Ternary logic is a multi-valued logic in which there are three states, thus the ternary numeral system is used to represent ternary logic equations. This article is a work in progress.

Contents

1 Formal definitions 1.1 Significations 1.2 Ternary operators 1.2.1 Inversion (NOT) 1.2.2 Conjunction (AND), Disjunction (OR), Implication (IF...THEN), and Equivalence (EQUALS) 1.2.3 Other operators 2 Ternary logic in philosophy 3 Trits, tribbles, and trytes 4 Basic Ternary Algebra: Unary Functions 4.1 Constant functions 4.2 One-to-one functions 4.3 Many-to-one functions 5 Binary functions 5.1 Commutativity 5.2 Preference functions 5.3 Tritmasks 5.4 Named functions 6 Advanced functions 6.1 Unbalanced arithmetic 6.1.1 Negation: 3's complement 6.1.2 Addition / subtraction 6.2 Balanced arithmetic 6.2.1 Negation: inversion 6.2.2 Addition / subtraction 6.3 Unknown-state logic 7 Implementation 7.1 Existing computers 8 Magnetism 9 Electromechanical relays 10 Rapid single flux quantum 11 Rectifiers 12 External links

Formal definitions

Significations

In ternary logic, the number symbols have the following meaning:

• 0 means true or false
• 1 means true
• 2 means false

(This is different from base-3 Ternary numerals, where 0, 1, and 2 mean the same thing they do in decimal).

Ternary operators

Inversion (NOT)

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

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

Ternary logic

x	y	Conjunctive (X Y)	Disjunctive (X Y)	Implication (X → Y)	Equivalence (X ↔ Y)
0	0	0	0	1	1
0	1	0	1	1	0
0	2	2	0	0	0
1	0	0	1	0	0
1	1	1	1	1	1
1	2	2	1	2	2
2	0	2	0	1	0
2	1	2	1	1	2
2	2	2	2	1	1

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.

	f1	f2	f3	f4	f5	f6	f7	"1"	f9	f10	f11	"0"	f13	f14	f15	f16	f17	f18	f19	NOT	f21	"2"	f23	f24	f25	f26	f27
0	0	0	0	2	2	2	1	1	1	0	0	0	2	2	2	1	1	1	0	0	0	2	2	2	1	1	1
1	1	1	1	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0	2	2	2	2	2	2	2	2	2
2	2	1	0	2	1	0	2	1	0	2	1	0	2	1	0	2	1	0	2	1	0	2	1	0	2	1	0

The number of functions for a given number of variables for trinary logic can be calculated by the equation , 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.

Ternary logic in philosophy

Trits, tribbles, and trytes

Basic Ternary Algebra: Unary Functions

Constant functions

 000 clear to 0
 111 clear to 1
 222 clear to 2

One-to-one functions

This might look better in tables and the symbols might look better in TeX.

 F#  Name    Diff:012 Inverse   Expression
 012 buffer       '''  012      A    A
 021 swap 1/2     '/\  021      ['A  ∪A
 102 swap 0/1     /\'  102      ]'A  ∩A
 120 rotate up    ///  201      ]A   ∩A
 201 rotate down  \\\  120      [A   ∪A
 210 swap 0/2     \'/  210      'A   A, or A'

Many-to-one functions

 F#  ITE  Expression
 001 210 \A     ↘A            Shift Down
 002 220 ]/'A   ∩↗A
 010 100 \]A    ↘∩A
 011 001 \/A    ↘↗A
 020 120 ]/['A  ∩↗∪A
 022 002 [\'A   ∪↘A
 100 010 \'A    ↘A
 101 101 [/['A  ∪↗∪A
 110 210 [/'A   ∪↗A
 112 221 /\A    ↗↘A
 121 121 ]\]A   ∩↘∩A
 122 012 /A     ↗A            Shift Up
 200 020 ]/A    ∩↗A
 202 102 [\]A   ∪↘∩A
 211 021 ]\'A   ∩↘A
 212 112 /['A   ↗∪A
 220 202 [\A    ∪↘A
 221 212 /'A    ↗A

Binary functions

Commutativity

Preference functions

Tritmasks

Named functions

Advanced functions

Unbalanced arithmetic

Negation: 3's complement

Addition / subtraction

Balanced arithmetic

Negation: inversion

Addition / subtraction

Unknown-state logic

Implementation

Existing computers

• Setun

Magnetism

Electromechanical relays

Rapid single flux quantum

Rectifiers

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.