Emil Leon Post is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions.
[2] 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.
"[6] Only in 1966, when Max Fisch and Atwell Turquette began publishing what they rediscovered in his unpublished manuscripts, did Peirce's triadic ideas become widely known.
It is only because, in macrocosmic experience, everything that we regard as an empirically meaningful statement seems to be at least potentially verifiable or falsifiable that we prefer the convention according to which we say that every such statement is either true or false, but in many cases we don't know which.Similarly, Stephen Cole Kleene used a third value to represent predicates that are "undecidable by [any] algorithms whether true or false"[14][8] As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system.
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.
Given 2 possible values TF of the single Boolean input, there are four different patterns of output TT, TF, FT, FF resulting from the following unary operators acting on each value: always T, Identity, NOT, always F. Given three possible values of a ternary variable, each times three possible results of a unary operation, there are 27 different output patterns: TTT, TTU, TTF, TUT, TUU, TUF, TFT, TFU, TFF, UTT, UTU, UTF, UUT, UUU, UUF, UFT, UFU, UFF, FTT, FTU, FTF, FUT, FUU, FUF, FFT, FFU, and FFF.)
If the truth values 1, 0, and -1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic (where x + y uses addition, xy uses multiplication, and x2 uses exponentiation), or by the minimum/maximum functions: In these truth tables, the unknown state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic.
In Kleene logic, the knowledge of whether any particular unknown state secretly represents true or false at any moment in time is not available.
[20] Material implication for Łukasiewicz logic truth table is In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as: It is also possible to derive a few other useful unary operators (first derived by Tarski in 1921):[citation needed] They have the following truth tables: M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize modal logic using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..." In Łukasiewicz's Ł3 the designated value is True, meaning that only a proposition having this value everywhere is considered a tautology.
However, using the operator I defined above, it is possible to state tautologies that are their analogues: The truth table for the material implication of R-mingle 3 (RM3) is A defining characteristic of RM3 is the lack of the axiom of Weakening: which, by adjointness, is equivalent to the projection from the product: RM3 is a non-cartesian symmetric monoidal closed category; the product, which is left-adjoint to the implication, lacks valid projections, and has U as the monoid identity.
Some 3VL modulars arithmetics have been introduced more recently, motivated by circuit problems rather than philosophical issues:[22] The database query language SQL implements ternary logic as a means of handling comparisons with NULL field content.