Page 280 - Encyclopedia of Philosophy of Language

Language and Logic
Among the best-known many-valued logics are Lukasiewicz's, Post's, Bochvar's, and Kleene's.
In Lukasiewicz's 3-valued logic (motivated by the idea, already suggested by Aristotle in de Inter- pretatione ix, that future contingent sentences are nei- ther true nor false but 'indeterminate') both the Law of the Excluded Middle ('LEM'; 'p or not p') and the Law of Non-Contradiction ('LNC'; 'not both p and not-p') fail, taking the intermediate value when 'p' does. But the Principle of Identity ('if p then p') holds, since the matrices assign 'true' to a conditional when both antecedent and consequent are assigned 'inter- mediate.' Generalization yields 4- and more-valued logics.
Post (motivated by the mathematical interest of generalizing the method of truth-tables) developed a class of many-valued systems notable for an unusual, cyclic negation: in the 3-valued case, the negation of a true sentence is assigned the intermediate value, the negation of an intermediate sentence, 'false,' and the negation of a false sentence, 'true.' LEM, LNC and the Principle of Identity all fail.
Bochvar (motivated by the idea that semantic para- doxes such as the liar—"This sentence is false'—can be resolved by acknowledgingthat such sentences are neither true nor false) developed a 3-valued logic in which any sentence compounded by means of his 'pri- mary' connectives of which any component is assigned 'paradoxical,' is itself assigned 'paradoxical'. Hence no classical theorems, expressed in terms of the pri- mary connectives, are uniformly assigned 'true.' Boch- var also introduces an assertion operator, read 'it is true that,' which leaves a true sentence true and a false sentence false, but makes a paradoxical sentence false; and the classical theorems are restored in terms of 'secondary' or 'external' connectives, defined via the primary connectives and the assertion operator, which are 2-valued.
In Kleene's 3-valued system (in which the third value is to be read either 'undefined' or 'unknown, value immaterial') the matrices of the 'weak' con- nectives are like those for Bochvar's primary con- nectives. The matrices for Kleene's 'strong' connectives are constructed on the principle that a compound sentence with an undefined or unknown component or components should be assigned 'true' ['false'] just hi case the assignments to the other com- ponents would be sufficient to determine the com- pound as true [false], whether the undefined or unknown component were true or false. So not only LEM and LNC, but also the Principle of Identity, fail; for 'if p then q' is assigned V if both 'p' and 'q' are assigned '«,' so 'if p then p' takes V when 'p' does.
2.3 Fuzzy Logic
In Zadeh's deviant set theory, intended to represent the extensions of vague as well as precise predicates,
membership comes in degrees, the degree of mem- bership of an object in a fuzzy set being represented by some real number between 0 and 1. 'Fuzzy logic' sometimes refers to the family of indenumerably- many-valued systems that results from using fuzzy set theory in a semantic characterization in which sentence connectives are associated in the usual way with set-theoretical operations (negation with comp- lementation, implication with inclusion, etc.), but sen- tence letters can take any of the indenumerably many values of the interval [0,1]. But Zadeh himself reserves the term 'fuzzy logic' for the result of a second stage of'fuzzification,' motivatedbytheideathatthemeta- linguistic predicates 'true' and 'false' are themselves vague. In this sense, 'fuzzy logic' refers to a family of systems in which the indenumerably many values of the base logic (Zadeh favors the indenumerably- many-valued extension of Lukasiewicz's 3-valued logic) are superseded by denumerably many fuzzy truth-values, fuzzy sub-sets of the set of values of the base logic, characterized linguistically as "true, false, not true, very true, not very true, . . . , ' etc. In fuzzy logic, according to Zadeh, such traditional concerns as axiomatization, proof procedures, etc., are 'per- ipheral'; for fuzzy logic is, he suggests, not just a logic of fuzzy concepts, but a logic which is itself fuzzy (for
a further discussion, see Sect. 3.2 below).
2.4 Quantum Logic
It has sometimes been thought that certain pec- uliarities—'causal anomalies' in Reichenbach's phrase—of quantum mechanics can be accom- modated only by a restriction of the logical apparatus representing quantum mechanical reasoning. The 'quantum logic' proposed by Reichenbach is Luka- siewicz's 3-valued system extended by new forms of negation, implication, and equivalence, and Des- touches-Fevrier's is also many-valued; but 'quantum logic' more often refers to the nontruth-functional system of Birkhoff and von Neumann, in which LEM and LNC hold, but the Distributive Laws (the prin- ciples that from '(A or B) and (A or C)' one may infer 'A or (B and C),' and from '(A or B)and C' one may infer '(A and C) or (B and C)') fail.
2.5 Intuitionist Logic
Intuitionism is a school in the philosophy of math- ematics founded by L. E. J. Brouwer, who held that mathematical entities are mental constructions, and that logic is secondary to mathematics. These ideas motivate, first, a restricted mathematics in which non- constructive existence proofs are not permitted, and, second, a restricted logic in which neither LEM nor the Principle of Double Negation ('DN'; 'if not-not-p then p') hold. There are rival systems of intuitionist logic, the best-known being Heyting's; others are more restricted yet.
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