History
Nicolai A. Vasiliev since 1910 and Jan Łukasiewicz called their own work non-Aristotelian logic. Alfred Korzybski carried the term into his system of General Semantics in 1933 (citing Łukasiewicz), and science fiction writer A. E. van Vogt later helped popularize it. Korzybski focused on the use of three or more truth values in the new systems of logic, although he connected this to his own rejection of Aristotle's principle of identity. Following Łukasiewicz's early work, Korzybski and later proponents of General Semantics associate these truth values with probabilities and the use of scientific induction. Łukasiewicz later seemed more cautious about this connection.
While Łukasiewicz seems to have spent more time on three-valued logic than any other system, he said that one could keep increasing the number of truth values indefinitely. Thus, he wrote: "if 0 is interpreted as falsehood, 1 as truth, and other numbers in the interval 0-1 as the degrees of probability corresponding to various possibilities, a many-valued logic is obtained which is expansion of three-valued logic and differs from the latter in certain details." Richard Threlkeld Cox later showed in Cox's theorem that any extension of Aristotelian logic to incorporate truth values between 0 and 1, in order to be consistent, must be equivalent to Bayesian probability.
Nicolai A. Vasiliev in 1910 rejected the law of contradiction as well as law of the excluded middle and proposed a logic he called imaginary which is tolerant to contradiction.
Hans Reichenbach described a system of logic that he explicitly linked with probability theory. He called his probability logic a generalization of two-valued logic. Reichenbach also suggested applying a three-valued logic to quantum mechanics. His probability logic does not receive much attention from modern logicians.
Aristotle allowed for the possibility of all these logics in De Interpretatione, Chapter 9. He wrote here that when it comes to statements about the future, "it is not necessary that of every affirmation and opposite negation one should be true and the other false." (Revised Oxford translation)
Lotfi Zadeh developed a system of "fuzzy logic" using a range of truth values from 0 to 1, but distinguished it sharply from probability theory.
Robert Anton Wilson in The New Inquisition developed a non-Aristotelian system of classification in which propositions can be assigned one of 7 values: true, false, indeterminate, meaningless, self-referential, game rule, or strange loop. Wilson did not devise a formal system for manipulating propositions once classified, but suggested that we can clarify our thinking by not restricting ourselves to simplistic true/false binaries.
Alternative terms for these logics in common academic usage include deviant logic and multi-valued logic (see Haack, 'Philosophy of Logic', 1980). Not all non-classical logics fall into this class, e.g. Modal logic is a non-classical logic which, however, has only two truth values.
Read more about this topic: Non-Aristotelian Logic
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