logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

and

model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

, a valuation can be: *In

propositional logic The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

, an assignment of

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...

s to

propositional variable In mathematical logic, a propositional variable (also called a sentence letter, sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building ...

s, with a corresponding assignment of truth values to all propositional formulas with those variables. *In

first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

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) 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 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 subformu ...

s using logical connectives and quantifiers. A

consists of a set (

domain of discourse In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also ...

) 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 The ''Sentences'' (. ) is a compendium of Christian theology written by Peter Lombard around 1150. It was the most important religious textbook of the Middle Ages. Background The sentence genre emerged from works like Prosper of Aquitaine's ...

(formulas with no

free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...

) 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 = v(\phi)

for a proposition

\phi

.Dirk van Dalen, (2004) ''Logic and Structure'', Springer Universitext, page 18 - Theorem 1.2.2.

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