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 premises ...

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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

, 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''). Computing In some progr ...

s to

propositional variable In mathematical logic, a propositional variable (also called a 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-blocks of propositi ...

s, with a corresponding assignment of truth values to all

propositional formula In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional fo ...

s 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 quantifie ...

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 structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

, 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 subformu ...

s using logical connectives and quantifiers. A

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 ''The Four Books of Sentences'' (''Libri Quattuor Sententiarum'') is a book of theology written by Peter Lombard in the 12th century. It is a systematic compilation of theology, written around 1150; it derives its name from the ''sententiae'' o ...

(formulas with no

free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

) 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,

!.html"_;"title="phi.html"_;"title="![\phi">![\phi!">phi.html"_;"title="![\phi">![\phi!v

_for_a_proposition_

\phi

.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

\phi

.Dirk van Dalen, (2004) ''Logic and Structure'', Springer Universitext, (''see section 1.2'')

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_

See also

References