Editing Paraconsistent logic (section)

== Relation to other logics ==
One important type of paraconsistent logic is [[relevance logic]]. A logic is ''relevant'' if it satisfies the following condition:

: if ''A'' → ''B'' is a theorem, then ''A'' and ''B'' share a [[logical constant|non-logical constant]].

It follows that a [[relevance logic]] cannot have (''p'' ∧ ¬''p'') → ''q'' as a theorem, and thus (on reasonable assumptions) cannot validate the inference from {''p'', ¬''p''} to ''q''.

Paraconsistent logic has significant overlap with [[many-valued logic]]; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). [[Dialetheic logic]]s, which are also many-valued, are paraconsistent, but the converse does not hold. The ideal 3-valued paraconsistent logic given below becomes the logic [[three-valued logic|RM3]] when the contrapositive is added.

[[Intuitionistic logic]] allows ''A'' ∨ ¬''A'' not to be equivalent to true, while paraconsistent logic allows ''A'' ∧ ¬''A'' not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the "[[duality (mathematics)|dual]]" of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly, the dual notion to paraconsistency is called ''paracompleteness'', and the "dual" of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called ''anti-intuitionistic'' or ''dual-intuitionistic logic'' (sometimes referred to as ''Brazilian logic'', for historical reasons).<ref>See Aoyama (2004).</ref>  The duality between the two systems is best seen within a [[sequent calculus]] framework. While in intuitionistic logic the sequent

: <math>\vdash A \lor \neg A</math>

is not derivable, in dual-intuitionistic logic

: <math>A \land \neg A \vdash</math>

is not derivable{{citation needed|reason=I think that's a typo. Not sure, though|date=April 2017}}. Similarly, in intuitionistic logic the sequent

: <math>\neg \neg A \vdash A</math>

is not derivable, while in dual-intuitionistic logic

: <math>A \vdash \neg \neg A</math>

is not derivable. Dual-intuitionistic logic contains a connective # known as ''pseudo-difference'' which is the dual of intuitionistic implication. Very loosely, {{nowrap|1=''A'' # ''B''}} can be read as "''A'' but not ''B''". However, # is not [[truth-functional]] as one might expect a 'but not' operator to be; similarly, the intuitionistic implication operator cannot be treated like "{{nowrap|1=¬ (''A'' ∧ ¬''B'')}}".  Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as {{nowrap|1=¬''A'' = (⊤ # ''A'')}}

A full account of the duality between paraconsistent and intuitionistic logic, including an explanation on why dual-intuitionistic and paraconsistent logics do not coincide, can be found in Brunner and Carnielli (2005).

These other logics avoid explosion: [[implicational propositional calculus]], [[positive propositional calculus]], [[equivalential calculus]] and [[minimal logic]]. The latter, minimal logic, is both paraconsistent and paracomplete (a subsystem of intuitionistic logic). The other three simply do not allow one to express a contradiction to begin with since they lack the ability to form negations.