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Can Logic Prove (or Justify) Itself?

Logic has developed much further since the days of Aristotle. While the fundamental concepts of “traditional logic” (as the logic of Aristotle has come to be called) are sound, a whole new way of looking at logic has emerged: modern logic. What separates the two is the method, or form, in which these concepts are expressed. Unlike traditional logic, modern logic relies heavily on symbolic techniques and mathematical methods (it looks and feels just like advanced mathematics). The purpose of relying on a formalized language of symbols is to avoid the ambiguities of ordinary language which is used in traditional logic. By just the form of this system, it is possible to make valid inferences. But even if the argument is completely logical, or valid, it says nothing of whether the conclusion is true or false. In other words, we can know if the argument is sound, regardless of the truth of the conclusion.

A productive paradox may now be seen in the history of thought. Arithmetic is where the content lies, and not logic; but logic prompts [establishes] certainty, and not arithmetic.

David Berlinski

If all the assertions which mathematics puts forward can be derived from one another by formal logic, mathematics cannot amount to anything more than an immense tautology. Logical inference can teach us nothing essentially new, and if everything is to proceed from the principle of identity, everything must be reducible to it. But can we really allow that these theorems which fill so many books serve no other purpose than to say in a roundabout fashion ‘A=A’?

Poincaré

Logic has always been a dangerous discipline, any number of logicians going mad after finding themselves hopelessly lost in the wilderness of their own thoughts. When years later they are rescued by the thumping helicopter of common sense, they pick gratefully at emergency rations, and smile for the cameras, but when asked what on earth they thought they were doing, they can do little more than shrug their bony shoulders, saying, if they say anything at all, that like Gödel himself they were looking for something they could not find.

*David Berlinski* (from *The Advent of the Algorithm*)

In 1931 Austrian logician Kurt Gödel announced his discovery that complete certainty was never to be encountered in mathematics by any route founded on traditional logic. Furthermore, Gödel found that any imaginable remedy for an inadequate system of logic will also prove inadequate in precisely the same way. In short, there will always be questions that arise in mathematics that cannot be settled with logical certainty. This result was remarkable enough, but what made Gödel’s achievement even more noteworthy is that he had used logic to incriminate logic.

Gödel’s theorem showed both that arithmetic is incomplete and that a proof of its consistency is beyond the powers of arithmetic itself. Some mathematicians—John von Neumann, for example—understood the proof at once and grasped its implications, but Gödel’s reasoning was so subtle, and his proof such a masterpiece of concision and paradox, that at least thirty years were to pass before the general mathematical community understood that something remarkable had been achieved.

David Berlinski

Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonizing, controlling force rather than a creative one. Even in the most purely logical realms, it is insight that first arrives at what is new.

Simple logical relationships are, in fact, insights. It’s only because they seem so self-evident and because the contrary defies our imagination that they are not recognized as insights.

At the beginning of the new millennium, we still do not know why mathematics is true and whether it is certain. But we know what we do not know in an immeasurably richer way than we did. And learning this has been a remarkable achievement—among the greatest and least-known of the modern era.

David Berlinski

Mathematics is the only science where one never knows what one is talking about nor whether what is said is true.

Bertrand Russell

Logic and truth, as a matter of fact, have very little to do
with each other. Logic is concerned merely with the fidelity and accuracy with
which a certain process is performed, a process which can be performed with any
materials, with any assumption. You can be as logical about griffins and
basilisks as about sheep and pigs. On the
assumption that a man has two ears, it is good logic that three men have
six ears, but on the assumption that a man has four ears, it is equally good
logic that three men have twelve. And the power of seeing how many
ears the average man, as a fact, possesses, the power of counting a
gentleman’s ears accurately and without mathematical confusion,
is not a logical thing but a primary and direct experience, like a physical
sense, like a religious vision. The power of counting ears may be limited
by a blow on the head; it may be disturbed and even augmented by two
bottles of champagne; but it cannot be affected by argument. Logic has
again and again been expended, and expended most brilliantly and
effectively, on things that do not exist at all. There is far more logic,
more sustained consistency of the mind, in the science of heraldry than in
the science of biology. There is more logic in *Alice in Wonderland* than
in the Statute Book or the Blue Books. The
relations of logic to truth depend, then, not upon its perfection as logic,
but upon certain pre-logical faculties and certain pre-logical discoveries,
upon the possession of those faculties, upon the power of making those
discoveries. If a man starts with certain assumptions, he may be a good
logician and a good citizen, a wise man, a successful figure. If he starts
with certain other assumptions, he may be an equally good logician and a
bankrupt, a criminal, a raving lunatic. Logic, then, is not necessarily an
instrument for finding truth; on the contrary, truth is necessarily an
instrument for using logic—for using it, that is, for the discovery of
further truth and for the profit of humanity. Briefly, you can only find
truth with logic if you have already found truth without it.

Thoughts about Logic & Proof

Accept my premises and I will lead you infallibly to my conclusions.

Your conclusions can be completely wrong even though your logic is completely right.

All argument begins with an assumption, or a set of assumptions; that is, with something you don’t dispute. You can, of course, dispute the assumptions at the beginning of your argument, but in that case you are beginning a different argument with another set of assumptions at the beginning of it. And so on ad infinitum.

Irving Bashevis Singer

Any premises that are capable of being put into words are also capable of being verbally questioned. Any argument whatsoever can thus be made into an infinite regress.

There always comes a point at which the question of sanity takes precedence over the question of demonstrable truth.

Christopher Derrick

As logic improves, less and less can be proved.

Bertrand Russell

Bertrand Russell

As nothing can be proved but by supposing something intuitively known, and evident without proof, so nothing can be defined but by the use of words too plain to admit a definition.

Samuel Johnson

No argument can establish the truth of its premises, since if it tried to do so it would be circular; and therefore no argument can establish the truth of its conclusions.

Bertrand Russell

C. S. Lewis

To prove that anything is true you need some truth to start with.

Insanity is often the logic of an accurate mind overtaxed.

Dr. Oliver Wendell Holmes

Einstein was only half joking when he said, “My office in Prague looked out over an asylum and there were times when I felt a certain kinship with the inmates. They were the madmen who did not concern themselves with physics, I was the madman who did.”

It is not enough for a system of ideas to be complete in theory. It must not be crippling in practice.

G. K. Chesterton

Logic must be governed by common sense. There’s not much use counting the steps of the logic if every step takes us further away from common sense.

Alan Wood

You can never prove your first statement or it would not be your first.

G. K. Chesterton

Logic is often more efficient as a weapon for destroying other logicians than as an instrument for discovering truth.

LOGIC: an unfair means sometimes used to win an argument.

J. B. Morton

Proof is an idol before which the mathematician tortures himself.

Sir Arthur Eddington

To find within a body of doctrine G (where G is, say, the theorems of arithmetic) a proof that G is consistent is impossible, for to accept the validity of such a proof is to concede to a part of G a special privilege which is clearly not justified if the coherence of G as a whole is in doubt.

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