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In realizability truth values are sets of programs, which can be understood as computational evidence of validity of a formula. For example, the truth value of the statement "for every number there is a prime larger than it" is the set of all programs that take as input a number , and output a prime larger than . In category theory, truth ...
The simplest approach to truth values means that the statement may be "true" in one case, but "false" in another. In one sense of the term tautology , it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms (may also be called a valuation or assignment depending upon the context).
A truth table is a semantic proof method used to determine the truth value of a propositional logic expression in every possible scenario. [93] By exhaustively listing the truth values of its constituent atoms, a truth table can show whether a proposition is true, false, tautological, or contradictory. [94] See § Semantic proof via truth tables.
Classical propositional logic is a truth-functional logic, [3] in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. [4] On the other hand, modal logic is non-truth-functional.
The symbols used will vary from author to author and between fields of endeavor. In general the abbreviations "T" and "F" stand for the evaluations TRUTH and FALSITY applied to the variables in the propositional formula (e.g. the assertion: "That cow is blue" will have the truth-value "T" for Truth or "F" for Falsity, as the case may be.).
Logical value – Value indicating the relation of a proposition to truth; Multi-valued logic – Propositional calculus in which there are more than two truth values; Münchhausen trilemma – Thought experiment used to demonstrate the impossibility of proving any truth; Pluralist theories of truth
Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
In particular, the truth value of can change from one model to another. On the other hand, the claim that two formulas are logically equivalent is a statement in metalanguage, which expresses a relationship between two statements and . The statements are logically equivalent if, in every model, they have the same truth value.