A ternary, three-valued or trivalent logic is a term to describe any of several multi-valued logic systems in which there are three truth values indicating true, false and some third value. This is contrasted with the more commonly known bivalent logics (such as boolean logic) which provide only for true and false. Multi-valued logics are logical calculi in which there are more than two possible truth values. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... Boolean logic is a complete system for logical operations. ...

Definitions

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: Ternary or trinary is the base-3 numeral system. ...

• 1 for true, 2 for false, and 0 for unknown, irrelevant, or both.^[1]
• 0 for false, 1 for true, with the third value being non-integer symbol such as # or ½.^{[citation needed]}
• Balanced ternary uses -1 for false, 1 for true and 0 for the third value; these values may also be simplified to -, +, and 0, respectively.^[2]

This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and 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. Balanced Ternary (aka Signary) is a method of numeric representation which expands on binary and is similar to trinary. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

True/false/unknown ternary logic

As one of the more intuitively obvious schemes, this configuration has found application in various areas, perhaps most notably the database query language SQL. SQL (commonly expanded to Structured Query Language — see History for the terms derivation) is the most popular computer language used to create, modify, retrieve and manipulate data from relational database management systems. ...

In SQL, "NULL" represents the 'unknown' state, and a number of operators are defined, such as >, <, = and <>. Whilst these operators do not require logic-valued inputs, they do produce logical outputs. Comparing something to NULL results in NULL itself being returned - even in the case of the test for equality. This is because it is impossible to tell whether one unknown state is really the same as another - hence the result of the comparison is itself unknown. To overcome this, an "IS NULL" operator was defined. ^[3] Columns in Relational database management systems (RDBMS) can optionally store NULL values. ... This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ...

Below is a truth table showing the results of some logic operations for this three-state system. Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

Electronics

The most widely implemented form of three-state logic is found in digital electronics. It is very important to note that this is not true ternary logic. It is mentioned here for completeness, being the only widespread three-state system in use. Outputs can have one of three states, yet inputs can only recognise two. Hence the kind of relations shown in the table above do not occur. In electronics a three-state, tri-state or 3-state digital logic gate is one in which the output circuit can be disconnected from the rest of the circuit, putting the output in a high impedance state. ... Digital Electronics is based on a number of discrete voltage levels, usually two, as distinct from analog electronics which uses voltages to represent variables directly. ...

Commonly referred to as tristate ^[4] logic (a trademark of National Semiconductor), it comprises the usual true and false states, with a third transparent high impedance state (or 'off-state') which effectively disconnects the logic output. This provides an effective way to connect several logic outputs to a single input, where all but one are put into the high impedance state, allowing the remaining output to operate in the normal binary sense. This is commonly used to connect banks of computer memory and other similar devices to a common data bus; a large number of devices can communicate over the same channel simply by ensuring only one is enabled at a time. Categories: Electronics companies of the United States | Companies based in California | Corporation stubs ... In digital circuits, high impedance a voltage that is lower than the threshhold for a digital 0. ... In computer architecture, a bus is a subsystem that transfers data or power between computer components inside a computer or between computers. ...

Although it could be argued that the high-impedance state is effectively an "unknown", there is absolutely no provision in the vast majority of normal electronics to interpret a high-impedance state as a state in itself. Inputs can only detect true and false; high-impedance is best described as invisible. Typically, most electronic logic configurations default to a true state when they sense no input - thus they also interpret a high impedance on an input as a true state, although this is by no means universal.

True ternary logic can be implemented in electronics, although the complexity of design has thus far made it uneconomical to pursue commercially and interest has been primarily confined to research, since 'normal' binary logic is very much cheaper to implement and in most cases can easily be configured to emulate ternary systems. However, there are useful applications in fuzzy logic and error correction, and several true ternary logic devices have been manufactured (see external links). Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... In computer science and information theory, error correction consists of using methods to detect and/or correct errors in the transmission or storage of data by the use of some amount of redundant data and (in the case of transmission) the selective retransmission of incorrect segments of the data. ...

References

1. ^ Hayes, Brian (November-December, 2001). "Third Base". American Scientist 89 (6): 490-494. Retrieved on 2006-12-25.
2. ^ Knuth, Donald E. (1981). The Art of Computer Programming Vol. 2. Reading, Mass.: Addison-Wesley Publishing Company, 190. 
3. ^ Lex de Haan and Gennick, Jonathan (July-August, 2005). "Nulls: Nothing to Worry About". Oracle Magazine.
4. ^ National Semiconductor, LS TTL Data Book, National Semiconductor Corporation, 1993.

2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... December 25 is the 359th day of the year (360th in leap years) in the Gregorian Calendar, with 6 days remaining in the year. ... Donald Knuth at a reception for the Open Content Alliance. ... Headquartered in Santa Clara, California, USA, National Semiconductor is one of the largest semiconductor manufacturers, specializing in analog devices and subsystems. ... 1993 (MCMXCIII) was a common year starting on Friday of the Gregorian calendar and marked the Beginning of the International Decade to Combat Racism and Racial Discrimination (1993-2003). ...

See also

• Digital circuit
• Ternary
• Ternary computer
• Boolean algebra
• Boolean function
• Binary logic
• Setun - an experimental Russian computer which was based on ternary logic

Digital circuits are electric circuits based on a number of discrete voltage levels. ... Ternary can mean: Ternary form, a form used for structuring music Ternary logic, a logic system with values true, false, and some other value Ternary numeral system, a base-3 counting system Ternary operation, an operation that takes three parameters Ternary plot or Ternary graph, a plot that shows the... Ternary computers use three-valued logic in their calculations. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... Setun (Russian: ) was a balanced ternary computer developed in 1958 in Russia. ...

External links

• Trinary Computer Systems
• TriINTERCAL - the intentionally-obfuscated INTERCAL programming language supports an implementation of ternary logic
• Boost.Tribool – an implementation of ternary logic in C++
• Team-R2D2 - a French institute that fabricated the first full-ternary logic chip (a 64-tert SRAM and 4-tert adder) in 2004