Twovalued

Twovalued, or two-valued, refers to logics and semantic frameworks that assign exactly two truth values, commonly true (T) and false (F), to every proposition. The most familiar example is classical or Boolean logic, used in mathematics, computer science, and philosophy.

In two-valued logic, formulas are evaluated by truth-functional connectives. A valuation assigns T or F to each atomic proposition; the truth value of composite formulas is determined by standard truth tables: NOT flips T/F; AND yields T only if both operands are T; OR yields T if at least one operand is T; IMPLIES is defined so that P→Q is equivalent to ¬P ∨ Q; biconditional P↔Q is true when P and Q have the same value.

The framework embodies the law of the excluded middle and the principle of bivalence: every proposition is either true or false, with no middle ground. Classical propositional logic is complete and sound with respect to these semantics. The algebraic counterpart is Boolean algebra, where truth values form a two-element Boolean algebra.

History and relation: two-valued logic has roots in ancient and medieval logic but was formalized in the 19th and 20th centuries with George Boole’s algebra of logic and later formalization by Frege, Russell, and others; the two values are typically represented as true and false, 1 and 0.

Applications and limitations: Two-valued logic underpins digital circuits and most programming languages, theorem proving, and databases. It is not well-suited for vagueness, truth-value gaps, or higher-order uncertainty, leading to three-valued and many-valued logics.