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Can Logic be Proved?

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 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.

The truth of the conclusion is dependent on the truth “value” (true-false) of the premises, and is completely independent of the argument’s validity. If you begin with true premises and maintain proper logical form, you will end with true conclusions. If you start with false beginnings, it is impossible to end with true conclusions, if the argument is logical, valid. Like computers (which are based on logic), garbage in, garbage out. Also, if the computer is broken, its logic faulty, then true premises may or may not deliver true conclusions. The bottom line is that modern symbolic logic can be a powerful tool in the right hands.

In the beginning, modern logic was not really intended to solve problems of logic at all; it was about math. Mathematicians, in their efforts to make the system of mathematics completely comprehensible, found themselves going through the back door of mathematics into logic. After all, it is logic on which all mathematics is founded. George Boole (1815-1864), the famous mathematician, was the first to take the decisive steps past traditional logic. In his efforts to give a systematic explanation to the foundations of mathematics he created the beginnings of modern logic. When early mathematicians first examined the foundations of mathematics they were able to create a formal system of logic to support it. Unfortunately the systems they produced all had problems. Inevitably a contradiction would appear in their system or it would rely on some support outside of the system for its authority (it left some element of the system undefined). Neither of these conditions were acceptable.

Naturally, these mathematician/philosophers wanted to tidy up their problem and provide an ultimate justification for mathematics. To get that air tight case, the foundation for the argument, logic, also had to be airtight. In essence, they wanted “proof” that mathematics, as a system, was valid. But as the algebra of logic evolved into a full fledged theory of modern logic, the situation just got worse. The theory evolved, but the problem evolved with it: it was never solved. No matter how elaborate the system of logic became, it always had either a contradiction or an undefined element that was supported from outside the system. As you can imagine, you just can’t have such an important, fundamental theory left hanging in the breeze. As a result, philosophers have been struggling with finding an acceptable answer ever since. It was so important that the search for a proof of logic (and ultimately mathematics) developed a theoretical basis, and has become a theory in and of itself.

The central issue of “proof theory” goes right to the heart of the problem: consistency and completeness. “Consistency” in this context means “not a contradiction” and “completeness” means “all elements of the system defined.” The reason the two requirements of consistency and completeness are necessary is inherent in the system of logic. Consistency and completeness deal with the notions of “contradiction” and “identity,” and these two concepts are crucial to logic itself.

Even before Aristotle certain logical rules became so highly regarded that they were known as the “laws of thought.” These rules included the rule of “identity,” of “contradiction” and the principle of the “excluded middle.” The rule of “identity” states that a thing is what it is (given by traditional logicians as “A is A”). The rule of contradiction states that a thing cannot both be and not be a particular thing at the same time (given by traditional logicians as “A is B and A is not B cannot both be true”). The law of excluded middle will be addressed later.

Completeness of a system merely asks that for each and every element of the system there exist a definition within the system which identifies it. If a system is incomplete it means it has at least one element of the system which is undefined within the system. This element lacks identity.

The requirement of consistency asks that for any element of the system there not exist its contradiction within the system. You can’t have a system which maintains something and at the same time maintains just the opposite. These are necessary requirements of a system of logic and are the principles of “proof theory.”

In an attempt to secure a foundation for mathematics, Bertrand Russell introduced the “theory of types.” Russell showed that you could take the element of the system which is “incomplete,” or not defined, and define it in another layer of mathematical logic. He called it the theory of types because there are levels, or “types,” of logic which avoid this paradox. Think of the theory of types as a logical onion, with logic layered upon logic. The problem was that if one begins this procedure it is impossible to stop. If one does not postulate an infinite number of “types” then the last level is as incomplete as the first. This is precisely what Russell did with his “axiom of infinity.” The axiom of infinity simply stated that one could continue layering logic upon logic forever.

Unfortunately, even Russell himself recognized that the axiom of infinity was an arbitrary rule not supported by the theory of logic which he, and those who went before him, created. It was assumed. Because it was assumed, there is no definition within the system to support it and, therefore the system was still incomplete. The undefined element in the system was the axiom of infinity. Logic, and mathematics, were still unsupported. The theory of types represents a level of development shared with nearly all of the systems of symbolic logic now employed.

In the early part of this century the Austrian mathematician and logician Kurt Godel followed David Hilbert of Germany in searching the logic of mathematics for a proof of the ideal mathematical system. What he ultimately found shattered this goal forever, for mathematics and all “formal” systems (including logic). Godel proved that a proof theory for formal systems does not, and cannot, exist within the system (known as Godel’s Incompleteness Theorem). He proved that if a formal system is consistent then it must be incomplete. Further, he proved that if a formal system is complete, with no undefined elements, then it must be inconsistent, or contradictory. He showed the consistency and completeness of no type system can be proved within itself. Logic has fundamental structural flaws which bring into question its ultimate authority to deliver valid arguments, and therefore truth. Mathematics does not have, and will never have, the foundation of certainty we have always assumed it possessed.

To philosophers, the truth of an argument’s conclusions is not related to its validity. The truth or falsity of a proposition is known as “value,” and it is studied as a separate part of logic. Even though value is studied separately, it faces the same dilemma. Alfred Tarski, the brilliant Polish-American mathematician and logician, developed a proof of value very similar to Godel’s. His proof showed that any definition of the term “true” (in terms of the language specified “in use”) would result inevitably in a contradiction. Further, he found that when “truth is understood as a property of sentences of the language in question, such acceptance of a semantic term without definition is inevitable.” This is also similar to Godel’s Incompleteness Theory, in that Tarski found that when the term truth was left undefined (incomplete) the contradiction could be avoided.

The conclusions reached by Godel and Tarski lead to the same consequences that followed Sextus Empiricus’s argument. If you are looking for final answers to the fundamental questions of how we discover truth and understanding, an undeniable reality becomes apparent: there simply isn’t a complete understanding of our most cherished and important concepts. Since logic is ultimately not based on a solid foundation, there is no way to completely justify it as the ultimate authority in determining truth. And when even the concept of truth has been shown indeterminate and undefined within the system, how can truth ever be achieved? Ultimately, it seems, it cannot. Nothing determined by logic is certain. This includes the truth of this philosophy, or any philosophy.