LOGIC

It's sometimes said that there is no such thing as an absolute proof. But just what do we mean by that, and how can we prove it? Well, a proof must be performed starting with a premise set addressing a realm of reality (or perhaps an imaginary realm of unreality), the complete set of all statements that can be made within that realm of reality, and a concise set of definitions to support the whole mess. A proof has no validity outside the realm of reality addressed by the premises.

For any set of premises, there are four possible kinds of conclusions. Some conclusions can be proven true. Some conclusions can be proven false. Some conclusions can't be proven either way. Some conclusions can be proven both true and false at the same time. A premise set for which there are no conclusions that can be proven both true and false at the same time is said to be consistent. A premise set for which there are no conclusions that can't be proven either way is said to be complete.

The ancient Greeks developed an excellent understanding of logic. Aristotle appears to have been the first person to codify the principles of logic. Euclid then used logic to elevate mathematics from a mere manipulative art to a truly scholarly pursuit. Both Aristotle and Euclid realized that logic can only get from premises to conclusions. It can't pull premises out of nowhere. They accepted certain axioms as self-evident and needing no proof, but many philosophers have found this concept rather bothersome.

Ever since the days of Aristotle and Euclid, logicians have been trying to find an all-encompassing premise set that is both complete and consistent, and can prove itself so. For instance, it is known that the premise set of Euclidean geometry is consistent but not complete, but this can not be proven within itself. A somewhat larger premise set is needed to prove the consistency and incompleteness of Euclidean geometry.

If such a perfect premise set can be found, all possible truths about all realms of reality could be known and proven absolutely. Nobody has yet succeeded. If we could succeed, we'd have found the real God, and all false religions would be forced to admit defeat and we'd be left with only the One True Religion, whatever that might be. But in order to do this, we've gotta find a way to pull premises out of nowhere.

Rene Descartes thought he had a way, with his famous statement, "I think, therefore I am." He's quite right that he can prove that he exists, and therefore existence is possible. But he didn't get much farther than that without getting himself tangled up into a morass of circular reasoning.

In the eighteenth century, some logicians, especially Immanuel Kant, began to have doubts that there could ever be one grandiose perfect all-encompassing logic system. Kant gave a number of reasons, but did not offer a proof that would be acceptable by modern standards of logical rigor.

In 1879, Gottlob Frege wrote a monumental tome in which he introduced many new logical methods and demonstrated their validity. Among other things, he showed that the principles of logic themselves can serve as premises.

But that leaves with a question: How do you prove that the principles of logic themselves are valid?

Several other mathematicians, most notably David Hilbert, thought they had a way. They would try to use the principles of logic to invoke themselves recursively to prove themselves, and at the same time try to avoid the evil curse of circular reasoning.

And then, horror of horrors, absurdities started showing up as a result of using Hilbert's methods. The most famous one was found by Bertrand Russell, who discovered that if you allow the meaningfulness of the set of all sets that are not members of themselves, it is both a member of itself and not a member of itself. This sent David Hilbert back to the drawing board, but he was never able to present an absolute proof of the validity of the principles of logic.

Mathematicians began to eagerly anticipate the day when all of mathematics could become absolutely provable. Theologians began worrying that anything that is absolutely provable is necessarily true in all possible realms of reality, and is therefore eternal and omnipresent. Naturally, this sorta blows the doctrine that only God can be eternal and omnipresent, so theologians were hoping somebody would find a pattern in these absolutely provable things that they could shoehorn their concept of God into.

Then in 1931, Kurt Godel proved first that there is no such thing as a premise set that is both complete and consistent. He then proved that a premise set can never prove itself either complete or incomplete. He then proved that a premise set can never prove itself either consistent or inconsistent except in the case where a specific inconsistency is actually found. Since the principles of logic can constitute a premise set, this means that the principles of logic can never prove their own validity. So far, nobody has ever found a flaw in any of his proofs.

So everybody was upset. Mathematicians couldn't have a way to prove everything, theologians didn't have a pattern by which they could prove God exists, and observational scientists didn't have a way to use all their observations as premises in formal logic systems.

Well, so much for absolute knowledge. Do you believe the principles of logic are valid, or do you believe there's a magic three-headed sky-zombie up there somewhere pulling the strings? I guess you can take your choice.

What does it take to prove something in an ordinary realm of reality? First, it takes great care in formulating the definitions to be used in making your statements. A bit of sloppiness in even one definition can lead you down the primrose path to absurdity very quickly. Most of us have probably seen these proofs that one equals two, and other ludicrous results. I once proved that civilians are required to salute officers left-handed. Civilians do not salute. Saluting right-handed is a subset of saluting, therefore civilians do not salute right-handed. Therefore, if civilians want to salute, they have to salute left-handed.

Next, you've got to be careful in selecting your premises and showing how your premises are supportable. No fair suddenly introducing a new and untried premise mid-proof. If you've discovered that your proof can't be done without expanding your logic system to include a new premise, you've got to start over from the beginning and re-examine all your steps to be sure they can accommodate this new premise. Then, when your proof is finished, you need to recognize that it is only valid within the realm of reality in which your premises are supportable.

If you use only premises that are analytically supportable, that is, they can be proved logically using only the principles of logic as prior premises, then your conclusion will be necessarily true if the principles of logic are valid, but that's really not quite an absolute proof. Oh by the way, can you absolutely prove that there's no such thing as an absolute proof?

If you use premises that are supportable by observations, keep in mind that those observations are made within a specific portion of the universe, and it would be very difficult to demonstrate that your proof has any validity outside that portion of the universe. Also, observations are imperfect, and the validity of the proof is limited to the degree to which the observations have been verified. For instance, if you state the premise "All cows are brown" and it's supported by having only observed five cows in Farmer Cramshaw's pasture, that's not very good support. It might be better to restrict the premise to "All five cows in Farmer Cramshaw's pasture are brown" thus making it clear up front that the restriction will then be applicable to what you'll end up proving.

You may support a premise by stating it as a postulate, thus effectively having your logic system address an imaginary model of a realm of reality, rather than an actual realm of reality. Now, you've got to realize that your proof is only valid in that portion of reality that's correctly represented by the model. Euclidean geometry is an example of a logic system that uses postulates as premises. Euclidean geometry has proven itself very useful, but there's no way you can ever say that the things you can prove with it are absolutely true. They might not be true in some remote unknown part of the universe. In fact, there are other geometries that have been invented that give more useful results in certain astronomical realms of reality.

Once you've got your definitions firm and your premises well supported, all you've got to do is turn the crank on Aristotle's rules. Well, not exactly. A lot of scientific work is based on gathering a preponderance of evidence instead of strict deduction. Scientists will debate, sometimes for many years, what the evidence really implies. The more controversial theories such evolution and plate tectonics and the existence of meteorites did not have an easy time becoming accepted. These theories are still in the process of being modified as new evidence is discovered.

So what do scientists do when new but fairly well verifiable observations conflict with old and fairly well established theories? Well, they debate a lot and they reinvestigate a lot. Some recent observations in physics and astronomy are suggesting that the Universal Fine Structure Constant (whatever that is) has changed by about a thousandth of a percent in the last nine billion years. I have no idea how they measure that, but they seem to be pretty confident of their measurements. But this violates several cherished theories about the existence of the universe! Anyhow, some explanation has to be found to either modify the theories or explain the measurements. One or the other has got to go. All you can do, logically, is continue to make more investigations until the discrepancy is resolved.

Just remember, no matter how much you can prove, there's more yet to be proved. And if you can prove it all, one Nobel Prize coming right up!