Valuation (logic)

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, and the corresponding assignment of a truth value to each sentence in the language for that structure.

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.

See also

* Interpretation (logic)

References

*Harvard reference
Surname1 = Rasiowa
Given1 = Helena
Surname2 = Sikorski
Given2 = Roman
title = The Mathematics of Metamathematics
publisher = PWN
year = 1970
place = Warsaw
edition = 3rd, chapter 6 "Algebra of formalized languages".


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