Truthvalues

Truthvalues are the semantic values assigned to declarative sentences by a valuation in a logical system. They indicate whether a sentence holds in a model, or in some logics, reflect other statuses such as uncertainty or degree of truth. Different logics specify different sets of truth values and rules for combining them under logical connectives.

In classical logic, the set of truth values is {true, false}. A valuation assigns each atomic proposition either true or false, and the connectives are defined by truth tables. The principle of bivalence holds that every proposition is either true or false.

Many-valued logics extend truth values beyond two. Examples include three-valued logics that add an unknown value, and four-valued logics that include true, false, both true and false, or neither. In these systems, a sentence may be designated as true, false, both, or neither, depending on the chosen semantics; truth values are often organized in an ordering to reflect partial truth or inconsistency.

Fuzzy and probabilistic approaches offer alternative interpretations of truth values. In fuzzy logic, truth values are real numbers in the interval [0,1], interpreted as degrees of truth. Logical connectives are defined by t-norms, t-conorms, and related operations. In probabilistic or belief-based logics, truth values may be probabilities or confidence measures, capturing uncertainty rather than absolute truth.

Truthvalues are central to the interpretation of logical systems and their applications in computer science, linguistics, mathematics, and formal semantics. They distinguish how different logics assign meaning to propositions and combine them to form complex statements. See also valuation, truth table, and semantic theory.