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
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 quantifi ...
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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
.
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 formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformul ...
s 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
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)