Truth-value

"True and false" redirects here. For the book, see True and False: Heresy and Common Sense for the Actor. For the Unix commands, see true and false (commands). For other uses, see True (disambiguation) and False (disambiguation).

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

Classical logic

 
true
 
··
conjunction
¬
 
false
··
disjunction
Negation permutes
truth with false and
conjunction with disjunction

In classical logic, with its intended semantics, the truth values are true (1 or T) and false (0 or ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws:

¬(Template:MvarTemplate:Mvar) ⇔ ¬Template:Mvar ∨ ¬Template:Mvar
¬(Template:MvarTemplate:Mvar) ⇔ ¬Template:Mvar ∧ ¬Template:Mvar

Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.

Multi-valued logic

Multi-valued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval Template:Closed-closed such structure is a total order; this may be expressed as existence of various degrees of truth.

Algebraic semantics

Main article: Algebraic logic

Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the necessary truth of formulae.

But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.

In other theories

Intuitionistic type theory uses types in the place of truth values.

Topos theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.

See also

Thinking portal

External links

  • Stanford Encyclopedia of Philosophy

Template:Logical truth

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