Traditionally, logical calculi are bivalentthat is, there are only two possible truth values for any proposition, true and false (which generally correspond to our intuitive notions of truth and falsity). But bivalence is only one possible range of truth values that may be assigned, and other logical systems have been developed with variations on bivalence, or with more than two possible truthvalue assignments.
In the classical bivalence scheme, true and false are determinate values: a proposition is either true or false (exclusively), and if the proposition does not have one of those values, by definition it must have the other. This is the justification for the Law of the excluded middle: P ∨ ¬P (i.e., either the proposition or its negation holds).
One point to remember is that logic is a system for preserving some property of propositions across transformations. In classical logic, this property is 'truth': In a valid argument, the truth of the derived proposition is guaranteed because the application of valid steps preserves the property. However, that property doesn't have be that of 'truth'; instead, it can some other concept.
For example, the preserved property could be justification (this is the foundational concept of intuitionistic logic). Thus, a proposition is not true or false; instead, it is justified or not. A key difference between justification and truth, in this case, is that the law of the excluded middle doesn't hold: a proposition that is not not justified is not necessarily justified; instead, it's only not proven that it's not justified. The key difference is the determinacy of the preserved property: One may prove that P is justified, that P is not justified, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.
Fuzzy logics were introduced by Lotfi Zadeh as a formalization of vagueness, i.e., the phenomenon that a predicate may apply to an object not absolutely, but to a certain degree, and that there may be borderline cases. Such logics can be used to deal with the sorites paradox. Instead of the two truth values 'true' and 'false,' fuzzy logic employs infinitely many values between 0, corresponding to 'absolutely false', and 1 corresponding to 'absolutely true'. A borderline case might then be assigned a truth value of 0.5. One can apply these systems of logic as the foundation of fuzzy set theories. Other examples of infinitelyvalued logics are probability logic[?] and neutrosophic logic.
The Polish logician and philosopher Jan Lukasiewicz[?] began to create systems of manyvalued logic in 1920, using a third value for "possible" to deal with Aristotle's paradox of the sea battle. Meanwhile the American mathematician Emil L. Post[?] (1921) also introduced the formulation of additional truth degrees. Gödel in 1932 showed that intuitionistic logic is not a finitelymany valued logic, and defined a system of Gödel logics[?] intermediate between classical and intuitionistic logic.
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