Girish Mahajan (Editor)

Valuation (logic)

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In logic and model theory, a valuation can be:

Contents

  • 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 is a valuation, that is, a mapping from the atoms to the set { t , f } , then the double-bracket notation is commonly used to denote a valuation; that is, v ( ϕ ) = [ [ ϕ ] ] v for a proposition ϕ .

    References

    Valuation (logic) Wikipedia
    (Text) CC BY-SA


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