Ternary logic - Definition and Overview

Ternary logic is a multi-valued logic in which there are three truth values indicating true, false and unknown.

Formal definitions

Significations

In ternary logic, the number symbols 0, 1, and 2 are usually assigned to the following truth values (or some permutation thereof):

• 0 means unknown
• 1 means true
• 2 means false

Ternary operators

Inversion (NOT)

The negation operator should interchange truth and falsehood, while leaving the third value unchanged, since the negation of an unknown statement is still unknown. Thus:

• NOT(0) = 0
• NOT(1) = 2
• NOT(2) = 1

Conjunction (AND), Disjunction (OR), Implication (IF...THEN), and Equivalence (EQUALS)

The binary operators can also be evaluated under this truth value interpretation. The result is:

Note that any two statements with the same truth value are equivalent, even if the truth value is unknown.

Other operators

All other ternary logic operators can be simulated by the four basic operators NOT, AND, OR and IF...THEN.

The 27 one-variable functions in ternary logic are represented in the following table. Some of them are given names:

The number of functions for a given number of variables for trinary logic can be calculated by the equation <math>3^{3^v}<math>, where v represents the number of variables. This gives us

• 27 one-variable functions in ternary logic (as compared to only 4 one-variable Boolean functions).
• 19,683 two-variable functions in trinary logic (compared with 16 for binary), and
• 7,625,597,484,987 three-variable functions.

Two-argument functions

Commutativity

As stated above, there are 19,683 two-argument ternary functions. However, only 729 (<math>3^{bc(2, 3)}<math>, where bc is the binomial coefficient) of these are commutative. Of the four functions defined above, OR, AND, and EQUIV are commutative, while IF/THEN is not. For comparison, there are 8 two-argument binary functions.

Implementation

• Setun a computer actually implemented in ternary logic.

External links

• Steve Grub's Trinary.cc (http://www.trinary.cc/)
• Trinary Computer Systems (http://xyzzy.freeshell.org/trinary/)
• TriINTERCAL (http://www.progsoc.uts.edu.au/~sbg/intercal/ick5.html)
• Trivalent Logic in the Language of Aymara (http://www.aymara.org/biblio/igr/igr3.html)
• A brief introduction to ternary logic (http://www.aymara.org/ternary/ternary.pdf)

See also: Digital circuit, Ternary, Boolean algebra, Boolean function, binary logic.