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Concept
Three-valued logic
Summary
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false.
Emil Leon Post is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions. The conceptual form and basic ideas of three-valued logic were initially published by Jan Łukasiewicz and Clarence Irving Lewis. These were then re-formulated by Grigore Constantin Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.
Around 1910, Charles Sanders Peirce defined a many-valued logic system. He never published it. In fact, he did not even number the three pages of notes where he defined his three-valued operators. Peirce soundly rejected the idea all propositions must be either true or false; boundary-propositions, he writes, are "at the limit between P and not P." However, as confident as he was that "Triadic Logic is universally true," he also jotted down that "All this is mighty close to nonsense." Only in 1966, when Max Fisch and Atwell Turquette began publishing what they rediscovered in his unpublished manuscripts, did Peirce's triadic ideas become widely known.
As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. A few of the more common examples are:
in balanced ternary, each digit has one of 3 values: −1, 0, or +1; these values may also be simplified to −, 0, +, respectively;
in the redundant binary representation, each digit can have a value of −1, 0, 0/1 (the value 0/1 has two different representations);
in the ternary numeral system, each digit is a trit (trinary digit) having a value of: 0, 1, or 2;
in the skew binary number system, only the least-significant non-zero digit can have a value of 2, and the remaining digits have a value of 0 or 1;
1 for true, 2 for false, and 0 for unknown, unknowable/undecidable, irrelevant, or both;
0 for false, 1 for true, and a third non-integer "maybe" symbol such as ?, #, 1⁄2, or xy.
Official source
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