Editing Paraconsistent logic (section)

== 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>