logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...

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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

and higher-order logics, a structure, (the

interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...

) 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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

, 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

v

is a valuation, that is, a mapping from the atoms to the set

\

, then the double-bracket notation is commonly used to denote a valuation; that is,

!">phi.html" ;"title="![\phi">![\phi!v

for a proposition

\phi

.Dirk van Dalen, (2004) ''Logic and Structure'', Springer Universitext, (''see section 1.2'')

References

*, 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 Semantic units Model theory Interpretation (philosophy)

Mathematical logic

Notation

See also

References