Valuation (logic)
In this article, we will explore a variety of aspects related to Valuation (logic), from its origins to its current impact on society. We will analyze its evolution over time, as well as its relevance in the current context. In addition, we will examine the different perspectives and opinions that exist around Valuation (logic), with the aim of offering a comprehensive vision that allows us to understand its true meaning. Through this in-depth analysis, we seek to give the reader a broader and more complete understanding of Valuation (logic), addressing all the relevant aspects that encompass this topic.
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, (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 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, for a propositional formula .[1]
![{\displaystyle \!]_{v}=v(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058ae7ef72626b33ee157be7ddab782d3583795a)
See also
References
- ^ Dirk van Dalen, (2004) Logic and Structure, Springer Universitext, page 18 - Theorem 1.2.2. ISBN 978-3-540-20879-2
- Rasiowa, Helena; Sikorski, Roman (1970), The Mathematics of Metamathematics (3rd ed.), Warsaw: PWN, chapter 6 Algebra of formalized languages.
- J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford University Press. p. 155. ISBN 978-0-19-853192-0.