In logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 and model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, a valuation can be:

• In propositional logic, an assignment of truth values to propositional variable
  Propositional variable
  In mathematical logic, a propositional variable is a variable which can either be true or false...
  
  s, with a corresponding assignment of truth values to all propositional formula
  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...
  
  s with those variables.
• In first-order logic
  First-order logic
  First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
  
   and higher-order logics, a structure
  Structure (mathematical logic)
  In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
  
  , (the interpretation
  Interpretation (logic)
  An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some 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
  Homomorphism
  In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
  
  , while valuation is simply a function
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
  .

Mathematical logic

In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a truth schema

T-schema

The T-schema or truth schema is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth...

. 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

Atomic formula

In mathematical logic, an atomic 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 subformulas. Atoms are thus the simplest well-formed formulas of the logic...

s using logical connectives and quantifiers. A structure

Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

 consists of a set (domain of discourse

Domain of discourse

In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

) 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

Sentence (mathematical logic)

In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...

 (formulas with no free variables) in the language.