Paraconsistent Logic - Relation To Other Logics

Relation To Other Logics

One important type of paraconsistent logic is relevance logic. A logic is relevant iff it satisfies the following condition:

if A → B is a theorem, then A and B share a 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 logics, which are also many-valued, are paraconsistent, but the converse does not hold.

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 "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). The duality between the two systems is best seen within a sequent calculus framework. While in intuitionistic logic the sequent

is not derivable, in dual-intuitionistic logic

is not derivable. Similarly, in intuitionistic logic the sequent

is not derivable, while in dual-intuitionistic logic

is not derivable. Dual-intuitionistic logic contains a connective # known as pseudo-difference which is the dual of intuitionistic implication. Very loosely, 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 "¬ (A ∧ ¬B)". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as ¬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).