nLab logical connective

Contents

Contents

Idea

In formal logic, a logical connective is a type formation rule which from given (sets and) propositions forms new propositions.

For example, the logical connective “AND” combines a pair of propositions P 1P_1, P 2P_2 to a new proposition P 1P 2P_1 \wedge P_2.

To the extent that propositions are regarded as functions from sets/types SS (whose elements/terms they are propositions about) to the set/type of truth values (the type of propositions)

Γ,s:SP(s):PropΓP:SProp \frac{ \Gamma ,\;\; s \colon S \;\;\; \vdash \;\;\; P(s) \colon Prop }{ \Gamma \;\;\; \vdash \;\;\; P \,\colon\, S \to Prop }

one may regard logical connectives as forming the resulting functions. This is what is expressed by the notorious truth tables of classical logic.

Examples

\phantom{-}symbol\phantom{-}\phantom{-}in logic\phantom{-}
A\phantom{A}\inA\phantom{A}element relation
A\phantom{A}:\,:A\phantom{A}typing relation
A\phantom{A}==A\phantom{A}equality
A\phantom{A}\vdashA\phantom{A}A\phantom{A}entailment / sequentA\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A}true / topA\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A}false / bottomA\phantom{A}
A\phantom{A}\RightarrowA\phantom{A}implication
A\phantom{A}\LeftrightarrowA\phantom{A}logical equivalence
A\phantom{A}¬\notA\phantom{A}negation
A\phantom{A}\neqA\phantom{A}negation of equality / apartnessA\phantom{A}
A\phantom{A}\notinA\phantom{A}negation of element relation A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}negation of negationA\phantom{A}
A\phantom{A}\existsA\phantom{A}existential quantificationA\phantom{A}
A\phantom{A}\forallA\phantom{A}universal quantificationA\phantom{A}
A\phantom{A}\wedgeA\phantom{A}logical conjunction
A\phantom{A}\veeA\phantom{A}logical disjunction
symbolin type theory (propositions as types)
A\phantom{A}\toA\phantom{A}function type (implication)
A\phantom{A}×\timesA\phantom{A}product type (conjunction)
A\phantom{A}++A\phantom{A}sum type (disjunction)
A\phantom{A}00A\phantom{A}empty type (false)
A\phantom{A}11A\phantom{A}unit type (true)
A\phantom{A}==A\phantom{A}identity type (equality)
A\phantom{A}\simeqA\phantom{A}equivalence of types (logical equivalence)
A\phantom{A}\sumA\phantom{A}dependent sum type (existential quantifier)
A\phantom{A}\prodA\phantom{A}dependent product type (universal quantifier)
symbolin linear logic
A\phantom{A}\multimapA\phantom{A}A\phantom{A}linear implicationA\phantom{A}
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}additive disjunctionA\phantom{A}
A\phantom{A}&\&A\phantom{A}A\phantom{A}additive conjunctionA\phantom{A}
A\phantom{A}\invampA\phantom{A}A\phantom{A}multiplicative disjunctionA\phantom{A}
A\phantom{A}!\;!A\phantom{A}A\phantom{A}exponential conjunctionA\phantom{A}

Last revised on February 18, 2023 at 14:05:06. See the history of this page for a list of all contributions to it.