In logic and model theory, a '''valuation''' can be: *In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables. *In first-order logic and higher-order logics, a structure, (the interpretation) and the corresponding assignment of a truth value to each sentence in the language for that structure (the valuation proper). The interpretation must be a homomorphism, while valuation is simply a function.
==Mathematical logic==
In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema. Valuations are also called truth assignments.
In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas.
In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. Formulas are built out of atomic formulas using logical connectives and quantifiers. A structure consists of a set (domain of discourse) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. Corresponding to each structure is a unique truth assignment for all sentences (formulas with no free variables) in the language.
== Notation == If <math>v</math> is a valuation, that is, a mapping from the atoms to the set <math>\{ t, f \}</math>, then the double-bracket notation is commonly used to denote a valuation; that is, <math>[\![\phi]\!]_v = v(\phi)</math> for a propositional formula <math>\phi</math>.<ref>Dirk van Dalen, (2004) ''Logic and Structure'', Springer Universitext, page 18 - Theorem 1.2.2. {{isbn|978-3-540-20879-2}}</ref>
== See also == * Algebraic semantics
== References == {{Reflist}}
*{{Citation | surname1 = Rasiowa | given1 = Helena | authorlink1 = Helena Rasiowa | surname2 = Sikorski | given2 = Roman | authorlink2 = Roman Sikorski | title = The Mathematics of Metamathematics | publisher = PWN | year = 1970 | place = Warsaw | edition = 3rd}}, chapter 6 ''Algebra of formalized languages''. * {{cite book|author1=J. Michael Dunn|author2=Gary M. Hardegree|title=Algebraic methods in philosophical logic|url=https://books.google.com/books?id=LTOfZn728-EC&pg=PA155|year=2001|publisher=Oxford University Press|isbn=978-0-19-853192-0|page=155}}
Category:Semantic units Category:Model theory Category:Interpretation (philosophy)