A Brief Logical History

I have just connected this blog to Google+ and LinkedIn. In case this works and you are just now seeing this and joining me, I began this particular series last week with The Learned Professors I continue my series of blog posts on my developments in logic. Welcome to the Revolution.

Mathematical or symbolic logic was developed by George Boole, although not it its present form. It was at first quite cumbersome. When the “inclusive or” (A or B or both) was adopted instead of the “exclusive or” (A or B but not both), it became possible to substantially simplify its expressions. Two major branches developed: Propositional logic, which dealt with the truth of simple sentences, and predicate logic, which dealt with classes and sets of objects.

Beginning in 1910, Bertrand Russell and Alfred North Whitehead attempted to develop logic as a formal system with a few axioms and deduce the rest of it using logical methods. This led to the development of alternate logics in which some of the rules of inference of traditional logic were modified and their consequences explored. Two of the principal varieties were intuitionistic logic, which was formalized by Arend Heyting in the 1930s, following the thinking of Earl Brouwer; and the Lewis Systems of modal logic, which is the logic of possibility and necessity. These systems were developed in the 1920s.

Also, in the 1920s, Jan Lukasiewicz, a Polish logician, developed a three valued propositional logic, and following his lead, other multi-valued logics were developed. Notable among the three valued versions are the Kleene and Priest logics, the Bochvar logic, and the Post logics. Of these, the Kleene and Priest logics are most closely related, and I will skip over the Bochvar logic and the Post logics.

None of these is quite adequate as an extension of classical logic. On the one hand, intuitionistic logic and the Lewis systems lack the important, powerful tool of truth tables in order to evaluate propositions and proposed theorems. On the other hand, the multi-valued logics lack good deductive rules of inference and are forced to use elaborate circumlocutions in order to prove or establish useful theorems and results. They are unbearably cumbersome and their explanations are turgid and full of symbolism and notation that only the initiate can read or comprehend.

An alternate approach, developed by Lotfi Zadeh, is “fuzzy logic”, which is similar to the logics developed by Lukasiewicz, but using numeric degrees of truth.

My approach is revolutionary because by providing a sound basis of inference in Lukasiewicz 3-value logic, it offers the possibility of unifying intuitionism, modal logic, the most important three valued logics, and concepts of fuzzy logic with the same power and ease of that classical logic. I mean to unify and simplify a good part of logic.

My biggest difficulty is getting anyone to pay attention. My “Hey Lookee Here What I Found!” has met with a resounding “Ho Hum”. However, lack of interest is not disproof. Since I haven’t been granted entry to the sacred halls of upper academia and have no mentor to guide me in the secrets of successfully getting published, and since I don’t know how to phrase or market my ideas to make them interesting, I’m doing this here in small chunks.

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