Editing Paraconsistent logic

{{Short description|Type of formal logic without explosion principle}}

{{More citations needed|date=April 2018}}

'''Paraconsistent logic''' is a type of [[non-classical logic]] that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. Specifically, paraconsistent logic is the subfield of [[logic]] that is concerned with studying and developing "inconsistency-tolerant" systems of logic, purposefully excluding the [[principle of explosion]].

Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of [[Aristotle]]);<ref>{{cite encyclopedia|url=http://plato.stanford.edu/entries/logic-paraconsistent/|title=Paraconsistent Logic|encyclopedia=[[Stanford Encyclopedia of Philosophy]]|access-date=1 December 2015|archive-url=https://web.archive.org/web/20151211014311/http://plato.stanford.edu/entries/logic-paraconsistent/|archive-date=2015-12-11|url-status=live}}</ref> however, the term ''paraconsistent'' ("beside the consistent") was first coined in 1976, by the [[Peru]]vian [[philosopher]] [[Francisco Miró Quesada Cantuarias]].<ref>Priest (2002), p. 288 and §3.3.</ref> The study of paraconsistent logic has been dubbed '''paraconsistency''',<ref>Carnielli, W.; Rodrigues, A.

"[http://philsci-archive.pitt.edu/14115/1/letj.pdf An epistemic approach to paraconsistency: a logic of evidence and truth]"

[[University of Pittsburgh|Pittsburg]]</ref> which encompasses the school of [[dialetheism]].

==Definition==
In [[classical logic]] (as well as [[intuitionistic logic]] and most other logics), contradictions [[Entailment|entail]] everything. This feature, known as the [[principle of explosion]] or ''ex contradictione sequitur quodlibet'' ([[Latin]], "from a contradiction, anything follows")<ref>{{cite journal|last1=Carnielli |first1=W. |author-link1=Walter Carnielli |last2=Marcos |last3=J. |year=2001 |url=https://dimap.ufrn.br/~jmarcos/papers/JM/01-CM-ECNSQL.pdf |title=Ex contradictione non sequitur quodlibet |journal=Bulletin of Advanced Reasoning and Knowledge |volume=1 |pages=89–109}}</ref> can be expressed formally as

{| class="wikitable" style="width:400px;"
|-
| 1
| align="center" | <math>P \land\neg P</math>
| colspan="2" | Premise
|-
| 2
| align="center" | <math>P\,</math>
| [[Conjunction elimination]]
| align="center" | from 1
|-
| 3
| align="center" | <math>P \lor A</math>
| [[Disjunction introduction]]
| align="center" | from 2
|-
| 4
| align="center" | <math>\neg P\,</math>
| [[Conjunction elimination]] 
| align="center" | from 1
|-
| 5
| align="center" | <math>A\,</math>
| [[Disjunctive syllogism]]
| align="center" | from 3 and 4
|}

Which means: if ''P'' and its negation ¬''P'' are both assumed to be true, then of the two claims ''P'' and (some arbitrary) ''A'', at least one is true.  Therefore, ''P'' or ''A'' is true.  However, if we know that either ''P'' or ''A'' is true, and also that ''P'' is false (that ¬''P'' is true) we can conclude that ''A'', which could be anything, is true. Thus if a [[theory (logic)|theory]] contains a single inconsistency, the theory is [[trivialism|trivial]] – that is, it has every sentence as a theorem.

The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.

==Comparison with classical logic==
The entailment relations of paraconsistent logics are [[propositional calculus|propositionally]] ''weaker'' than [[classical logic]]; that is, they deem ''fewer'' propositional inferences valid. The point is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate every entailment that classical logic does. In some sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more ''expressive'' than their classical counterparts including the hierarchy of [[metalanguage]]s due to [[Alfred Tarski]] and others. According to [[Solomon Feferman]]: "natural language abounds with directly or indirectly [[self-referential]] yet apparently harmless expressions—all of which are excluded from the Tarskian framework."<ref>{{cite journal|author=Feferman, Solomon|year=1984|title=Toward Useful Type-Free Theories, I|doi=10.2307/2274093|pages=75–111|volume=49|issue=1|journal=The Journal of Symbolic Logic|jstor=2274093|s2cid=10575304 }}</ref> This expressive limitation can be overcome in paraconsistent logic.

== Motivation ==
A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent [[information]] in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them.

Research into paraconsistent logic has also led to the establishment of the philosophical school of [[dialetheism]] (most notably advocated by [[Graham Priest]]), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues.<ref name="Fisher2007">{{cite book|author=Jennifer Fisher|title=On the Philosophy of Logic|url=https://books.google.com/books?id=k8L_YW-lEEQC&pg=PT142|year=2007|publisher=Cengage Learning|isbn=978-0-495-00888-0|pages=132–134}}</ref> Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing [[trivialism]], i.e. accepting that all contradictions (and equivalently all statements) are true.<ref name="GabbayWoods2007">{{cite book|editor1=Dov M. Gabbay|editor2=John Woods|title=The Many Valued and Nonmonotonic Turn in Logic|chapter-url=https://books.google.com/books?id=3TNj1ZkP3qEC&pg=PA131|year=2007|publisher=Elsevier|isbn=978-0-444-51623-7|page=131|author=Graham Priest|chapter=Paraconsistency and Dialetheism}}</ref> However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer a weaker standard like [[empirical adequacy]], as proposed by [[Bas van Fraassen]].<ref name="Allhoff2010">{{cite book|editor=Fritz Allhoff|title=Philosophies of the Sciences: A Guide|chapter-url=https://books.google.com/books?id=Vu6cVCLvPt0C&pg=PA55|year=2010|publisher=John Wiley & Sons|isbn=978-1-4051-9995-7|page=55|author=Otávio Bueno|chapter=Philosophy of Logic}}</ref>

==Philosophy==
In classical logic, Aristotle's three laws, namely, the excluded middle (''p'' or ¬''p''), non-contradiction ¬ (''p'' ∧ ¬''p'') and identity (''p'' iff ''p''), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency.

On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at the object language level.

==Tradeoffs==
Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles:<ref>See the article on the [[principle of explosion]] for more on this.</ref>
{| class="wikitable" style="margin: auto;"
![[Disjunction introduction]]
|<math>A \vdash A \lor B</math>
|-
![[Disjunctive syllogism]]
|<math>A \lor B, \neg A \vdash B</math>
|}

Both of these principles have been challenged.

One approach is to reject disjunction introduction but keep disjunctive [[syllogism]] and transitivity. In this approach, rules of [[natural deduction]] hold, except for [[disjunction introduction]] and [[excluded middle]]; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold: [[double negation]] as well as [[associativity]], [[commutativity]], [[distributivity]], [[De Morgan's laws|De Morgan]], and [[idempotence]] inferences (for conjunction and disjunction).  Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A.

Another approach is to reject disjunctive syllogism. From the perspective of [[dialetheism]], it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ''¬ A'', then ''A'' is excluded and ''B'' can be inferred from ''A ∨ B''. However, if ''A'' may hold as well as ''¬A'', then the argument for the inference is weakened.

Yet another approach is to do both simultaneously. In many systems of [[relevant logic]], as well as [[linear logic]], there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism.  Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.

Furthermore, the rule of proof of negation (below) just by itself is inconsistency non-robust in the sense that the negation of every proposition can be proved from a contradiction.

{| class="wikitable" style="margin: auto;"
![[Negation#Rules of inference|Proof of Negation]]
|If <math> A \vdash B \land \neg B</math>, then <math> \vdash \neg A</math>
|}
Strictly speaking, having just the rule above is paraconsistent because it is not the case that ''every'' proposition can be proved from a contradiction. However, if the rule [[double negation elimination]] (<math>\neg \neg A \vdash A</math>) is added as well, then every proposition can be proved from a contradiction. Double negation elimination does not hold for [[intuitionistic logic]].

== {{anchor|Example}}Logic of Paradox ==
One example of paraconsistent logic is the system known as LP ("'''Logic of Paradox'''"), first proposed by the [[Argentina|Argentinian]] logician [[Florencio González Asenjo]] in 1966 and later popularized by [[Graham Priest|Priest]] and others.<ref>Priest (2002), p. 306.</ref>

One way of presenting the semantics for LP is to replace the usual [[function (mathematics)|functional]] valuation with a [[relation (mathematics)|relational]] one.<ref>LP is also commonly presented as a [[many-valued logic]] with three truth values (''true'', ''false'', and ''both'').</ref>  The binary relation <math>V\,</math> relates a [[Well-formed formula|formula]] to a [[truth value]]: <math>V(A,1)\,</math> means that <math>A\,</math> is true, and <math>V(A,0)\,</math> means that <math>A\,</math> is false. A formula must be assigned ''at least'' one truth value, but there is no requirement that it be assigned ''at most'' one truth value. The semantic clauses for [[negation]] and [[disjunction]] are given as follows:
* <math>V( \neg A,1) \Leftrightarrow V(A,0)</math>
* <math>V( \neg A,0) \Leftrightarrow V(A,1)</math>
* <math>V(A \lor B,1) \Leftrightarrow V(A,1) \text{ or } V(B,1)</math>
* <math>V(A \lor B,0) \Leftrightarrow V(A,0) \text{ and } V(B,0)</math>
(The other [[logical connective]]s are defined in terms of negation and disjunction as usual.)
Or to put the same point less symbolically:
* ''not A'' is true [[if and only if]] ''A'' is false
* ''not A'' is false if and only if ''A'' is true
* ''A or B'' is true if and only if ''A'' is true or ''B'' is true
* ''A or B'' is false if and only if ''A'' is false and ''B'' is false
(Semantic) [[logical consequence]] is then defined as truth-preservation:
:  <math>\Gamma\vDash A</math> if and only if <math>A\,</math> is true whenever every element of <math>\Gamma\,</math> is true.
Now consider a valuation <math>V\,</math> such that <math>V(A,1)\,</math> and <math>V(A,0)\,</math> but it is not the case that <math>V(B,1)\,</math>. It is easy to check that this valuation constitutes a [[counterexample]] to both explosion and disjunctive syllogism. However, it is also a counterexample to [[modus ponens]] for the [[material conditional]] of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction.<ref>See, for example, Priest (2002), §5.</ref>

As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as [[De Morgan's laws]] and the usual [[Natural deduction|introduction and elimination rules]] for negation, [[Logical conjunction|conjunction]], and disjunction. Surprisingly, the [[logical truth]]s (or [[Tautology (logic)|tautologies]]) of LP are precisely those of classical propositional logic.<ref>See Priest (2002), p. 310.</ref>  (LP and classical logic differ only in the ''[[inference]]s'' they deem valid.)  Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as [[first-degree entailment]] (FDE). Unlike LP, FDE contains no logical truths.

LP is only one of ''many'' paraconsistent logics that have been proposed.<ref>Surveys of various approaches to paraconsistent logic can be found in Bremer (2005) and Priest (2002), and a large family of paraconsistent logics is developed in detail in Carnielli, Congilio and Marcos (2007).</ref> It is presented here merely as an illustration of how a paraconsistent logic can work.

== 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.

== An ideal three-valued paraconsistent logic ==
Here is an example of a [[three-valued logic]] which is paraconsistent and ''ideal'' as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23.<ref>{{Cite web |url=https://www.cs.tau.ac.il/~aa/articles/ideal.pdf |title=Ideal Paraconsistent Logics |access-date=2018-08-21 |archive-url=https://web.archive.org/web/20170809143217/http://www.cs.tau.ac.il/~aa/articles/ideal.pdf |archive-date=2017-08-09 |url-status=live }}</ref> The three truth-values are: ''t'' (true only), ''b'' (both true and false), and ''f'' (false only).

{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|+
! width="25" | P
! width="25" | ¬P
|-
! scope="row" {{yes|t}}
| {{no|f}}
|-
! scope="row" | b
| b
|-
! scope="row" {{no|f}}
| {{yes|t}}
|}
|
{| class="wikitable" style="text-align:center;"
|+ 
! rowspan="2" colspan="2" | P → Q
! colspan="3" | Q
|-
! width="25" {{yes|t}}
! width="25" | b
! width="25" {{no|f}}
|-
! scope="row" rowspan="3" width="25" | P
! scope="row" width="25" {{yes|t}}
| {{yes|t}}
| b
| {{no|f}}
|-
! scope="row" | b
| {{yes|t}}
| b
| {{no|f}}
|-
! scope="row" {{no|f}}
| {{yes|t}}
| {{yes|t}}
| {{yes|t}}
|}
|
{| class="wikitable" style="text-align:center;"
|+
! rowspan="2" colspan="2" | P ∨ Q
! colspan="3" | Q
|-
! width="25" {{yes|t}}
! width="25" | b
! width="25" {{no|f}}
|-
! scope="row" rowspan="3" width="25" | P
! scope="row" width="25" {{yes|t}}
| {{yes|t}}
| {{yes|t}}
| {{yes|t}}
|-
! scope="row" | b
| {{yes|t}}
| b
| b
|-
! scope="row" {{no|f}}
| {{yes|t}}
| b
| {{no|f}}
|}
|
{| class="wikitable" style="text-align:center;"
|+
! rowspan="2" colspan="2" | P ∧ Q
! colspan="3" | Q
|-
! width="25" {{yes|t}}
! width="25" | b
! width="25" {{no|f}}
|-
! scope="row" rowspan="3" width="25" | P
! scope="row" width="25" {{yes|t}}
| {{yes|t}}
| b
| {{no|f}}
|-
! scope="row" | b
| b
| b
| {{no|f}}
|-
! scope="row" {{no|f}}
| {{no|f}}
| {{no|f}}
| {{no|f}}
|}
|}

A formula is true if its truth-value is either ''t'' or ''b'' for the valuation being used. A formula is a tautology of paraconsistent logic if it is true in every valuation which maps atomic propositions to {''t'', ''b'', ''f''}. Every tautology of paraconsistent logic is also a tautology of classical logic. For a valuation, the set of true formulas is closed under [[modus ponens]] and the [[deduction theorem]]. Any tautology of classical logic which contains no negations is also a tautology of paraconsistent logic (by merging ''b'' into ''t''). This logic is sometimes referred to as "Pac" or "LFI1".

=== Included ===
Some tautologies of paraconsistent logic are:
* All axiom schemas for paraconsistent logic:
:<math>P \to (Q \to P)</math> ** for deduction theorem and ?→{''t'',''b''} = {''t'',''b''}
:<math>(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))</math> ** for deduction theorem (note: {''t'',''b''}→{''f''} = {''f''} follows from the deduction theorem)
:<math>\lnot (P \to Q) \to P</math> ** {''f''}→? = {''t''}
:<math>\lnot (P \to Q) \to \lnot Q</math> ** ?→{''t''} = {''t''}
:<math>P \to (\lnot Q \to \lnot (P \to Q))</math> ** {''t'',''b''}→{''b'',''f''} = {''b'',''f''}
:<math>\lnot \lnot P \to P</math> ** ~{''f''} = {''t''}
:<math>P \to \lnot \lnot P</math> ** ~{''t'',''b''} = {''b'',''f''} (note: ~{''t''} = {''f''} and ~{''b'',''f''} = {''t'',''b''} follow from the way the truth-values are encoded)
:<math>P \to (P \lor Q)</math> ** {''t'',''b''}v? = {''t'',''b''}
:<math>Q \to (P \lor Q)</math> ** ?v{''t'',''b''} = {''t'',''b''}
:<math>\lnot (P \lor Q) \to \lnot P</math> ** {''t''}v? = {''t''}
:<math>\lnot (P \lor Q) \to \lnot Q</math> ** ?v{''t''} = {''t''}
:<math>(P \to R) \to ((Q \to R) \to ((P \lor Q) \to R))</math> ** {''f''}v{''f''} = {''f''}
:<math>\lnot P \to (\lnot Q \to \lnot (P \lor Q))</math> ** {''b'',''f''}v{''b'',''f''} = {''b'',''f''}
:<math>(P \land Q) \to P</math> ** {''f''}&? = {''f''}
:<math>(P \land Q) \to Q</math> ** ?&{''f''} = {''f''}
:<math>\lnot P \to \lnot (P \land Q)</math> ** {''b'',''f''}&? = {''b''.''f''}
:<math>\lnot Q \to \lnot (P \land Q)</math> ** ?&{''b'',''f''} = {''b'',''f''}
:<math>(\lnot P \to R) \to ((\lnot Q \to R) \to (\lnot (P \land Q) \to R))</math> ** {''t''}&{''t''} = {''t''}
:<math>P \to (Q \to (P \land Q))</math> ** {''t'',''b''}&{''t'',''b''} = {''t'',''b''}
:<math>(P \to Q) \to ((\lnot P \to Q) \to Q)</math> ** ? is the union of {''t'',''b''} with {''b'',''f''}
* Some other theorem schemas:
:<math>P \to P</math>
:<math>(\lnot P \to P) \to P</math>
:<math>((P \to Q) \to P) \to P</math>
:<math>P \lor \lnot P</math>
:<math>\lnot (P \land \lnot P)</math>
:<math>(\lnot P \to Q) \to (P \lor Q)</math>
:<math>((\lnot P \to Q) \to Q) \to (((P \land \lnot P) \to Q) \to (P \to Q))</math> ** every truth-value is either ''t'', ''b'', or ''f''.
:<math>((P \to Q) \to R) \to (Q \to R)</math>

=== Excluded ===
Some tautologies of classical logic which are ''not'' tautologies of paraconsistent logic are:
:<math>\lnot P \to (P \to Q)</math> ** no explosion in paraconsistent logic
:<math>(\lnot P \to Q) \to ((\lnot P \to \lnot Q) \to P)</math>
:<math>(P \to Q) \to ((P \to \lnot Q) \to \lnot P)</math>
:<math>(P \lor Q) \to (\lnot P \to Q)</math> ** disjunctive syllogism fails in paraconsistent logic
:<math>(P \to Q) \to (\lnot Q \to \lnot P)</math> ** contrapositive fails in paraconsistent logic
:<math>(\lnot P \to \lnot Q) \to (Q \to P)</math>
:<math>((\lnot P \to Q) \to Q) \to (P \to Q)</math>
:<math>(P \land \lnot P) \to (Q \land \lnot Q)</math> ** not all contradictions are equivalent in paraconsistent logic
:<math>(P \to Q) \to (\lnot Q \to (P \to R))</math>
:<math>((P \to Q) \to R) \to (\lnot P \to R)</math>
:<math>((\lnot P \to R) \to R) \to (((P \to Q) \to R) \to R)</math> ** counter-factual for {''b'',''f''}→? = {''t'',''b''} (inconsistent with ''b''→''f'' = ''f'')

=== Strategy ===
Suppose we are faced with a contradictory set of premises &Gamma; and wish to avoid being reduced to triviality. In classical logic, the only method one can use is to reject one or more of the premises in &Gamma;. In paraconsistent logic, we may try to compartmentalize the contradiction. That is, weaken the logic so that &Gamma;→''X'' is no longer a tautology provided the propositional variable ''X'' does not appear in &Gamma;. However, we do not want to weaken the logic any more than is necessary for that purpose. So we wish to retain modus ponens and the deduction theorem as well as the axioms which are the introduction and elimination rules for the logical connectives (where possible).

To this end, we add a third truth-value ''b'' which will be employed within the compartment containing the contradiction. We make ''b'' a fixed point of all the logical connectives.
:<math> b = \lnot b = (b \to b) = (b \lor b) = (b \land b) </math>

We must make ''b'' a kind of truth (in addition to ''t'') because otherwise there would be no tautologies at all.

To ensure that modus ponens works, we must have
:<math> (b \to f) = f ,</math>
that is, to ensure that a true hypothesis and a true implication lead to a true conclusion, we must have that a not-true (''f'') conclusion and a true (''t'' or ''b'') hypothesis yield a not-true implication.

If all the propositional variables in &Gamma; are assigned the value ''b'', then &Gamma; itself will have the value ''b''. If we give ''X'' the value ''f'', then
:<math> (\Gamma \to X) = (b \to f) = f </math>.
So &Gamma;→''X'' will not be a tautology.

Limitations:
(1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic. Having ''b'' would change the language from that of classical logic. Having ''t'' or ''f'' would allow the explosion again because
:<math> \lnot t \to X </math> or <math> f \to X </math>
would be tautologies. Note that ''b'' is not a fixed point of those constants since ''b'' ≠ ''t'' and ''b'' ≠ ''f''.

(2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas.

(3) The loss of disjunctive syllogism may result in insufficient commitment to developing the 'correct' alternative, possibly crippling mathematics.

(4) To establish that a formula &Gamma; is equivalent to &Delta; in the sense that either can be substituted for the other wherever they appear as a subformula, one must show
:<math>(\Gamma \to \Delta) \land (\Delta \to \Gamma) \land (\lnot \Gamma \to \lnot \Delta) \land (\lnot \Delta \to \lnot \Gamma)</math>.
This is more difficult than in classical logic because the contrapositives do not necessarily follow.

== Applications ==
Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:<ref name="See Bremer">Most of these are discussed in Bremer (2005) and Priest (2002).</ref>
* [[Semantics]]: Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of [[truth]] that does not fall prey to paradoxes such as [[Liar paradox|the Liar]]. However, such systems must also avoid [[Curry's paradox]], which is much more difficult as it does not essentially involve negation.
* [[Set theory]] and the [[foundations of mathematics]]
* [[Epistemology]] and [[belief revision]]: Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems.
* [[Knowledge management]] and [[artificial intelligence]]: Some [[computer scientist]]s have utilized paraconsistent logic as a means of coping gracefully with inconsistent<ref>See, for example, [[truth maintenance systems]] or the articles in Bertossi et al. (2004).</ref> or contradictory<ref>Gershenson, C. (1999). Modelling emotions with multidimensional logic. In Proceedings of the 18th International Conference of the North American Fuzzy Information Processing Society (NAFIPS ’99), pp. 42–46, New York City, NY. IEEE Press. http://cogprints.org/1479/</ref> information. Mathematical framework and rules of paraconsistent logic have been proposed as the [[activation function]] of an [[artificial neuron]] in order to build a [[neural network]] for [[function approximation]], [[model identification]], and [[Control (management)|control]] with success.<ref>{{Cite journal|last1=de Carvalho Junior|first1=A.|last2=Justo|first2=J. F.|last3=Angelico|first3=B. A.|last4=de Oliveira|first4=A. M.|last5=da Silva Filho|first5=J. I.|date=2021|title=Rotary Inverted Pendulum Identification for Control by Paraconsistent Neural Network|journal=IEEE Access|volume=9|pages=74155–74167|doi=10.1109/ACCESS.2021.3080176|bibcode=2021IEEEA...974155D |issn=2169-3536|doi-access=free}}</ref>
* [[Deontic logic]] and [[metaethics]]: Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts.
* [[Software engineering]]: Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the [[documentation]], [[use cases]], and [[Source code|code]] of large [[software systems]].<ref name="Hewitt 2008b">Hewitt (2008b)</ref><ref name="Hewitt 2008a">Hewitt (2008a)</ref><ref>Carl Hewitt. "Formalizing common sense reasoning for scalable inconsistency-robust information coordination using Direct Logic Reasoning and the Actor Model". in Vol. 52 of ''Studies in Logic''. College Publications. {{isbn|1848901593}}. 2015.</ref>
* [[Expert system]]. The Para-analyzer algorithm based on paraconsistent annotated logic by 2-value annotations (PAL2v), also called paraconsistent annotated evidential logic (PAL ''E''t), derived from paraconsistent logic, has been used in decision-making systems, such as to support medical diagnosis.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref>
* [[Electronics]] design routinely uses a [[four-valued logic]], with "hi-impedance (z)" and "don't care (x)" playing similar roles to "don't know" and "both true and false" respectively, in addition to true and false. This logic was developed independently of philosophical logics.
* [[Control system]]: A model reference control built with recurrent paraconsistent neural network for a rotary inverted pendulum presented better robustness and lower control effort compared to a classical well tuned pole placement controller.<ref>{{cite journal|author1=Carvalho, A.|author2=Angelico, B. A.|author3=Justo, J. F.|author4=Oliveira, A. M.|author5=Silva, J. I. D.|title=Model reference control by recurrent neural network built with paraconsistent neurons for trajectory tracking of a rotary inverted pendulum|journal=Applied Soft Computing| year=2023|volume=133|pages=109927|issn=1568-4946|doi=10.1016/j.asoc.2022.109927}}</ref>
*[[Digital filter]]: PAL2v Filter Algorithm, using a paraconsistent artificial neural cell of learning by contradiction extraction (PANLctx) in the composition of a paraconsistent analysis network (PANnet), based on the PAL2V rules and equations, can be used as an estimator, average extractor, filtering and in signal treatment for industrial automation and robotics.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref><ref>{{cite book |last1=de Carvalho Jr. |first1=Arnaldo |last2=Da Silva Filho |first2=João Inácio |last3=de Freitas Minicz |first3=Márcio |last4=Matuck |first4=Gustavo R. |last5=Côrtes |first5=Hyghor Miranda |last6=Garcia |first6=Dorotéa Vilanova |last7=Tasinaffo |first7=Paulo Marcelo |last8=Abe |first8=Jair Minoro |title=Advances in Applied Logics |chapter=A Paraconsistent Artificial Neural Cell of Learning by Contradiction Extraction (PANCLCTX) with Application Examples |series=Intelligent Systems Reference Library |date=2023 |volume=243 |pages=63–79 |doi=10.1007/978-3-031-35759-6_5|isbn=978-3-031-35758-9 }}</ref><ref>{{cite journal |last1=Carvalho |first1=Arnaldo |last2=Justo |first2=João F. |last3=Angélico |first3=Bruno A. |last4=de Oliveira |first4=Alexandre M. |last5=da Silva Filho |first5=João Inacio |title=Paraconsistent State Estimator for a Furuta Pendulum Control |journal=SN Computer Science |date=22 October 2022 |volume=4 |issue=1 |doi=10.1007/s42979-022-01427-z|s2cid=253064746 }}</ref>
*[[Contradiction]] Extractor. A recurrent algorithm based on the PAL2v rules and equations has been used to extract contradictions in a set of statistical data.<ref>{{cite journal |last1=de Carvalho Junior |first1=Arnaldo |last2=Justo |first2=João Francisco |last3=de Oliveira |first3=Alexandre Maniçoba |last4=da Silva Filho |first4=João Inacio |title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives |journal=Engineering Applications of Artificial Intelligence |date=1 January 2024 |volume=127 |issue=B |pages=107342 |doi=10.1016/j.engappai.2023.107342|s2cid=264898768 }}</ref>
* [[Quantum physics]]
* [[Black hole]] physics
* [[Hawking radiation]]
* [[Quantum computing]]
* [[Spintronics]]
* [[Quantum entanglement]]
* [[Quantum coupling]]
* [[Uncertainty principle]]

== Criticism ==
Logic, as it is classically understood, rests on three main rules ([[Law of thought|Laws of Thought]]): The [[Law of identity|Law of Identity]] (''LOI''), the [[Law of noncontradiction|Law of Non-Contradiction]] (''LNC''), and the [[Law of excluded middle|Law of the Excluded Middle]] (''LEM''). Paraconsistent logic deviates from classical logic by refusing to accept ''LNC''. However, the ''LNC'' can be seen as closely interconnected with the ''LOI'' as well as the ''LEM'':

   ''LoI'' states that ''A'' is ''A'' (''A''≡''A''). This means that ''A'' is distinct from its opposite or negation (''not A'', or ¬''A''). In classical logic this distinction is supported by the fact that when ''A'' is true, its opposite is not. However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''), which blurs their distinction. And without distinction, it becomes challenging to define identity. Dropping the ''LNC'' thus runs risk to also eliminate the ''LoI''.

   ''LEM'' states that either ''A'' or ''not A'' are true (''A''∨¬''A''). However, without the ''LNC'', both ''A'' and ''not A'' can be true (''A''∧¬''A''). Dropping the ''LNC'' thus runs risk to also eliminate the ''LEM''

Hence, dropping the ''LNC'' in a careless manner risks losing both the ''LOI'' and ''LEM'' as well. And dropping ''all'' three classical laws does not just change the ''kind'' of logic—it leaves us without any functional system of logic altogether. Loss of ''all'' logic eliminates the possibility of structured reasoning, A careless paraconsistent logic therefore might run risk of disapproving of any means of thinking other than chaos. Paraconsistent logic aims to evade this danger using careful and precise technical definitions. As a consequence, most criticism of paraconsistent logic also tends to be highly technical in nature (e.g. surrounding questions such as whether a paradox can be true). 

However, even on a highly technical level, paraconsistent logic can be challenging to argue against. It is obvious that paraconsistent logic leads to contradictions. However, the paraconsistent logician embraces contradictions, including any contradictions that are a part or the result of paraconsistent logic. As a consequence, much of the critique has focused on the applicability and comparative effectiveness of paraconsistent logic. This is an important debate since embracing paraconsistent logic comes at the risk of losing a large amount of [[Theorem|theorems]] that form the basis of [[mathematics]] and [[physics]].

Logician [[Stewart Shapiro]] aimed to make a case for paraconsistent logic as part of his argument for a pluralistic view of logic (the view that different logics are equally appropriate, or equally correct). He found that a case could be made that either, [[Intuitionistic logic|intuitonistic logic]] as the "One True Logic", or a pluralism of [[Intuitionistic logic|intuitonistic logic]] and [[classical logic]] is interesting and fruitful. However, when it comes to paraconsistent logic, he found "no examples that are ... compelling (at least to me)".<ref>{{Cite book |last=Shapiro |first=Stewart |title=Varieties of Logic |publisher=Oxford University Press |year=2014 |isbn=978-0-19-882269-1 |location=Oxford, UK |pages=82}}</ref>

In "Saving Truth from Paradox", [[Hartry Field]] examines the value of paraconsistent logic as a solution to [[Paradox|paradoxa]].<ref>{{Cite book |last=Field |first=Hartry |title=Saving Truth from Paradox |publisher=Oxford University Press |year=2008 |isbn=978-0-19-923074-7 |location=New York}}</ref> Field argues for a view that avoids both truth gluts (where a statement can be both true and false) and truth gaps (where a statement is neither true nor false). One of Field's concerns is the problem of a paraconsistent [[metatheory]]: If the logic itself allows contradictions to be true, then the metatheory that describes or governs the logic might also have to be paraconsistent. If the metatheory is paraconsistent, then the justification of the logic (why we should accept it) might be suspect, because any argument made within a paraconsistent framework could potentially be both valid and invalid. This creates a challenge for proponents of paraconsistent logic to explain how their logic can be justified without falling into paradox or losing explanatory power. [[Stewart Shapiro]] expressed similar concerns: "there are certain notions and concepts that the dialetheist invokes (informally), but which she cannot adequately express, unless the meta-theory is (completely) consistent. The insistence on a consistent meta-theory would undermine the key aspect of dialetheism"<ref>{{Cite book |last=Shapiro |first=Stewart |title=Simple Truth, Contradiction, Conistency |publisher=Oxford University Press |isbn=978-0-19-920419-9 |editor-last=Priest |editor-first=Graham |location=New York |publication-date=2004 |pages=338 |editor-last2=Beall |editor-first2=JC |editor-last3=Armour-Garb |editor-first3=Bradley}}</ref>

In his book "In Contradiction", which  argues in favor of paraconsistent dialetheism, [[Graham Priest]] admits to metatheoretic difficulties: "Is there a metatheory for paraconsistent logics that is acceptable in paraconsistent terms? The answer to this question is not at all obvious."<ref>{{Cite book |last=Priest |first=Graham |title=In Contradiction. A Study of the Transconsistent |publisher=Oxford University Press |year=1987 |isbn=0-19-926330-2 |location=New York |pages=258}}</ref>

Littmann and [[Keith Simmons (philosopher)|Keith Simmons]] argued that dialetheist theory is unintelligible: "Once we realize that the theory includes not only the statement '(L) is both true and false' but also the statement '(L) isn't both true and false' we may feel at a loss."<ref>{{Cite book |last1=Littmann |first1=Greg |title=A Critique of Dialetheism |last2=Simmons |first2=Keith |publisher=Oxford University Press |year=2004 |isbn=978-0-19-920419-9 |editor-last=Priest |editor-first=Graham |location=New York |pages=314–335 |editor-last2=Beall |editor-first2=JC |editor-last3=Armour-Garb |editor-first3=Bradley}}</ref> 

Some philosophers have argued against dialetheism on the grounds that the counterintuitiveness of giving up any of the three principles above outweighs any counterintuitiveness that the principle of explosion might have.

Others, such as [[David Lewis (philosopher)|David Lewis]], have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true.<ref>See Lewis (1982).</ref>  A related objection is that "negation" in paraconsistent logic is not really ''[[negation]]''; it is merely a [[Square of opposition|subcontrary]]-forming operator.<ref>See Slater (1995), Béziau (2000).</ref>

== Alternatives ==
Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use [[multi-valued logic]] with [[Bayesian inference]] and the [[Dempster-Shafer theory]], allowing that no non-tautological belief is completely (100%) irrefutable because it must be based upon incomplete, abstracted, interpreted, likely unconfirmed, potentially uninformed, and possibly incorrect knowledge (of course, this very assumption, if non-tautological, entails its own refutability, if by "refutable" we mean "not completely [100%] irrefutable").

== Notable figures ==
Notable figures in the history and/or modern development of paraconsistent logic include:

* [[Alan Ross Anderson]] (United States, 1925–1973). One of the founders of [[relevance logic]], a kind of paraconsistent logic.
* [[Florencio González Asenjo]] ([[Argentina]], 1927-2013)
* [[Diderik Batens]] (Belgium)
* [[Nuel Belnap]] (United States, b. 1930) developed logical connectives of a [[four-valued logic]].
* [[Jean-Yves Béziau]] (France/Switzerland, b. 1965). Has written extensively on the general structural features and philosophical foundations of paraconsistent logics.
* [[Ross Brady]] (Australia)
* [[Bryson Brown]] (Canada)
* [[Walter Carnielli]] ([[Brazil]]). The developer of the  ''possible-translations semantics'',  a new semantics which makes paraconsistent logics applicable and philosophically understood.
* [[Newton da Costa]] ([[Brazil]], 1929-2024). One of the first to develop formal systems of paraconsistent logic.
* [[Itala M. L. D'Ottaviano]] ([[Brazil]])
* [[J. Michael Dunn]] (United States). An important figure in relevance logic.
* [[Carl Hewitt]]
* [[Stanisław Jaśkowski]] ([[Poland]]). One of the first to develop formal systems of paraconsistent logic.
* [[R. E. Jennings]] (Canada)
* [[David Kellogg Lewis]] (USA, 1941–2001). Articulate critic of paraconsistent logic.
* [[Jan Łukasiewicz]] ([[Poland]], 1878–1956)
* [[Robert K. Meyer]] (United States/Australia)
* [[Chris Mortensen (philosopher)|Chris Mortensen]] (Australia). Has written extensively on [[paraconsistent mathematics]].
* [[Lorenzo Peña]] (Spain, b. 1944). Has developed an original line of paraconsistent logic, gradualistic logic (also known as ''transitive logic'', TL), akin to [[fuzzy logic]].
* [[Val Plumwood]] [formerly Routley] (Australia, b. 1939). Frequent collaborator with Sylvan.
* [[Graham Priest]] (Australia). Perhaps the most prominent advocate of paraconsistent logic in the world today.
* [[Francisco Miró Quesada]] ([[Peru]]). Coined the term ''paraconsistent logic''.
* [[B. H. Slater]] (Australia). Another articulate critic of paraconsistent logic.
* [[Richard Sylvan]] [formerly Routley] (New Zealand/Australia, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest.
* [[Nicolai A. Vasiliev]] (Russia, 1880–1940). First to construct logic tolerant to contradiction (1910).

==See also==
{{Portal|Philosophy}}
*[[Deviant logic]]
*[[Formal logic]]
*[[Fuzzy logic]]
*[[Probability logic]]
*[[Intuitionistic logic]]
*[[Table of logic symbols]]

== Notes ==
{{Reflist|2}}

==Resources==
* {{cite book |title=Handbook of Paraconsistency |editor=Jean-Yves Béziau |editor-link=Jean-Yves Béziau |editor2=Walter Carnielli |editor2-link=Walter Carnielli |editor3=Dov Gabbay |editor3-link=Dov Gabbay |location=London |publisher=King's College |year=2007  |isbn=978-1-904987-73-4}}
* {{cite journal | author=Aoyama, Hiroshi | title= LK, LJ, Dual Intuitionistic Logic, and Quantum Logic | journal=Notre Dame Journal of Formal Logic | year=2004 | volume=45 | issue= 4 | pages= 193–213  | doi=10.1305/ndjfl/1099238445| doi-access=free }}
* {{cite book |editor-last=Bertossi |editor-first=Leopoldo| title=Inconsistency Tolerance | year=2004 | publisher=Springer | location=Berlin  | isbn=3-540-24260-0  }}
* {{cite journal |author1=Brunner, Andreas  |author2=Carnielli, Walter  |name-list-style=amp | title=  Anti-intuitionism and paraconsistency | journal= [[Journal of Applied Logic]] | year=2005 | volume=3 | issue= 1 | pages= 161–184  | doi=10.1016/j.jal.2004.07.016| doi-access=free }}
* {{cite book | last=Béziau | first=Jean-Yves | editor=D. Batens |display-editors=et al.| title=Frontiers of Paraconsistent Logic | year=2000 | publisher=Research Studies Press | location=Baldock | isbn=0-86380-253-2 | pages=95–111 | chapter=What is Paraconsistent Logic?}}
* {{cite book | last=Bremer | first=Manuel | title=An Introduction to Paraconsistent Logics | year=2005 | publisher=Peter Lang | location=Frankfurt | isbn=3-631-53413-2}}
* {{cite book | last=Brown | first=Bryson | editor=Dale Jacquette| title=A Companion to Philosophical Logic | url=https://archive.org/details/companiontophilo00jacq | url-access=limited | year=2002 | publisher=Blackwell Publishers | location=Malden, Massachusetts | isbn=0-631-21671-5 | pages=[https://archive.org/details/companiontophilo00jacq/page/n642 628]–650 | chapter=On Paraconsistency}}
* {{cite book | last=Carnielli | first=Walter |author2=Coniglio, Marcelo E. |author3=Marcos, J. | editor=D. Gabbay | editor-link=Dov Gabbay |editor2=F. Guenthner | title=Handbook of Philosophical Logic, Volume 14 | edition=2nd | year=2007 | publisher=[[Kluwer Academic Publishers]] | location=The Netherlands | isbn=978-1-4020-6323-7 | pages=1–93 | chapter=Logics of Formal Inconsistency}}
* {{cite journal|author=Feferman, Solomon | year=1984 | title=Toward Useful Type-Free Theories, I |doi=10.2307/2274093 |pages=75–111|volume=49|issue=1|journal=The Journal of Symbolic Logic| jstor=2274093 | s2cid=10575304 }}
* {{cite conference | author=Hewitt, Carl |year=2008a | title=Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency |book-title=Coordination, Organizations, Institutions, and Norms in Agent Systems III |editor1=Jaime Sichman |editor2=Pablo Noriega |editor3=Julian Padget |editor4=Sascha Ossowski |series=Lecture Notes in Computer Science |volume=4780 |publisher=Springer-Verlag |doi=10.1007/978-3-540-79003-7}}
* {{cite arXiv | author=Hewitt, Carl |year=2008b | title=Common sense for concurrency and inconsistency tolerance using Direct Logic and the Actor model |eprint=0812.4852 | class=cs.LO}}
* {{cite book | last=Lewis | first=David | title=Papers in Philosophical Logic | url=https://archive.org/details/papersphilosophi01lewi | url-access=limited | orig-year=1982 | year=1998 | publisher=Cambridge University Press | location=Cambridge | isbn=0-521-58788-3 | pages=[https://archive.org/details/papersphilosophi01lewi/page/n102 97]–110 | chapter=Logic for Equivocators}}
* {{cite journal | last=Peña | first=Lorenzo | author-link=Lorenzo Peña | title=Graham Priest's 'Dialetheism': Is it altogether true? | orig-year=1996 | url=http://lp.jurid.net/articles/logica/dialethe.htm | hdl=10261/9714 | access-date=2009-05-03 | journal=Sorites | year=1996 | volume=7 | pages=28–56 | archive-url=https://web.archive.org/web/20110704184243/http://lp.jurid.net/articles/logica/dialethe.htm | archive-date=2011-07-04 | url-status=dead }}
* {{cite book | last=Priest | first=Graham |editor= D. Gabbay |editor-link= Dov Gabbay |editor2=F. Guenthner| title=Handbook of Philosophical Logic |volume=6 | edition=2nd | year=2002 | publisher=[[Kluwer Academic Publishers]] | location=The Netherlands  | isbn=1-4020-0583-0  | pages=287–393 | chapter=Paraconsistent Logic.}}
* {{cite encyclopedia |author1=Priest, Graham  |author2=Tanaka, Koji  |name-list-style=amp | orig-year=1996| year=2009 | title=Paraconsistent Logic | encyclopedia=[[Stanford Encyclopedia of Philosophy]] | url=http://plato.stanford.edu/entries/logic-paraconsistent/ | access-date= June 17, 2010}} (First published Tue Sep 24, 1996; substantive revision Fri Mar 20, 2009)
* {{cite journal | author=Slater, B. H. | title= Paraconsistent Logics? | journal=Journal of Philosophical Logic | year=1995 | volume=24 | pages= 451–454 | doi=10.1007/BF01048355 | issue=4| s2cid= 12125719 }}
* {{cite book | last=Woods | first=John | title=Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences  | year=2003 | publisher=[[Cambridge University Press]] | location=Cambridge | isbn=0-521-00934-0}}
*{{cite journal|author1=De Carvalho, A.|author2=Justo, J. F.|author3=De Oliveira, A. M.|author4=Da Silva Filho, J. I.|title=A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives|journal=Engineering Applications of Artificial Intelligence|year=2024|volume=127B|pages=107342|issn=0952-1976|doi=10.1016/j.engappai.2023.107342}}

==External links==
* {{cite IEP |url-id=para-log |title=Paraconsistent Logic}}
* {{cite SEP |url-id=logic-paraconsistent |title=Paraconsistent Logic}}
* {{cite SEP |url-id=mathematics-inconsistent |title=Inconsistent Mathematics}}
* [http://www.paraconsistency.org "World Congress on Paraconsistency, Ghent 1997, Juquehy 2000, Toulouse, 2003, Melbourne 2008, Kolkata, 2014"]
* [https://arxiv.org/abs/0805.1481/ Paraconsistent First-Order Logic with infinite hierarchy levels of contradiction LP#. Axiomatical system HST#, as paraconsistent generalization of Hrbacek set theory HST]
* O. Arieli, A. Avron, A. Zamansky, [https://www.cs.tau.ac.il/~aa/articles/ideal.pdf "Ideal Paraconsistent Logics"]

{{Non-classical logic}}

{{Authority control}}

[[Category:Belief revision]]
[[Category:Non-classical logic]]
[[Category:Paraconsistent logic| ]]
[[Category:Philosophical logic]]
[[Category:Systems of formal logic]]