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 
See also
- Algebraic semantics
References
- Dirk van Dalen, (2004) Logic and Structure, Springer Universitext, (see section 1.2) ISBN 978-3-540-20879-2
- Rasiowa, Helena; Sikorski, Roman (1970), The Mathematics of Metamathematics (3rd ed.), Warsaw: PWN, chapter 6 Algebra of formalized languages.
- J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford University Press. p. 155. ISBN 978-0-19-853192-0.
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 logicIn 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 NotationIf v displaystyle v is a valuation that is a mapping from the atoms to the set t f displaystyle t f then the double bracket notation is commonly used to denote a valuation that is v ϕ ϕ v displaystyle v phi phi v for a proposition ϕ displaystyle phi See alsoAlgebraic semanticsReferencesDirk van Dalen 2004 Logic and Structure Springer Universitext see section 1 2 ISBN 978 3 540 20879 2 Rasiowa Helena Sikorski Roman 1970 The Mathematics of Metamathematics 3rd ed Warsaw PWN chapter 6 Algebra of formalized languages J Michael Dunn Gary M Hardegree 2001 Algebraic methods in philosophical logic Oxford University Press p 155 ISBN 978 0 19 853192 0
