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 twovalued 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:Mvar∧Template:Mvar) ⇔ ¬Template:Mvar ∨ ¬Template:Mvar
 ¬(Template:Mvar∨Template:Mvar) ⇔ ¬Template:Mvar ∧ ¬Template:Mvar
Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.
Multivalued logic
Multivalued 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:Closedclosed such structure is a total order; this may be expressed as existence of various degrees of truth.
Algebraic semantics
Not all logical systems are truthvaluational 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 nontruthvaluational 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
External links
 Stanford Encyclopedia of Philosophy


 Overview 

 Academic areas  

 Foundations  


                

Template:Logical truth
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.