Three-valued logic
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 indeterminate 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. Conceptual form and basic ideas were initially created by Jan Łukasiewicz and C. I. Lewis. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.
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:
Inside a ternary computer, ternary values are represented by ternary signals.
This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, true}, and extends conventional Boolean connectives to a trivalent context. Ternary predicate logics exist as well; these may have readings of the quantifier different from classical (binary) predicate logic and may include alternative quantifiers as well.
Where Boolean logic has 22 = 4 unary operators, the addition of a third value in ternary logic leads to a total of 33 = 27 distinct operators on a single input value. Similarly, where Boolean logic has 222 = 16 distinct binary operators (operators with 2 inputs), ternary logic has 332 = 19,683 such operators. Where we can easily name a significant fraction of the Boolean operators (not, and, or, nand, nor, exclusive or, equivalence, implication), it is unreasonable to attempt to name all but a small fraction of the possible ternary operators.
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