Logical value
Encyclopedia
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 mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

 to truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...

.

In classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

, with its intended semantics, the truth values are true and false; that is, classical logic is a two-valued logic. Intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...

 lacks a complete set of truth values because its semantics, the Brouwer–Heyting–Kolmogorov interpretation, is specified in terms of provability conditions, and not directly in terms of the truth of formulae. Multi-valued logic
Multi-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...

s (such as fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...

 and relevance logic
Relevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...

) allow for more than two truth values, possibly containing some internal structure.

Even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics
Algebraic semantics
An programming language theory, the algebraic semantics of a programming language is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner....

. For example, the algebraic semantics of intuitionistic logic is given in terms of Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...

s.

Topos theory uses truth values in a special sense: the truth values of a topos are the global element
Global element
In category theory, a global element of an object A from a category is a morphismwhere 1 is a terminal object of the category. Roughly speaking, global elements are a generalization of the notion of “elements” from the category of sets, and they can be used to import set-theoretic...

s of the subobject classifier
Subobject classifier
In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements...

. Having truth values in this sense does not make a logic truth valuational.

See also

  • Agnosticism
    Agnosticism
    Agnosticism is the view that the truth value of certain claims—especially claims about the existence or non-existence of any deity, but also other religious and metaphysical claims—is unknown or unknowable....

  • Boolean domain
    Boolean domain
    In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true...

  • Degrees of truth
  • False dilemma
    False dilemma
    A false dilemma is a type of logical fallacy that involves a situation in which only two alternatives are considered, when in fact there are additional options...

  • Logical connective
    Logical connective
    In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

  • Logical truth
    Logical truth
    Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...

  • Negation
    Negation
    In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

  • Slingshot argument
    Slingshot argument
    In logic, a slingshot argument is one of a group of arguments claiming to show that all true sentences stand for the same thing.This type of argument was dubbed the "slingshot" by philosophers Jon Barwise and John Perry due to its disarming simplicity. Versions of the slingshot argument have been...

  • Supervaluationism
    Supervaluationism
    In logic, supervaluationism is a semantics for dealing with irreferential singular terms and vagueness. Consider the sentence 'Pegasus likes licorice' in which the name 'Pegasus' fails to refer. What should its truth value be? There is nothing in the myth that would justify any assignment of values...

  • Semantic theory of truth
    Semantic theory of truth
    A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.-Origin:The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish...

  • Truth-value semantics
    Truth-value semantics
    In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap...

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