In
logic, a
threevalued logic (also
trinary logic,
trivalent,
ternary, or
trilean,sometimes abbreviated
3VL) is any of several
manyvalued 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 reformulated by
Grigore Moisil in an axiomatic algebraic form, and also extended to
nvalued logics in 1945.
Representation of values
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;^{[1]}
 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 mostsignificant nonzero digit has a value 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;^{[2]}^{[not in citation given]}
 0 for false, 1 for true, and a third noninteger "maybe" symbol such as ?, #, ½,^{[3]} or xy.
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;
^{[citation needed]} these may have readings of the
quantifier different from classical (binary) predicate logic and may include alternative quantifiers as well.
Logics
Where
Boolean logic has 2
^{2} = 4
unary operators, the addition of a third value in ternary logic leads to a total of 3
^{3} = 27 distinct operators on a single input value. Similarly, where Boolean logic has 2
^{22} = 16 distinct binary operators (operators with 2 inputs), ternary logic has 3
^{32} = 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.
^{[4]}
Kleene and Priest logics
Below is a set of
truth tables showing the logic operations for
Kleene's "strong logic of indeterminacy" and Priest's "logic of paradox".
(F, false; U, unknown; T, true)
 AND(A, B)
A ∧ B  B 
F  U  T 
A  F  F  F  F 
U  F  U  U 
T  F  U  T 
 OR(A, B)
A ∨ B  B 
F  U  T 
A  F  F  U  T 
U  U  U  T 
T  T  T  T 

 (−1, false; 0, unknown; +1, true)
 MIN(A, B)
A ∧ B  B 
−1  0  +1 
A  −1  −1  −1  −1 
0  −1  0  0 
+1  −1  0  +1 
 MAX(A, B)
A ∨ B  B 
−1  0  +1 
A  −1  −1  0  +1 
0  0  0  +1 
+1  +1  +1  +1 


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. The difference lies in the definition of tautologies. Where Kleene logic's only designated truth value is T, Priest logic's designated truth values are both T and U. In Kleene logic, the knowledge of whether any particular unknown state secretly represents true or false at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one unknown operand. For example, since true OR true equals true, and true OR false also equals true, one can infer that true OR unknown equals true, as well. In this example, since either bivalent state could be underlying the unknown state, but either state also yields the same result, a definitive true results in all three cases.
If numeric values, e.g.
balanced ternary values, are assigned to
false,
unknown and
true such that
false is less than
unknown and
unknown is less than
true, then A AND B AND C... = MIN(A, B, C …) and A OR B OR C … = MAX(A, B, C...).
Material implication for Kleene logic can be defined as:
$A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{NOT}}(A)\ {\mbox{OR}}\ B$, and its truth table is
IMP_{K}(A, B), OR(¬A, B)
A → B  B 
T  U  F 
A  T  T  U  F 
U  T  U  U 
F  T  T  T 
 IMP_{K}(A, B), MAX(−A, B)
A → B  B 
+1  0  −1 
A  +1  +1  0  −1 
0  +1  0  0 
−1  +1  +1  +1 

which differs from that for Łukasiewicz logic (described below).
Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a wellformed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only
designated truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the
Logic of Paradox (LP) has the same truth tables as Kleene logic, but it has two
designated truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules.)
^{[5]}
Łukasiewicz logic[edit]
The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is true. This section follows the presentation from Malinowski's chapter of the Handbook of the History of Logic, vol 8.^{[6]}
IMP_{Ł}(A, B)
A → B  B 
T  U  F 
A  T  T  U  F 
U  T  T  U 
F  T  T  T 
 IMP_{Ł}(A, B)
A → B  B 
+1  0  −1 
A  +1  +1  0  −1 
0  +1  +1  0 
−1  +1  +1  +1 

In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
 A ∨ B = (A → B) → B
 A ∧ B = ¬(¬A ∨ ¬ B)
 A ↔ B = (A → B) ∧ (B → A)
It's also possible to derive a few other useful unary operators (first derived by Tarski in 1921):
 MA = ¬A → A
 LA = ¬M¬A
 IA = MA ∧ ¬LA
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 threevalued 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. For example,
A → A and
A ↔ A are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the
law of excluded middle,
A ∨ ¬A, and the
law of noncontradiction,
¬(A ∧ ¬A) are not tautologies in Ł3. However, using the operator
I defined above, it is possible to state tautologies that are their analogues:
Bochvar logic
Application in SQL Modular algebras
Some 3VL
modular algebras have been introduced more recently, motivated by circuit problems rather than philosophical issues:
^{[7]}
 Cohn algebra
 Pradhan algebra
 Dubrova and Muzio algebra
The database structural query language
SQL implements ternary logic as a means of handling comparisons with
NULL field content. The original intent of NULL in SQL was to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs.
^{[8]} The SQL standard defines an optional feature called F571, which adds some unary operators, among which is
IS UNKNOWN
corresponding to the Łukasiewicz
I in this article. The addition of
IS UNKNOWN
to the other operators of SQL's threevalued logic makes the SQL threevalued logic
functionally complete,
^{[9]} meaning its logical operators can express (in combination) any conceivable threevalued logical function.