Truth value

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In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false).^[1][2]

Computing

In some programming languages, any

evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false".

Classical logic

	⊤		·∧·
	true		conjunction
¬	↕		↕
	⊥		·∨·
	false		disjunction
Negation interchanges true with false and conjunction with disjunction.

In

:

¬(p ∧ q) ⇔ ¬p ∨ ¬q

¬(p ∨ q) ⇔ ¬p ∧ ¬q

Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as valuation.

Intuitionistic and constructive logic

In

, statements are assigned a truth value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if one can build a proof of the statement from those axioms. A statement is false if one can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value. Unproven statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). Indeed, one can prove that they have no third truth value, a result dating back to Glivenko in 1928.^[3]

Instead, statements simply remain of unknown truth value, until they are either proven or disproven.

There are various ways of interpreting intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation. See also Intuitionistic logic § Semantics.

Multi-valued logic

.

Algebraic semantics

Not all

of formulae.

But even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Boolean algebra semantics of classical propositional calculus.

In other theories

Intuitionistic type theory uses types in the place of truth values.

Topos theory uses truth values in a special sense: the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational.

See also

• Agnosticism
• Bayesian probability
• Circular reasoning
• Degree of truth
• False dilemma
• History of logic § Algebraic period
• Paradox
• Semantic theory of truth
• Slingshot argument
• Supervaluationism
• Truth-value semantics
• Verisimilitude

References

External links

• Shramko, Yaroslav; Wansing, Heinrich. "Truth Values". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.