Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle);[1] however, the term paraconsistent ("beside the consistent") was first coined in 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias.
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.
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.
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.
[6] 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.
[8] 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.
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.
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, and idempotence inferences (for conjunction and disjunction).
Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.
Or to put the same point less symbolically: (Semantic) logical consequence is then defined as truth-preservation: Now consider a valuation
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.
[12] 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 introduction and elimination rules for negation, conjunction, and disjunction.
Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as first-degree entailment (FDE).
The other three simply do not allow one to express a contradiction to begin with since they lack the ability to form negations.
A formula is a tautology of paraconsistent logic if it is true in every valuation which maps atomic propositions to {t, b, f}.
For a valuation, the set of true formulas is closed under modus ponens and the deduction theorem.
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).
(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 Γ is equivalent to Δ in the sense that either can be substituted for the other wherever they appear as a subformula, one must show This is more difficult than in classical logic because the contrapositives do not necessarily follow.
And dropping all three classical laws does not just change the kind of logic—it leaves us without any functional system of logic altogether.
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).
This is an important debate since embracing paraconsistent logic comes at the risk of losing a large amount of theorems that form the basis of mathematics and physics.
[30] In "Saving Truth from Paradox", Hartry Field examines the value of paraconsistent logic as a solution to paradoxa.
"[33] Littmann and 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.
Others, such as 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.
[36] 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").