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Truth value

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Title: Truth value  
Author: World Heritage Encyclopedia
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Subject: Truth function, False (logic), Logical truth, Boolean algebra, Contraposition
Collection: Concepts in Logic, Epistemology, Propositions, Truth, Value
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Truth value

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth.

Contents

  • Classical logic 1
  • Intuitionistic and constructive logic 2
  • Multi-valued logic 3
  • Algebraic semantics 4
  • In other theories 5
  • See also 6
  • References 7
  • External links 8

Classical logic

 
true
 
··
conjunction
¬
 
false
··
disjunction
Negation interchanges
true with false and
conjunction with disjunction

In

External links

  1. ^ Proof that intuitionistic logic has no third truth value, Glivenko 1928

References

See also

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.

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

In other theories

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.

Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions. For example, intuitionistic logic 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 necessary truth of formulae.

Algebraic semantics

Multi-valued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example, on the unit interval [0,1] such structure is a total order; this may be expressed as existence of various degrees of truth.

Multi-valued logic

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

Instead statements simply remain of unknown truth value, until they are either proved or disproved.

Unproved statements in Intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). Indeed, you can prove that they have no third truth value, a result dating back to Glivenko in 1928[1]

In intuitionistic logic, and more generally, constructive mathematics, 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 you can build a proof of the statement from those axioms. A statement is false if you can deduce a contradiction from it. This leaves open the possibility of statements that have not yet been assigned a truth value.

Intuitionistic and constructive logic

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

¬(pq) ⇔ ¬p ∧ ¬q
q ∨ ¬p) ⇔ ¬qp
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