In
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 prem ...
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
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 a ...
, (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 morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
, 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 a ...
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
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).html" ;"title="phi">![\phi!.html" ;"title="phi.html" ;"title="![\phi">![\phi!">phi.html" ;"title="![\phi">![\phi!v 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)