Three-valued logic

Three-valued logic (3VL), also known as ternary logic or trivalent logic, is a form of logic in which there are three truth values indicating true, false, and some indeterminate third value. This is in contrast to the more familiar binary logic (or Boolean logic) that uses only two values, true and false. Three-valued logic is used in various disciplines, including computer science, philosophy, and mathematics, to deal with situations where information might be incomplete or uncertain.

Overview

Three-valued logic was first introduced by the Polish logician Jan Łukasiewicz in the 1920s as a way to analyze philosophical paradoxes. The third value, often represented as 'I', stands for "indeterminate" or "unknown" and is used to address scenarios where neither truth nor falsity can be determined. This concept has been applied in various fields to model uncertainty, manage database systems with incomplete information, and design digital circuits that can handle unknown input values.

Truth Values

In three-valued logic, the three truth values are typically represented as:

• True (T)
• False (F)
• Indeterminate (I)

Different systems of three-valued logic may interpret the indeterminate value in various ways, such as "unknown", "irrelevant", or "both true and false".

Logical Operators

Three-valued logic extends the traditional binary logical operators (such as AND, OR, and NOT) to handle three truth values. The behavior of these operators can vary depending on the specific system of 3VL being used. However, a common approach is to define them in a way that reduces to classical binary logic when the indeterminate value is not involved.

Examples

• AND: The result is true if both operands are true; false if at least one operand is false; otherwise, indeterminate.
• OR: The result is true if at least one operand is true; false if both operands are false; otherwise, indeterminate.
• NOT: The result is the opposite truth value for true and false; remains indeterminate if the operand is indeterminate.

Applications

Three-valued logic has found applications in various areas:

• In computer science, particularly in database management systems (DBMS), to deal with null values which represent unknown or missing information.
• In the design of digital circuits and computational logic, where the third value can represent a high-impedance state in tri-state logic.
• In philosophy and linguistics, to analyze statements that cannot be classified as simply true or false, such as paradoxes or sentences with vague terms.

Challenges

While three-valued logic provides a useful framework for dealing with uncertainty, it also introduces complexity in the interpretation and implementation of logical operations. Determining how to appropriately apply the indeterminate value in different contexts requires careful consideration.

See Also

• Binary logic
• Fuzzy logic
• Jan Łukasiewicz
• Logic in computer science
• Null (SQL)