• Source: Valuation (logic)

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



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 also


Algebraic semantics




References



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.

Kata Kunci Pencarian:

• Daftar perusahaan rintisan unikorn
• Otto Neurath
• Valuation (logic)
• Valuation
• Tautology (logic)
• Paraconsistent logic
• Modal logic
• Intuitionistic logic
• Truth value
• Predicate (mathematical logic)
• Residual income valuation
• List of logic symbols