In
logic and
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, 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
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
, (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, 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
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
consists of a set (
domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The domain ...
) 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
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,
for a proposition
.
[Dirk van Dalen, (2004) ''Logic and Structure'', Springer Universitext, (''see section 1.2'') ]
See also
* Algebraic semantics (mathematical logic)">Algebraic semantics
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)