's truth. As you suggest, we might have high confidence in some propositions before we even reflect upon them. As you suggest, our confidence in those propositions might turn out to have been misplaced because we might be able disprove it using premises in which we are even more confident, e.g., perhaps we trust present experience over memory, and our overall picture of physical reality over isolated deviations from that reality.

So, while there might be uncertainty in your proof, you can easily prove to yourself that you are the father of three or that you love your family. If nothing else, you are validating the definitions of fatherhood as they apply to your experiences. Meanwhile, religious claims have infinite uncertainty and cannot be validated against experience.

Franklin Mason said...

Well said, good Doctor. Allow me to think for a day or two about how best to respond. I have some idea about what I wish to say, but I do not wish to be hasty. (I am Entish in that regard.)