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# 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 $P_1$, $P_2$ to a new proposition $P_1 \wedge P_2$.

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

$\frac{ \Gamma ,\;\; s \colon S \;\;\; \vdash \;\;\; P(s) \colon Prop }{ \Gamma \;\;\; \vdash \;\;\; P \,\colon\, S \to Prop }$

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{-}$$\phantom{-}$in logic$\phantom{-}$
$\phantom{A}$$\in$$\phantom{A}$element relation
$\phantom{A}$$\,:$$\phantom{A}$typing relation
$\phantom{A}$$=$$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$$\phantom{A}$logical equivalence
$\phantom{A}$$\not$$\phantom{A}$negation
$\phantom{A}$$\neq$$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$$\phantom{A}$logical disjunction
symbolin type theory (propositions as types)
$\phantom{A}$$\to$$\phantom{A}$function type (implication)
$\phantom{A}$$\times$$\phantom{A}$product type (conjunction)
$\phantom{A}$$+$$\phantom{A}$sum type (disjunction)
$\phantom{A}$$0$$\phantom{A}$empty type (false)
$\phantom{A}$$1$$\phantom{A}$unit type (true)
$\phantom{A}$$=$$\phantom{A}$identity type (equality)
$\phantom{A}$$\simeq$$\phantom{A}$equivalence of types (logical equivalence)
$\phantom{A}$$\sum$$\phantom{A}$dependent sum type (existential quantifier)
$\phantom{A}$$\prod$$\phantom{A}$dependent product type (universal quantifier)
symbolin linear logic
$\phantom{A}$$\multimap$$\phantom{A}$$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$$\phantom{A}$exponential conjunction$\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.