5849:: 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.
566:), 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.
36:
5913:, 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 irrefutable").
5553:: 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.
6140:
526:, i.e. accepting that all contradictions (and equivalently all statements) are true. 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
6746:
de
Carvalho Jr., Arnaldo; Da Silva Filho, João Inácio; de Freitas Minicz, Márcio; Matuck, Gustavo R.; Côrtes, Hyghor Miranda; Garcia, Dorotéa Vilanova; Tasinaffo, Paulo Marcelo; Abe, Jair Minoro (2023). "A Paraconsistent Artificial Neural Cell of Learning by Contradiction Extraction (PANCLCTX) with
477:
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
510:
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
4929:
Suppose we are faced with a contradictory set of premises Γ 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 Γ. In paraconsistent logic, we may try to compartmentalize the contradiction. That is,
5853:
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"
5797:
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
4938:
does not appear in Γ. 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).
5801:
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
5399:
569:
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.
755:, 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.
5802:
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
522:), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing
453:
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.
6815:
de
Carvalho Junior, Arnaldo; Justo, João Francisco; de Oliveira, Alexandre Maniçoba; da Silva Filho, João Inacio (1 January 2024). "A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives".
6710:
de
Carvalho Junior, Arnaldo; Justo, João Francisco; de Oliveira, Alexandre Maniçoba; da Silva Filho, João Inacio (1 January 2024). "A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives".
6638:
de
Carvalho Junior, Arnaldo; Justo, João Francisco; de Oliveira, Alexandre Maniçoba; da Silva Filho, João Inacio (1 January 2024). "A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives".
6536:
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.
2047:. 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.
6674:
Carvalho, A.; Angelico, B. A.; Justo, J. F.; Oliveira, A. M.; Silva, J. I. D. (2023). "Model reference control by recurrent neural network built with paraconsistent neurons for trajectory tracking of a rotary inverted pendulum".
5821:
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,
5547:: 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.
5845:. 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
135:
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
5875:
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.
4100:
3573:
2027:
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).
5541:, 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.
5883:, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that "negation" in paraconsistent logic is not really
1365:
1279:
1803:" 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
5306:
4890:
6464:
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
4305:
3159:
2532:
5465:
have utilized paraconsistent logic as a means of coping gracefully with inconsistent or contradictory information. Mathematical framework and rules of paraconsistent logic have been proposed as the
5034:
4374:
3232:
7431:
De
Carvalho, A.; Justo, J. F.; De Oliveira, A. M.; Da Silva Filho, J. I. (2024). "A comprehensive review on paraconsistent annotated evidential logic: Algorithms, Applications, and Perspectives".
4667:
7231:
Hewitt, Carl (2008a). "Large-scale
Organizational Computing requires Unstratified Reflection and Strong Paraconsistency". In Jaime Sichman; Pablo Noriega; Julian Padget; Sascha Ossowski (eds.).
6780:
Carvalho, Arnaldo; Justo, João F.; Angélico, Bruno A.; de
Oliveira, Alexandre M.; da Silva Filho, João Inacio (22 October 2022). "Paraconsistent State Estimator for a Furuta Pendulum Control".
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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.
5861:
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."
758:
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.
3941:
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667:
802:
3872:
1894:
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1970:
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873:
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5872:: "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."
3344:
3296:
1461:
618:
6616:
Carl Hewitt. "Formalizing common sense reasoning for scalable inconsistency-robust information coordination using Direct Logic Reasoning and the Actor Model". in Vol. 52 of
5300:(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
2947:
2891:
3902:
2428:
828:
240:
5240:
5092:
7522:
Paraconsistent First-Order Logic with infinite hierarchy levels of contradiction LP#. Axiomatical system HST#, as paraconsistent generalization of Hrbacek set theory HST
1503:
1636:
1600:
1564:
1034:
977:
348:
310:
5266:
3779:
1528:
1482:
1055:
998:
933:
383:
270:
5527:. The Para-analyzer algorithm based on paraconsistent annotated logic by 2-value annotations (PAL2v), also called paraconsistent annotated evidential logic (PAL
7554:
498:
yet apparently harmless expressions—all of which are excluded from the Tarskian framework." This expressive limitation can be overcome in paraconsistent logic.
578:
Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles:
5294:(2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas.
1773:, which are also many-valued, are paraconsistent, but the converse does not hold. The ideal 3-valued paraconsistent logic given below becomes the logic
5297:(3) The loss of disjunctive syllogism may result in insufficient commitment to developing the 'correct' alternative, possibly crippling mathematics.
5901:
Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use
4001:
5798:
criticism of paraconsistent logic also tends to be highly technical in nature (e.g. surrounding questions such as whether a paradox can be true).
3483:
2348:
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 {
5964:(France/Switzerland, b. 1965). Has written extensively on the general structural features and philosophical foundations of paraconsistent logics.
1679:
they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as
5834:
is interesting and fruitful. However, when it comes to paraconsistent logic, he found "no examples that are ... compelling (at least to me)".
7184:
7017:
6960:
6910:
6885:
6860:
6764:
6384:
6355:
6326:
6252:
5559:
Extractor. A recurrent algorithm based on the PAL2v rules and equations has been used to extract contradictions in a set of statistical data.
2360:}. Every tautology of paraconsistent logic is also a tautology of classical logic. For a valuation, the set of true formulas is closed under
53:
1285:
2032:
1199:
5394:{\displaystyle (\Gamma \to \Delta )\land (\Delta \to \Gamma )\land (\lnot \Gamma \to \lnot \Delta )\land (\lnot \Delta \to \lnot \Gamma )}
162:
2063:
as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23. The three truth-values are:
7508:
7490:
7472:
7362:
6197:
5198:
Limitations: (1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic. Having
713:
inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A.
1690:
paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work.
6202:
4806:
7547:
7421:
7346:
7280:
7157:
7130:
7111:
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6935:
119:
4239:
2013:". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as
100:
7855:
147:
Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of
72:
3078:
2451:
2036:
677:
One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. In this approach, rules of
7294:
6625:
6121:(New Zealand/Australia, 1935–1996). Important figure in relevance logic and a frequent collaborator with Plumwood and Priest.
4956:
4311:
3178:
875:) is added as well, then every proposition can be proved from a contradiction. Double negation elimination does not hold for
57:
6491:
5451:: Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems.
79:
2001:
as one might expect a 'but not' operator to be; similarly, the intuitionistic implication operator cannot be treated like "
7845:
5097:
that is, to ensure that a true hypothesis and a true implication lead to a true conclusion, we must have that a not-true (
4616:
1769:; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent).
7248:
Hewitt, Carl (2008b). "Common sense for concurrency and inconsistency tolerance using Direct Logic and the Actor model".
506:
A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent
7850:
7840:
7540:
4740:
4674:
4550:
3668:
2368:. Any tautology of classical logic which contains no negations is also a tautology of paraconsistent logic (by merging
1139:
1079:
689:; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold:
7701:
5865:
5531:
t), derived from paraconsistent logic, has been used in decision-making systems, such as to support medical diagnosis.
4493:
4435:
2660:
840:
86:
140:
that is concerned with studying and developing "inconsistency-tolerant" systems of logic, purposefully excluding the
1807:, and the "dual" of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called
7594:
7413:
7338:
7176:
6066:
6062:
6003:
5712:), which blurs their distinction. And without distinction, it becomes challenging to define identity. Dropping the
5439:
46:
6524:
6098:
5880:
4380:
4119:
3947:
3421:
3359:
68:
6024:
3024:
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7835:
6237:
5935:
5134:
2606:
896:
682:
584:
314:
7079:
5505:: Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the
1680:
7599:
7517:"World Congress on Paraconsistency, Ghent 1997, Juquehy 2000, Toulouse, 2003, Melbourne 2008, Kolkata, 2014"
6075:(Spain, b. 1944). Has developed an original line of paraconsistent logic, gradualistic logic (also known as
5622:
5458:
3908:
3592:
720:, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if
631:
352:
274:
5413:
Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:
772:
6400:
6173:
5910:
5630:
5499:: Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts.
5478:
3827:
2040:
1867:
1673:) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the
911:
907:
463:
194:
141:
6550:
de Carvalho Junior, A.; Justo, J. F.; Angelico, B. A.; de Oliveira, A. M.; da Silva Filho, J. I. (2021).
4196:
3785:
2555:
1943:
1905:
1829:
1653:
As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as
846:
7804:
7584:
5598:
5420:: Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of
2784:
2742:
1800:
1374:
are defined in terms of negation and disjunction as usual.) Or to put the same point less symbolically:
624:
387:
5405:
This is more difficult than in classical logic because the contrapositives do not necessarily follow.
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7637:
7632:
7579:
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5827:
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5454:
3311:
3263:
1780:
1647:
1440:
876:
591:
186:
6034:
2914:
2858:
514:
Research into paraconsistent logic has also led to the establishment of the philosophical school of
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7647:
7563:
6997:
6124:
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5466:
3878:
2395:
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1654:
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807:
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216:
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5216:
93:
7809:
7724:
7589:
7396:
7249:
7219:
7211:
7167:
Carnielli, Walter; Coniglio, Marcelo E.; Marcos, J. (2007). "Logics of Formal Inconsistency". In
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2056:
1774:
1371:
527:
6046:
5967:
5429:
835:
Strictly speaking, having just the rule above is paraconsistent because it is not the case that
7503:
7272:
7149:
6951:
Littmann, Greg; Simmons, Keith (2004). Priest, Graham; Beall, JC; Armour-Garb, Bradley (eds.).
5990:, a new semantics which makes paraconsistent logics applicable and philosophically understood.
1487:
7799:
7764:
7752:
7729:
7716:
7706:
7693:
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7342:
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2365:
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1658:
1605:
1569:
1533:
1003:
946:
678:
329:
7102:
Béziau, Jean-Yves (2000). "What is Paraconsistent Logic?". In D. Batens; et al. (eds.).
6372:
289:
7757:
7440:
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7306:
7264:
7236:
7203:
7141:
7088:
7040:
7001:
6825:
6789:
6752:
6720:
6684:
6648:
6571:
6281:
6095:(Australia). Perhaps the most prominent advocate of paraconsistent logic in the world today.
5979:
5593:
5518:
5245:
3758:
1998:
1820:
1799:
not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the "
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1727:
531:
495:
491:
7485:
7744:
7660:
7655:
7617:
7499:
7481:
7357:
6256:
6192:
6013:
5993:
5929:
5921:
Notable figures in the history and/or modern development of paraconsistent logic include:
5850:
5831:
5818:
5614:
5562:
5448:
1735:
1699:
690:
686:
470:
182:
7290:
6072:
1512:
1466:
1039:
982:
917:
367:
254:
6567:
6118:
5945:
5869:
5610:
5550:
5544:
5474:
1750:
as a theorem, and thus (on reasonable assumptions) cannot validate the inference from {
1639:
1382:
748:
702:
443:
6552:"Rotary Inverted Pendulum Identification for Control by Paraconsistent Neural Network"
5857:
In his book "In Contradiction", which argues in favor of paraconsistent dialetheism,
1819:, for historical reasons). The duality between the two systems is best seen within a
7829:
7265:
7142:
6837:
6801:
6732:
6660:
6153:
6092:
5858:
5556:
5524:
5506:
5492:
2044:
1666:
1642:
to both explosion and disjunctive syllogism. However, it is also a counterexample to
900:
698:
694:
519:
487:
7400:
7223:
6301:
4946:
which will be employed within the compartment containing the contradiction. We make
478:
logic. It is due to such conservativeness that paraconsistent languages can be more
7665:
7571:
7444:
7314:
6829:
6724:
6652:
6233:
6158:
6086:
5838:
5444:
5425:
2361:
1643:
752:
483:
7527:
6484:
6901:
Shapiro, Stewart (2004). Priest, Graham; Beall, JC; Armour-Garb, Bradley (eds.).
6000:, 1929-2024). One of the first to develop formal systems of paraconsistent logic.
5793:
of logic—it leaves us without any functional system of logic altogether. Loss of
7683:
7609:
6756:
6576:
6551:
6080:
6019:
5951:
5807:
5583:
5534:
5514:
1070:
940:
717:
710:
515:
507:
170:
159:
35:
17:
7045:
7028:
6793:
6688:
6127:(Russia, 1880–1940). First to construct logic tolerant to contradiction (1910).
7622:
7330:
7240:
7168:
7093:
7074:
7005:
6135:
5846:
5567:
5496:
5435:
4095:{\displaystyle ((\lnot P\to Q)\to Q)\to (((P\land \lnot P)\to Q)\to (P\to Q))}
523:
447:
190:
7452:
6696:
6585:
3568:{\displaystyle (\lnot P\to R)\to ((\lnot Q\to R)\to (\lnot (P\land Q)\to R))}
1975:
is not derivable. Dual-intuitionistic logic contains a connective # known as
716:
Another approach is to reject disjunctive syllogism. From the perspective of
7776:
7627:
5939:
5637:). Paraconsistent logic deviates from classical logic by refusing to accept
5510:
5417:
1675:
892:
148:
5432:, which is much more difficult as it does not essentially involve negation.
5688:). In classical logic this distinction is supported by the fact that when
7410:
Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences
7375:(First published Tue Sep 24, 1996; substantive revision Fri Mar 20, 2009)
7233:
Coordination, Organizations, Institutions, and Norms in Agent Systems III
5885:
1066:
764:
205:, "from a contradiction, anything follows") can be expressed formally as
6751:. Intelligent Systems Reference Library. Vol. 243. pp. 63–79.
542:
In classical logic Aristotle's three laws, namely, the excluded middle (
7769:
7392:
7215:
6293:
6234:
An epistemic approach to paraconsistency: a logic of evidence and truth
5842:
5811:
5803:
7532:
7310:
7235:. Lecture Notes in Computer Science. Vol. 4780. Springer-Verlag.
6031:). One of the first to develop formal systems of paraconsistent logic.
1360:{\displaystyle V(A\lor B,0)\Leftrightarrow V(A,0){\text{ and }}V(B,0)}
747:
Yet another approach is to do both simultaneously. In many systems of
7140:
Brown, Bryson (2002). "On Paraconsistency". In Dale Jacquette (ed.).
6050:
6028:
6007:
5997:
5983:
1274:{\displaystyle V(A\lor B,1)\Leftrightarrow V(A,1){\text{ or }}V(B,1)}
839:
proposition can be proved from a contradiction. However, if the rule
7207:
6285:
5112:
If all the propositional variables in Γ are assigned the value
7521:
5609:
Logic, as it is classically understood, rests on three main rules (
906:
One way of presenting the semantics for LP is to replace the usual
7467:
7254:
5421:
202:
137:
7516:
7194:
Feferman, Solomon (1984). "Toward Useful Type-Free Theories, I".
6272:
Feferman, Solomon (1984). "Toward Useful Type-Free Theories, I".
1899:
is not derivable. Similarly, in intuitionistic logic the sequent
1061:
one truth value, but there is no requirement that it be assigned
7781:
6102:
4669:** not all contradictions are equivalent in paraconsistent logic
1791:
not to be equivalent to true, while paraconsistent logic allows
887:
One example of paraconsistent logic is the system known as LP ("
156:
7536:
6514:
Most of these are discussed in Bremer (2005) and Priest (2002).
6115:(Australia). Another articulate critic of paraconsistent logic.
5202:
would change the language from that of classical logic. Having
4885:{\displaystyle ((\lnot P\to R)\to R)\to (((P\to Q)\to R)\to R)}
1979:
which is the dual of intuitionistic implication. Very loosely,
29:
6538:
4934:
is no longer a tautology provided the propositional variable
4300:{\displaystyle (\lnot P\to Q)\to ((\lnot P\to \lnot Q)\to P)}
482:
than their classical counterparts including the hierarchy of
6043:(USA, 1941–2001). Articulate critic of paraconsistent logic.
5954:(United States, b. 1930) developed logical connectives of a
5841:
examines the value of paraconsistent logic as a solution to
2292:
2220:
2148:
2376:). This logic is sometimes referred to as "Pac" or "LFI1".
155:("beside the consistent") was first coined in 1976, by the
5047:) because otherwise there would be no tautologies at all.
3154:{\displaystyle (P\to R)\to ((Q\to R)\to ((P\lor Q)\to R))}
2527:{\displaystyle (P\to (Q\to R))\to ((P\to Q)\to (P\to R))}
7148:. Malden, Massachusetts: Blackwell Publishers. pp.
6089:(Australia, b. 1939). Frequent collaborator with Sylvan.
6016:(United States). An important figure in relevance logic.
1638:. It is easy to check that this valuation constitutes a
494:: "natural language abounds with directly or indirectly
7295:"Graham Priest's 'Dialetheism': Is it altogether true?"
6955:. New York: Oxford University Press. pp. 314–335.
5029:{\displaystyle b=\lnot b=(b\to b)=(b\lor b)=(b\land b)}
4369:{\displaystyle (P\to Q)\to ((P\to \lnot Q)\to \lnot P)}
3227:{\displaystyle \lnot P\to (\lnot Q\to \lnot (P\lor Q))}
7029:"LK, LJ, Dual Intuitionistic Logic, and Quantum Logic"
4430:** disjunctive syllogism fails in paraconsistent logic
462:
The entailment relations of paraconsistent logics are
5309:
5248:
5219:
5137:
5059:
4959:
4809:
4743:
4677:
4619:
4553:
4496:
4438:
4383:
4314:
4242:
4199:
4122:
4004:
3950:
3911:
3881:
3830:
3788:
3761:
3671:
3595:
3486:
3424:
3362:
3314:
3266:
3181:
3081:
3027:
2973:
2917:
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2787:
2745:
2663:
2609:
2558:
2454:
2398:
1946:
1937:
is not derivable, while in dual-intuitionistic logic
1908:
1870:
1832:
1823:
framework. While in intuitionistic logic the sequent
1608:
1572:
1536:
1515:
1490:
1469:
1443:
1288:
1202:
1142:
1082:
1042:
1006:
985:
949:
920:
849:
810:
775:
634:
594:
370:
332:
292:
257:
219:
7106:. Baldock: Research Studies Press. pp. 95–111.
4662:{\displaystyle (P\land \lnot P)\to (Q\land \lnot Q)}
406:
are both assumed to be true, then of the two claims
165:. The study of paraconsistent logic has been dubbed
7792:
7743:
7715:
7692:
7674:
7646:
7608:
7570:
7433:
Engineering Applications of Artificial Intelligence
7329:Priest, Graham (2002). "Paraconsistent Logic.". In
6855:. Oxford, UK: Oxford University Press. p. 82.
6818:
Engineering Applications of Artificial Intelligence
6713:
Engineering Applications of Artificial Intelligence
6641:
Engineering Applications of Artificial Intelligence
5928:(United States, 1925–1973). One of the founders of
5692:is true, its opposite is not. However, without the
2853:} follow from the way the truth-values are encoded)
744:, then the argument for the inference is weakened.
60:. Unsourced material may be challenged and removed.
7271:. Cambridge: Cambridge University Press. pp.
6930:. New York: Oxford University Press. p. 258.
6905:. New York: Oxford University Press. p. 338.
5393:
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2805:
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2526:
2422:
1964:
1926:
1888:
1850:
1765:Paraconsistent logic has significant overlap with
1683:(FDE). Unlike LP, FDE contains no logical truths.
1630:
1594:
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377:
342:
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264:
234:
7073:Brunner, Andreas & Carnielli, Walter (2005).
4795:{\displaystyle ((P\to Q)\to R)\to (\lnot P\to R)}
4729:{\displaystyle (P\to Q)\to (\lnot Q\to (P\to R))}
4605:{\displaystyle ((\lnot P\to Q)\to Q)\to (P\to Q)}
3723:{\displaystyle (P\to Q)\to ((\lnot P\to Q)\to Q)}
2094:
1188:{\displaystyle V(\neg A,0)\Leftrightarrow V(A,1)}
1128:{\displaystyle V(\neg A,1)\Leftrightarrow V(A,0)}
7379:Slater, B. H. (1995). "Paraconsistent Logics?".
7263:Lewis, David (1998) . "Logic for Equivocators".
6928:In Contradiction. A Study of the Transconsistent
5050:To ensure that modus ponens works, we must have
4539:{\displaystyle (\lnot P\to \lnot Q)\to (Q\to P)}
4481:{\displaystyle (P\to Q)\to (\lnot Q\to \lnot P)}
2706:{\displaystyle P\to (\lnot Q\to \lnot (P\to Q))}
2320:
2289:
2278:
2248:
2217:
2206:
2176:
2145:
2134:
2110:
27:Type of formal logic without explosion principle
7337:. Vol. 6 (2nd ed.). The Netherlands:
5645:can be seen as closely interconnected with the
4488:** contrapositive fails in paraconsistent logic
2340:A formula is true if its truth-value is either
2284:
2212:
2140:
1861:is not derivable, in dual-intuitionistic logic
674:Both of these principles have been challenged.
511:inconsistent theories and to reason with them.
450:– that is, it has every sentence as a theorem.
446:contains a single inconsistency, the theory is
6348:The Many Valued and Nonmonotonic Turn in Logic
5789:three classical laws does not just change the
5275:is not a fixed point of those constants since
4950:a fixed point of all the logical connectives.
4186:Some tautologies of classical logic which are
2384:Some tautologies of paraconsistent logic are:
1698:One important type of paraconsistent logic is
442:, which could be anything, is true. Thus if a
7548:
5424:that does not fall prey to paradoxes such as
8:
7356:Priest, Graham & Tanaka, Koji (2009) .
5826:as the "One True Logic", or a pluralism of
5777:in a careless manner risks losing both the
5680:is distinct from its opposite or negation (
5109:) hypothesis yield a not-true implication.
4423:{\displaystyle (P\lor Q)\to (\lnot P\to Q)}
4171:{\displaystyle ((P\to Q)\to R)\to (Q\to R)}
3990:{\displaystyle (\lnot P\to Q)\to (P\lor Q)}
3455:{\displaystyle \lnot Q\to \lnot (P\land Q)}
3393:{\displaystyle \lnot P\to \lnot (P\land Q)}
2388:All axiom schemas for paraconsistent logic:
7555:
7541:
7533:
7173:Handbook of Philosophical Logic, Volume 14
6527:or the articles in Bertossi et al. (2004).
6250:"Ex contradictione non sequitur quodlibet"
3058:{\displaystyle \lnot (P\lor Q)\to \lnot Q}
3004:{\displaystyle \lnot (P\lor Q)\to \lnot P}
2051:An ideal three-valued paraconsistent logic
1065:one truth value. The semantic clauses for
422:is true. However, if we know that either
7253:
7092:
7044:
6575:
5308:
5247:
5218:
5180:{\displaystyle (\Gamma \to X)=(b\to f)=f}
5136:
5116:, then Γ itself will have the value
5058:
4958:
4808:
4742:
4676:
4618:
4552:
4495:
4437:
4382:
4313:
4241:
4198:
4190:tautologies of paraconsistent logic are:
4121:
4003:
3949:
3910:
3880:
3829:
3787:
3760:
3670:
3594:
3485:
3423:
3361:
3313:
3265:
3180:
3080:
3026:
2972:
2916:
2860:
2786:
2744:
2662:
2640:{\displaystyle \lnot (P\to Q)\to \lnot Q}
2608:
2557:
2453:
2397:
1945:
1907:
1869:
1831:
1706:if it satisfies the following condition:
1627:
1607:
1591:
1571:
1555:
1535:
1519:
1514:
1494:
1489:
1473:
1468:
1442:
1334:
1287:
1248:
1201:
1141:
1081:
1046:
1041:
1025:
1005:
989:
984:
968:
948:
924:
919:
848:
809:
774:
633:
593:
374:
369:
339:
331:
291:
261:
256:
218:
120:Learn how and when to remove this message
7123:An Introduction to Paraconsistent Logics
6065:(Australia). Has written extensively on
5428:. However, such systems must also avoid
5210:would allow the explosion again because
4942:To this end, we add a third truth-value
2265:
2193:
2121:
2081:
760:
580:
207:
7075:"Anti-intuitionism and paraconsistency"
6903:Simple Truth, Contradiction, Conistency
6346:. In Dov M. Gabbay; John Woods (eds.).
6184:
4234:** no explosion in paraconsistent logic
1434:is then defined as truth-preservation:
193:everything. This feature, known as the
189:and most other logics), contradictions
6321:. Cengage Learning. pp. 132–134.
6260:Proc. 2nd Conf. on Reasoning and Logic
3936:{\displaystyle \lnot (P\land \lnot P)}
3632:{\displaystyle P\to (Q\to (P\land Q))}
662:{\displaystyle A\lor B,\neg A\vdash B}
6880:. New York: Oxford University Press.
6379:. John Wiley & Sons. p. 55.
6377:Philosophies of the Sciences: A Guide
5767:thus runs risk to also eliminate the
5716:thus runs risk to also eliminate the
2550:} follows from the deduction theorem)
2430:** for deduction theorem and ?→{
2270:
2198:
2126:
1665:, and disjunction. Surprisingly, the
1057:is false. A formula must be assigned
797:{\displaystyle A\vdash B\land \neg B}
7:
6446:See, for example, Priest (2002), §5.
6248:Carnielli, W. and Marcos, J. (2001)
3867:{\displaystyle ((P\to Q)\to P)\to P}
2033:implicational propositional calculus
2031:These other logics avoid explosion:
1889:{\displaystyle A\land \neg A\vdash }
414:, at least one is true. Therefore,
199:ex contradictione sequitur quodlibet
58:adding citations to reliable sources
7509:Stanford Encyclopedia of Philosophy
7491:Stanford Encyclopedia of Philosophy
7473:Internet Encyclopedia of Philosophy
7363:Stanford Encyclopedia of Philosophy
6421:LP is also commonly presented as a
6198:Stanford Encyclopedia of Philosophy
4227:{\displaystyle \lnot P\to (P\to Q)}
3816:{\displaystyle (\lnot P\to P)\to P}
2586:{\displaystyle \lnot (P\to Q)\to P}
1965:{\displaystyle A\vdash \neg \neg A}
1927:{\displaystyle \neg \neg A\vdash A}
1851:{\displaystyle \vdash A\lor \neg A}
868:{\displaystyle \neg \neg A\vdash A}
7526:O. Arieli, A. Avron, A. Zamansky,
7144:A Companion to Philosophical Logic
7033:Notre Dame Journal of Formal Logic
5868:argued that dialetheist theory is
5385:
5382:
5376:
5373:
5361:
5358:
5352:
5349:
5337:
5331:
5319:
5313:
5220:
5141:
4966:
4816:
4777:
4699:
4650:
4629:
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4509:
4500:
4469:
4460:
4405:
4357:
4345:
4279:
4270:
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4200:
4053:
4011:
3954:
3924:
3912:
3888:
3792:
3696:
3535:
3514:
3490:
3434:
3425:
3372:
3363:
3203:
3194:
3182:
3049:
3028:
2995:
2974:
2806:{\displaystyle P\to \lnot \lnot P}
2797:
2794:
2764:{\displaystyle \lnot \lnot P\to P}
2749:
2746:
2682:
2673:
2631:
2610:
2559:
1956:
1953:
1912:
1909:
1877:
1842:
1777:when the contrapositive is added.
1659:introduction and elimination rules
1491:
1484:is true whenever every element of
1444:
1149:
1089:
853:
850:
814:
788:
647:
333:
226:
169:, which encompasses the school of
25:
7175:(2nd ed.). The Netherlands:
7104:Frontiers of Paraconsistent Logic
6984:See Slater (1995), Béziau (2000).
6344:"Paraconsistency and Dialetheism"
5932:, a kind of paraconsistent logic.
2534:** for deduction theorem (note: {
899:in 1966 and later popularized by
163:Francisco Miró Quesada Cantuarias
7054:Bertossi, Leopoldo, ed. (2004).
6138:
5837:In "Saving Truth from Paradox",
5271:would be tautologies. Note that
5043:a kind of truth (in addition to
4930:weaken the logic so that Γ→
34:
7335:Handbook of Philosophical Logic
6497:from the original on 2017-08-09
6223:Priest (2002), p. 288 and §3.3.
6205:from the original on 2015-12-11
5988:possible-translations semantics
4102:** every truth-value is either
3339:{\displaystyle (P\land Q)\to Q}
3291:{\displaystyle (P\land Q)\to P}
2077:
2037:positive propositional calculus
1456:{\displaystyle \Gamma \vDash A}
613:{\displaystyle A\vdash A\lor B}
458:Comparison with classical logic
45:needs additional citations for
7445:10.1016/j.engappai.2023.107342
7381:Journal of Philosophical Logic
6830:10.1016/j.engappai.2023.107342
6725:10.1016/j.engappai.2023.107342
6653:10.1016/j.engappai.2023.107342
6232:Carnielli, W.; Rodrigues, A. "
5388:
5379:
5370:
5364:
5355:
5346:
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4771:
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3708:
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3690:
3687:
3684:
3678:
3672:
3626:
3623:
3611:
3608:
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3599:
3562:
3559:
3553:
3550:
3538:
3532:
3529:
3526:
3520:
3511:
3508:
3505:
3502:
3496:
3487:
3449:
3437:
3431:
3387:
3375:
3369:
3330:
3327:
3315:
3282:
3279:
3267:
3221:
3218:
3206:
3200:
3191:
3188:
3148:
3145:
3139:
3136:
3124:
3121:
3118:
3115:
3109:
3103:
3100:
3097:
3094:
3088:
3082:
3046:
3043:
3031:
2992:
2989:
2977:
2942:{\displaystyle Q\to (P\lor Q)}
2936:
2924:
2921:
2886:{\displaystyle P\to (P\lor Q)}
2880:
2868:
2865:
2791:
2755:
2700:
2697:
2691:
2685:
2679:
2670:
2667:
2628:
2625:
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2613:
2577:
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2521:
2518:
2512:
2506:
2503:
2500:
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2455:
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2411:
2405:
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1624:
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1164:
1161:
1146:
1122:
1110:
1104:
1101:
1086:
1022:
1010:
965:
953:
438:is true) we can conclude that
1:
7528:"Ideal Paraconsistent Logics"
7267:Papers in Philosophical Logic
7196:The Journal of Symbolic Logic
6485:"Ideal Paraconsistent Logics"
6274:The Journal of Symbolic Logic
3897:{\displaystyle P\lor \lnot P}
2423:{\displaystyle P\to (Q\to P)}
2079:
823:{\displaystyle \vdash \neg A}
235:{\displaystyle P\land \neg P}
5235:{\displaystyle \lnot t\to X}
2059:which is paraconsistent and
1602:but it is not the case that
391:
364:
356:
326:
318:
286:
278:
251:
213:
7702:Ontology (computer science)
7010:Handbook of Paraconsistency
6757:10.1007/978-3-031-35759-6_5
6577:10.1109/ACCESS.2021.3080176
5785:as well. And dropping
5087:{\displaystyle (b\to f)=f,}
3751:Some other theorem schemas:
3730:** ? is the union of {
2071:(both true and false), and
841:double negation elimination
518:(most notably advocated by
7872:
7595:Intuitionistic type theory
7504:"Inconsistent Mathematics"
7414:Cambridge University Press
7339:Kluwer Academic Publishers
7177:Kluwer Academic Publishers
7012:. London: King's College.
6794:10.1007/s42979-022-01427-z
6749:Advances in Applied Logics
6689:10.1016/j.asoc.2022.109927
6539:http://cogprints.org/1479/
6455:See Priest (2002), p. 310.
6375:. In Fritz Allhoff (ed.).
6318:On the Philosophy of Logic
6067:paraconsistent mathematics
5631:Law of the Excluded Middle
5440:foundations of mathematics
1815:(sometimes referred to as
914:one. The binary relation
891:"), first proposed by the
7241:10.1007/978-3-540-79003-7
7125:. Frankfurt: Peter Lang.
7094:10.1016/j.jal.2004.07.016
6953:A Critique of Dialetheism
6878:Saving Truth from Paradox
6851:Shapiro, Stewart (2014).
6525:truth maintenance systems
6425:with three truth values (
6350:. Elsevier. p. 131.
6059:(United States/Australia)
5986:). The developer of the
5936:Florencio González Asenjo
5195:will not be a tautology.
5101:) conclusion and a true (
2306:
2273:
2234:
2201:
2162:
2129:
2102:
1813:dual-intuitionistic logic
1509:Now consider a valuation
1498:{\displaystyle \Gamma \,}
897:Florencio González Asenjo
490:and others. According to
243:
7080:Journal of Applied Logic
7046:10.1305/ndjfl/1099238445
7027:Aoyama, Hiroshi (2004).
6620:. College Publications.
6315:Jennifer Fisher (2007).
5743:). However, without the
5623:Law of Non-Contradiction
5537:design routinely uses a
4892:** counter-factual for {
2281:
2209:
2137:
2089:
2086:
2055:Here is an example of a
1694:Relation to other logics
1631:{\displaystyle V(B,1)\,}
1595:{\displaystyle V(A,0)\,}
1559:{\displaystyle V(A,1)\,}
1418:is false if and only if
1394:is false if and only if
1029:{\displaystyle V(A,0)\,}
972:{\displaystyle V(A,1)\,}
683:disjunction introduction
585:Disjunction introduction
550:), non-contradiction ¬ (
343:{\displaystyle \neg P\,}
315:Disjunction introduction
7856:Systems of formal logic
7600:Constructive set theory
7333:; F. Guenthner (eds.).
7171:; F. Guenthner (eds.).
7121:Bremer, Manuel (2005).
7056:Inconsistency Tolerance
6926:Priest, Graham (1987).
6747:Application Examples".
6399:See the article on the
6004:Itala M. L. D'Ottaviano
5806:that form the basis of
5459:artificial intelligence
1681:first-degree entailment
1404:is true if and only if
430:is true, and also that
353:Conjunction elimination
305:{\displaystyle P\lor A}
275:Conjunction elimination
7486:"Paraconsistent Logic"
7468:"Paraconsistent Logic"
7358:"Paraconsistent Logic"
6876:Field, Hartry (2008).
6677:Applied Soft Computing
6412:Priest (2002), p. 306.
6401:principle of explosion
6342:Graham Priest (2007).
6262:(Bucharest, July 2000)
6193:"Paraconsistent Logic"
6174:Table of logic symbols
6099:Francisco Miró Quesada
5911:Dempster-Shafer theory
5479:function approximation
5395:
5262:
5261:{\displaystyle f\to X}
5236:
5181:
5088:
5030:
4886:
4796:
4730:
4663:
4606:
4540:
4482:
4424:
4370:
4301:
4228:
4172:
4096:
3991:
3937:
3898:
3868:
3817:
3775:
3774:{\displaystyle P\to P}
3724:
3633:
3569:
3456:
3394:
3340:
3292:
3228:
3155:
3059:
3005:
2943:
2887:
2807:
2765:
2707:
2641:
2587:
2528:
2424:
2041:equivalential calculus
1966:
1928:
1890:
1852:
1632:
1596:
1560:
1524:
1499:
1478:
1457:
1361:
1275:
1189:
1129:
1073:are given as follows:
1051:
1030:
994:
973:
929:
869:
824:
798:
663:
614:
379:
344:
306:
266:
236:
195:principle of explosion
142:principle of explosion
69:"Paraconsistent logic"
7585:Constructive analysis
6373:"Philosophy of Logic"
6371:Otávio Bueno (2010).
5599:Uncertainty principle
5396:
5263:
5237:
5182:
5089:
5031:
4908:} (inconsistent with
4887:
4797:
4731:
4664:
4607:
4541:
4483:
4425:
4371:
4302:
4229:
4173:
4097:
3992:
3938:
3899:
3869:
3818:
3776:
3725:
3634:
3570:
3457:
3395:
3341:
3293:
3229:
3156:
3060:
3006:
2944:
2888:
2808:
2766:
2708:
2642:
2588:
2529:
2425:
1997:". However, # is not
1967:
1929:
1891:
1853:
1633:
1597:
1561:
1525:
1500:
1479:
1458:
1362:
1276:
1190:
1130:
1052:
1031:
995:
974:
930:
870:
825:
799:
732:can be inferred from
664:
625:Disjunctive syllogism
615:
473:; that is, they deem
410:and (some arbitrary)
388:Disjunctive syllogism
380:
345:
307:
267:
237:
151:); however, the term
7846:Paraconsistent logic
7638:Fuzzy set operations
7633:Fuzzy finite element
7580:Intuitionistic logic
7408:Woods, John (2003).
7341:. pp. 287–393.
7058:. Berlin: Springer.
6169:Intuitionistic logic
6107:paraconsistent logic
5773:Hence, dropping the
5589:Quantum entanglement
5503:Software engineering
5483:model identification
5473:in order to build a
5455:Knowledge management
5307:
5246:
5217:
5135:
5057:
4957:
4807:
4741:
4675:
4617:
4551:
4494:
4436:
4381:
4312:
4240:
4197:
4120:
4002:
3948:
3909:
3879:
3828:
3786:
3759:
3669:
3593:
3484:
3422:
3360:
3312:
3264:
3179:
3079:
3025:
2971:
2915:
2859:
2785:
2743:
2661:
2607:
2556:
2452:
2396:
1944:
1906:
1868:
1830:
1781:Intuitionistic logic
1728:non-logical constant
1648:material conditional
1606:
1570:
1534:
1513:
1488:
1467:
1441:
1286:
1200:
1140:
1080:
1040:
1004:
983:
947:
918:
877:intuitionistic logic
847:
808:
773:
740:may hold as well as
632:
592:
368:
330:
290:
255:
217:
187:intuitionistic logic
133:Paraconsistent logic
54:improve this article
7851:Philosophical logic
7841:Non-classical logic
7815:Non-monotonic logic
7564:Non-classical logic
6782:SN Computer Science
6568:2021IEEEA...974155D
6125:Nicolai A. Vasiliev
6105:). Coined the term
6041:David Kellogg Lewis
6025:Stanisław Jaśkowski
5893:-forming operator.
5828:intuitonistic logic
5824:intuitonistic logic
5727:states that either
5676:). This means that
5467:activation function
5463:computer scientists
2267:
2195:
2123:
2083:
1809:anti-intuitionistic
1718:is a theorem, then
1523:{\displaystyle V\,}
1477:{\displaystyle A\,}
1432:logical consequence
1372:logical connectives
1050:{\displaystyle A\,}
993:{\displaystyle A\,}
928:{\displaystyle V\,}
378:{\displaystyle A\,}
265:{\displaystyle P\,}
7810:Intermediate logic
7590:Heyting arithmetic
7393:10.1007/BF01048355
6853:Varieties of Logic
6523:See, for example,
6474:See Aoyama (2004).
6255:2012-10-16 at the
5926:Alan Ross Anderson
5907:Bayesian inference
5903:multi-valued logic
5391:
5258:
5232:
5177:
5084:
5026:
4882:
4792:
4726:
4659:
4602:
4536:
4478:
4420:
4366:
4297:
4224:
4168:
4092:
3987:
3933:
3894:
3864:
3813:
3771:
3720:
3629:
3565:
3452:
3390:
3336:
3288:
3224:
3151:
3055:
3001:
2939:
2883:
2803:
2761:
2703:
2637:
2583:
2524:
2420:
2266:
2194:
2122:
2082:
2057:three-valued logic
1962:
1924:
1886:
1848:
1734:It follows that a
1686:LP is only one of
1628:
1592:
1556:
1520:
1495:
1474:
1453:
1357:
1271:
1185:
1125:
1047:
1026:
990:
969:
925:
865:
820:
794:
659:
610:
528:empirical adequacy
402:and its negation ¬
375:
340:
302:
262:
232:
7823:
7822:
7805:Inquisitive logic
7800:Dynamic semantics
7753:Three-state logic
7707:Ontology language
7186:978-1-4020-6323-7
7179:. pp. 1–93.
7019:978-1-904987-73-4
6975:See Lewis (1982).
6962:978-0-19-920419-9
6912:978-0-19-920419-9
6887:978-0-19-923074-7
6862:978-0-19-882269-1
6766:978-3-031-35758-9
6423:many-valued logic
6403:for more on this.
6386:978-1-4051-9995-7
6357:978-0-444-51623-7
6328:978-0-495-00888-0
6164:Probability logic
6146:Philosophy portal
5956:four-valued logic
5889:; it is merely a
5579:Quantum computing
5574:Hawking radiation
5539:four-valued logic
5471:artificial neuron
2366:deduction theorem
2338:
2337:
2334:
2333:
2262:
2261:
2190:
2189:
2118:
2117:
1977:pseudo-difference
1771:Dialetheic logics
1767:many-valued logic
1337:
1251:
910:valuation with a
833:
832:
765:Proof of Negation
681:hold, except for
679:natural deduction
672:
671:
530:, as proposed by
396:
395:
130:
129:
122:
104:
16:(Redirected from
7863:
7758:Tri-state buffer
7557:
7550:
7543:
7534:
7513:
7500:Zalta, Edward N.
7495:
7482:Zalta, Edward N.
7477:
7456:
7427:
7404:
7374:
7372:
7370:
7352:
7325:
7323:
7322:
7313:. Archived from
7286:
7270:
7259:
7257:
7244:
7227:
7190:
7163:
7147:
7136:
7117:
7098:
7096:
7069:
7050:
7048:
7023:
7002:Walter Carnielli
6998:Jean-Yves Béziau
6985:
6982:
6976:
6973:
6967:
6966:
6948:
6942:
6941:
6923:
6917:
6916:
6898:
6892:
6891:
6873:
6867:
6866:
6848:
6842:
6841:
6812:
6806:
6805:
6777:
6771:
6770:
6743:
6737:
6736:
6707:
6701:
6700:
6671:
6665:
6664:
6635:
6629:
6618:Studies in Logic
6614:
6608:
6605:
6599:
6596:
6590:
6589:
6579:
6547:
6541:
6534:
6528:
6521:
6515:
6512:
6506:
6505:
6503:
6502:
6496:
6489:
6481:
6475:
6472:
6466:
6462:
6456:
6453:
6447:
6444:
6438:
6419:
6413:
6410:
6404:
6397:
6391:
6390:
6368:
6362:
6361:
6339:
6333:
6332:
6312:
6306:
6305:
6269:
6263:
6246:
6240:
6230:
6224:
6221:
6215:
6214:
6212:
6210:
6189:
6148:
6143:
6142:
6141:
6077:transitive logic
5980:Walter Carnielli
5962:Jean-Yves Béziau
5879:Others, such as
5763:). Dropping the
5594:Quantum coupling
5519:software systems
5400:
5398:
5397:
5392:
5267:
5265:
5264:
5259:
5241:
5239:
5238:
5233:
5186:
5184:
5183:
5178:
5093:
5091:
5090:
5085:
5035:
5033:
5032:
5027:
4891:
4889:
4888:
4883:
4801:
4799:
4798:
4793:
4735:
4733:
4732:
4727:
4668:
4666:
4665:
4660:
4611:
4609:
4608:
4603:
4545:
4543:
4542:
4537:
4487:
4485:
4484:
4479:
4429:
4427:
4426:
4421:
4375:
4373:
4372:
4367:
4306:
4304:
4303:
4298:
4233:
4231:
4230:
4225:
4177:
4175:
4174:
4169:
4101:
4099:
4098:
4093:
3996:
3994:
3993:
3988:
3942:
3940:
3939:
3934:
3903:
3901:
3900:
3895:
3873:
3871:
3870:
3865:
3822:
3820:
3819:
3814:
3780:
3778:
3777:
3772:
3729:
3727:
3726:
3721:
3638:
3636:
3635:
3630:
3574:
3572:
3571:
3566:
3461:
3459:
3458:
3453:
3399:
3397:
3396:
3391:
3345:
3343:
3342:
3337:
3297:
3295:
3294:
3289:
3233:
3231:
3230:
3225:
3160:
3158:
3157:
3152:
3064:
3062:
3061:
3056:
3010:
3008:
3007:
3002:
2948:
2946:
2945:
2940:
2892:
2890:
2889:
2884:
2812:
2810:
2809:
2804:
2770:
2768:
2767:
2762:
2712:
2710:
2709:
2704:
2646:
2644:
2643:
2638:
2592:
2590:
2589:
2584:
2533:
2531:
2530:
2525:
2429:
2427:
2426:
2421:
2268:
2196:
2124:
2084:
2078:
2024:
2012:
1999:truth-functional
1989:can be read as "
1988:
1971:
1969:
1968:
1963:
1933:
1931:
1930:
1925:
1895:
1893:
1892:
1887:
1857:
1855:
1854:
1849:
1821:sequent calculus
1805:paracompleteness
1655:De Morgan's laws
1637:
1635:
1634:
1629:
1601:
1599:
1598:
1593:
1565:
1563:
1562:
1557:
1529:
1527:
1526:
1521:
1504:
1502:
1501:
1496:
1483:
1481:
1480:
1475:
1462:
1460:
1459:
1454:
1366:
1364:
1363:
1358:
1338:
1335:
1280:
1278:
1277:
1272:
1252:
1249:
1194:
1192:
1191:
1186:
1134:
1132:
1131:
1126:
1056:
1054:
1053:
1048:
1035:
1033:
1032:
1027:
999:
997:
996:
991:
978:
976:
975:
970:
934:
932:
931:
926:
889:Logic of Paradox
883:Logic of Paradox
874:
872:
871:
866:
829:
827:
826:
821:
803:
801:
800:
795:
761:
728:is excluded and
668:
666:
665:
660:
619:
617:
616:
611:
581:
558:) and identity (
532:Bas van Fraassen
496:self-referential
492:Solomon Feferman
434:is false (that ¬
398:Which means: if
384:
382:
381:
376:
349:
347:
346:
341:
311:
309:
308:
303:
271:
269:
268:
263:
241:
239:
238:
233:
208:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:Logic of Paradox
7871:
7870:
7866:
7865:
7864:
7862:
7861:
7860:
7836:Belief revision
7826:
7825:
7824:
7819:
7788:
7739:
7711:
7688:
7670:
7661:Relevance logic
7656:Structural rule
7642:
7618:Degree of truth
7604:
7566:
7561:
7498:
7480:
7466:
7463:
7430:
7424:
7407:
7378:
7368:
7366:
7355:
7349:
7328:
7320:
7318:
7289:
7283:
7262:
7247:
7230:
7208:10.2307/2274093
7193:
7187:
7166:
7160:
7139:
7133:
7120:
7114:
7101:
7072:
7066:
7053:
7026:
7020:
7008:, eds. (2007).
6996:
6993:
6988:
6983:
6979:
6974:
6970:
6963:
6950:
6949:
6945:
6938:
6925:
6924:
6920:
6913:
6900:
6899:
6895:
6888:
6875:
6874:
6870:
6863:
6850:
6849:
6845:
6814:
6813:
6809:
6779:
6778:
6774:
6767:
6745:
6744:
6740:
6709:
6708:
6704:
6673:
6672:
6668:
6637:
6636:
6632:
6615:
6611:
6606:
6602:
6597:
6593:
6562:: 74155–74167.
6549:
6548:
6544:
6535:
6531:
6522:
6518:
6513:
6509:
6500:
6498:
6494:
6487:
6483:
6482:
6478:
6473:
6469:
6463:
6459:
6454:
6450:
6445:
6441:
6420:
6416:
6411:
6407:
6398:
6394:
6387:
6370:
6369:
6365:
6358:
6341:
6340:
6336:
6329:
6314:
6313:
6309:
6286:10.2307/2274093
6271:
6270:
6266:
6257:Wayback Machine
6247:
6243:
6231:
6227:
6222:
6218:
6208:
6206:
6191:
6190:
6186:
6182:
6144:
6139:
6137:
6134:
6079:, TL), akin to
6063:Chris Mortensen
6057:Robert K. Meyer
6047:Jan Łukasiewicz
6014:J. Michael Dunn
5994:Newton da Costa
5930:relevance logic
5919:
5917:Notable figures
5899:
5851:Stewart Shapiro
5832:classical logic
5819:Stewart Shapiro
5649:as well as the
5641:. However, the
5615:Law of Identity
5611:Laws of Thought
5607:
5563:Quantum physics
5449:belief revision
5430:Curry's paradox
5411:
5305:
5304:
5244:
5243:
5215:
5214:
5133:
5132:
5055:
5054:
4955:
4954:
4927:
4805:
4804:
4739:
4738:
4673:
4672:
4615:
4614:
4549:
4548:
4492:
4491:
4434:
4433:
4379:
4378:
4310:
4309:
4238:
4237:
4195:
4194:
4184:
4118:
4117:
4000:
3999:
3946:
3945:
3907:
3906:
3877:
3876:
3826:
3825:
3784:
3783:
3757:
3756:
3667:
3666:
3591:
3590:
3482:
3481:
3462:** ?&{
3420:
3419:
3358:
3357:
3346:** ?&{
3310:
3309:
3262:
3261:
3177:
3176:
3077:
3076:
3023:
3022:
2969:
2968:
2913:
2912:
2857:
2856:
2783:
2782:
2741:
2740:
2659:
2658:
2605:
2604:
2554:
2553:
2450:
2449:
2394:
2393:
2382:
2053:
2014:
2002:
1980:
1942:
1941:
1904:
1903:
1866:
1865:
1828:
1827:
1817:Brazilian logic
1736:relevance logic
1700:relevance logic
1696:
1604:
1603:
1568:
1567:
1532:
1531:
1511:
1510:
1486:
1485:
1465:
1464:
1463:if and only if
1439:
1438:
1336: and
1284:
1283:
1198:
1197:
1138:
1137:
1078:
1077:
1038:
1037:
1002:
1001:
981:
980:
945:
944:
916:
915:
885:
845:
844:
806:
805:
771:
770:
691:double negation
687:excluded middle
630:
629:
590:
589:
576:
540:
504:
471:classical logic
464:propositionally
460:
366:
365:
328:
327:
288:
287:
253:
252:
215:
214:
183:classical logic
179:
167:paraconsistency
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
7869:
7867:
7859:
7858:
7853:
7848:
7843:
7838:
7828:
7827:
7821:
7820:
7818:
7817:
7812:
7807:
7802:
7796:
7794:
7790:
7789:
7787:
7786:
7785:
7784:
7774:
7773:
7772:
7762:
7761:
7760:
7749:
7747:
7741:
7740:
7738:
7737:
7732:
7727:
7721:
7719:
7713:
7712:
7710:
7709:
7704:
7698:
7696:
7690:
7689:
7687:
7686:
7680:
7678:
7676:Paraconsistent
7672:
7671:
7669:
7668:
7663:
7658:
7652:
7650:
7644:
7643:
7641:
7640:
7635:
7630:
7625:
7620:
7614:
7612:
7606:
7605:
7603:
7602:
7597:
7592:
7587:
7582:
7576:
7574:
7572:Intuitionistic
7568:
7567:
7562:
7560:
7559:
7552:
7545:
7537:
7531:
7530:
7524:
7519:
7514:
7496:
7478:
7462:
7461:External links
7459:
7458:
7457:
7428:
7422:
7405:
7387:(4): 451–454.
7376:
7353:
7347:
7326:
7287:
7281:
7260:
7245:
7228:
7191:
7185:
7164:
7158:
7137:
7131:
7118:
7112:
7099:
7087:(1): 161–184.
7070:
7064:
7051:
7039:(4): 193–213.
7024:
7018:
6992:
6989:
6987:
6986:
6977:
6968:
6961:
6943:
6936:
6918:
6911:
6893:
6886:
6868:
6861:
6843:
6807:
6772:
6765:
6738:
6702:
6666:
6630:
6609:
6607:Hewitt (2008a)
6600:
6598:Hewitt (2008b)
6591:
6542:
6529:
6516:
6507:
6476:
6467:
6457:
6448:
6439:
6414:
6405:
6392:
6385:
6363:
6356:
6334:
6327:
6307:
6264:
6241:
6225:
6216:
6183:
6181:
6178:
6177:
6176:
6171:
6166:
6161:
6156:
6150:
6149:
6133:
6130:
6129:
6128:
6122:
6119:Richard Sylvan
6116:
6110:
6096:
6090:
6084:
6070:
6060:
6054:
6044:
6038:
6035:R. E. Jennings
6032:
6022:
6017:
6011:
6001:
5991:
5977:
5971:
5965:
5959:
5949:
5946:Diderik Batens
5943:
5933:
5918:
5915:
5898:
5895:
5870:unintelligible
5606:
5603:
5602:
5601:
5596:
5591:
5586:
5581:
5576:
5571:
5565:
5560:
5554:
5551:Digital filter
5548:
5545:Control system
5542:
5532:
5522:
5500:
5490:
5475:neural network
5452:
5442:
5433:
5410:
5407:
5403:
5402:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5354:
5351:
5348:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5269:
5268:
5257:
5254:
5251:
5231:
5228:
5225:
5222:
5189:
5188:
5176:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5095:
5094:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5062:
5037:
5036:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4926:
4923:
4922:
4921:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4802:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4736:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4670:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4612:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4546:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4499:
4489:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4431:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4376:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4307:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4235:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4183:
4180:
4179:
4178:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4125:
4115:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
3997:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3943:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3904:
3893:
3890:
3887:
3884:
3874:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3823:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3781:
3770:
3767:
3764:
3753:
3752:
3748:
3747:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3664:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3588:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3479:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3417:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3355:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3307:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3259:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3174:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3074:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3020:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2966:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2910:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2854:
2802:
2799:
2796:
2793:
2790:
2780:
2760:
2757:
2754:
2751:
2748:
2738:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2656:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2602:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2551:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2447:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2390:
2389:
2381:
2378:
2336:
2335:
2332:
2331:
2328:
2325:
2322:
2318:
2317:
2314:
2311:
2308:
2304:
2303:
2300:
2297:
2294:
2291:
2287:
2286:
2283:
2280:
2276:
2275:
2272:
2263:
2260:
2259:
2256:
2253:
2250:
2246:
2245:
2242:
2239:
2236:
2232:
2231:
2228:
2225:
2222:
2219:
2215:
2214:
2211:
2208:
2204:
2203:
2200:
2191:
2188:
2187:
2184:
2181:
2178:
2174:
2173:
2170:
2167:
2164:
2160:
2159:
2156:
2153:
2150:
2147:
2143:
2142:
2139:
2136:
2132:
2131:
2128:
2119:
2116:
2115:
2112:
2108:
2107:
2104:
2100:
2099:
2096:
2092:
2091:
2088:
2075:(false only).
2052:
2049:
1973:
1972:
1961:
1958:
1955:
1952:
1949:
1935:
1934:
1923:
1920:
1917:
1914:
1911:
1897:
1896:
1885:
1882:
1879:
1876:
1873:
1859:
1858:
1847:
1844:
1841:
1838:
1835:
1732:
1731:
1695:
1692:
1667:logical truths
1661:for negation,
1657:and the usual
1640:counterexample
1626:
1623:
1620:
1617:
1614:
1611:
1590:
1587:
1584:
1581:
1578:
1575:
1554:
1551:
1548:
1545:
1542:
1539:
1518:
1507:
1506:
1493:
1472:
1452:
1449:
1446:
1428:
1427:
1413:
1399:
1389:
1383:if and only if
1368:
1367:
1356:
1353:
1350:
1347:
1344:
1341:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1281:
1270:
1267:
1264:
1261:
1258:
1255:
1250: or
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1195:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1135:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1045:
1024:
1021:
1018:
1015:
1012:
1009:
988:
967:
964:
961:
958:
955:
952:
923:
884:
881:
864:
861:
858:
855:
852:
831:
830:
819:
816:
813:
793:
790:
787:
784:
781:
778:
767:
749:relevant logic
736:. However, if
703:distributivity
670:
669:
658:
655:
652:
649:
646:
643:
640:
637:
627:
621:
620:
609:
606:
603:
600:
597:
587:
575:
572:
539:
536:
503:
500:
459:
456:
394:
393:
390:
385:
373:
363:
359:
358:
355:
350:
338:
335:
325:
321:
320:
317:
312:
301:
298:
295:
285:
281:
280:
277:
272:
260:
250:
246:
245:
242:
231:
228:
225:
222:
212:
178:
175:
153:paraconsistent
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7868:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7833:
7831:
7816:
7813:
7811:
7808:
7806:
7803:
7801:
7798:
7797:
7795:
7791:
7783:
7780:
7779:
7778:
7775:
7771:
7768:
7767:
7766:
7763:
7759:
7756:
7755:
7754:
7751:
7750:
7748:
7746:
7745:Digital logic
7742:
7736:
7733:
7731:
7728:
7726:
7723:
7722:
7720:
7718:
7714:
7708:
7705:
7703:
7700:
7699:
7697:
7695:
7691:
7685:
7682:
7681:
7679:
7677:
7673:
7667:
7664:
7662:
7659:
7657:
7654:
7653:
7651:
7649:
7648:Substructural
7645:
7639:
7636:
7634:
7631:
7629:
7626:
7624:
7621:
7619:
7616:
7615:
7613:
7611:
7607:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7581:
7578:
7577:
7575:
7573:
7569:
7565:
7558:
7553:
7551:
7546:
7544:
7539:
7538:
7535:
7529:
7525:
7523:
7520:
7518:
7515:
7511:
7510:
7505:
7501:
7497:
7493:
7492:
7487:
7483:
7479:
7475:
7474:
7469:
7465:
7464:
7460:
7454:
7450:
7446:
7442:
7438:
7434:
7429:
7425:
7423:0-521-00934-0
7419:
7415:
7412:. Cambridge:
7411:
7406:
7402:
7398:
7394:
7390:
7386:
7382:
7377:
7365:
7364:
7359:
7354:
7350:
7348:1-4020-0583-0
7344:
7340:
7336:
7332:
7327:
7317:on 2011-07-04
7316:
7312:
7308:
7304:
7300:
7296:
7292:
7291:Peña, Lorenzo
7288:
7284:
7282:0-521-58788-3
7278:
7274:
7269:
7268:
7261:
7256:
7251:
7246:
7242:
7238:
7234:
7229:
7225:
7221:
7217:
7213:
7209:
7205:
7202:(1): 75–111.
7201:
7197:
7192:
7188:
7182:
7178:
7174:
7170:
7165:
7161:
7159:0-631-21671-5
7155:
7151:
7146:
7145:
7138:
7134:
7132:3-631-53413-2
7128:
7124:
7119:
7115:
7113:0-86380-253-2
7109:
7105:
7100:
7095:
7090:
7086:
7082:
7081:
7076:
7071:
7067:
7065:3-540-24260-0
7061:
7057:
7052:
7047:
7042:
7038:
7034:
7030:
7025:
7021:
7015:
7011:
7007:
7003:
6999:
6995:
6994:
6990:
6981:
6978:
6972:
6969:
6964:
6958:
6954:
6947:
6944:
6939:
6937:0-19-926330-2
6933:
6929:
6922:
6919:
6914:
6908:
6904:
6897:
6894:
6889:
6883:
6879:
6872:
6869:
6864:
6858:
6854:
6847:
6844:
6839:
6835:
6831:
6827:
6824:(B): 107342.
6823:
6819:
6811:
6808:
6803:
6799:
6795:
6791:
6787:
6783:
6776:
6773:
6768:
6762:
6758:
6754:
6750:
6742:
6739:
6734:
6730:
6726:
6722:
6719:(B): 107342.
6718:
6714:
6706:
6703:
6698:
6694:
6690:
6686:
6682:
6678:
6670:
6667:
6662:
6658:
6654:
6650:
6647:(B): 107342.
6646:
6642:
6634:
6631:
6627:
6623:
6619:
6613:
6610:
6604:
6601:
6595:
6592:
6587:
6583:
6578:
6573:
6569:
6565:
6561:
6557:
6553:
6546:
6543:
6540:
6533:
6530:
6526:
6520:
6517:
6511:
6508:
6493:
6486:
6480:
6477:
6471:
6468:
6461:
6458:
6452:
6449:
6443:
6440:
6436:
6432:
6428:
6424:
6418:
6415:
6409:
6406:
6402:
6396:
6393:
6388:
6382:
6378:
6374:
6367:
6364:
6359:
6353:
6349:
6345:
6338:
6335:
6330:
6324:
6320:
6319:
6311:
6308:
6303:
6299:
6295:
6291:
6287:
6283:
6280:(1): 75–111.
6279:
6275:
6268:
6265:
6261:
6258:
6254:
6251:
6245:
6242:
6239:
6235:
6229:
6226:
6220:
6217:
6204:
6200:
6199:
6194:
6188:
6185:
6179:
6175:
6172:
6170:
6167:
6165:
6162:
6160:
6157:
6155:
6154:Deviant logic
6152:
6151:
6147:
6136:
6131:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6104:
6100:
6097:
6094:
6093:Graham Priest
6091:
6088:
6085:
6082:
6078:
6074:
6071:
6068:
6064:
6061:
6058:
6055:
6052:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6026:
6023:
6021:
6018:
6015:
6012:
6009:
6005:
6002:
5999:
5995:
5992:
5989:
5985:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5953:
5950:
5947:
5944:
5941:
5937:
5934:
5931:
5927:
5924:
5923:
5922:
5916:
5914:
5912:
5908:
5904:
5896:
5894:
5892:
5888:
5887:
5882:
5877:
5873:
5871:
5867:
5866:Keith Simmons
5864:Littmann and
5862:
5860:
5859:Graham Priest
5855:
5852:
5848:
5844:
5840:
5835:
5833:
5829:
5825:
5820:
5815:
5813:
5809:
5805:
5799:
5796:
5792:
5788:
5784:
5780:
5776:
5771:
5770:
5766:
5762:
5758:
5755:can be true (
5754:
5750:
5746:
5742:
5738:
5734:
5730:
5726:
5723:
5721:
5719:
5715:
5711:
5707:
5704:can be true (
5703:
5699:
5695:
5691:
5687:
5683:
5679:
5675:
5671:
5667:
5663:
5659:
5656:
5654:
5652:
5648:
5644:
5640:
5636:
5632:
5628:
5624:
5620:
5616:
5612:
5604:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5580:
5577:
5575:
5572:
5569:
5566:
5564:
5561:
5558:
5557:Contradiction
5555:
5552:
5549:
5546:
5543:
5540:
5536:
5533:
5530:
5526:
5525:Expert system
5523:
5520:
5516:
5512:
5508:
5507:documentation
5504:
5501:
5498:
5494:
5493:Deontic logic
5491:
5489:with success.
5488:
5484:
5480:
5476:
5472:
5468:
5464:
5460:
5456:
5453:
5450:
5446:
5443:
5441:
5437:
5434:
5431:
5427:
5423:
5419:
5416:
5415:
5414:
5408:
5406:
5367:
5343:
5325:
5303:
5302:
5301:
5298:
5295:
5292:
5290:
5286:
5282:
5278:
5274:
5255:
5249:
5229:
5223:
5213:
5212:
5211:
5209:
5205:
5201:
5196:
5194:
5174:
5171:
5165:
5159:
5153:
5147:
5131:
5130:
5129:
5127:
5123:
5120:. If we give
5119:
5115:
5110:
5108:
5104:
5100:
5081:
5078:
5075:
5069:
5063:
5053:
5052:
5051:
5048:
5046:
5042:
5039:We must make
5020:
5017:
5014:
5008:
5002:
4999:
4996:
4990:
4984:
4978:
4972:
4969:
4963:
4960:
4953:
4952:
4951:
4949:
4945:
4940:
4937:
4933:
4924:
4919:
4915:
4911:
4907:
4903:
4899:
4895:
4876:
4867:
4858:
4852:
4834:
4825:
4819:
4803:
4786:
4780:
4765:
4756:
4750:
4737:
4717:
4711:
4702:
4687:
4681:
4671:
4653:
4647:
4644:
4632:
4626:
4623:
4613:
4596:
4590:
4578:
4569:
4563:
4547:
4530:
4524:
4512:
4503:
4490:
4472:
4463:
4448:
4442:
4432:
4414:
4408:
4393:
4390:
4387:
4377:
4360:
4348:
4339:
4324:
4318:
4308:
4291:
4282:
4273:
4255:
4249:
4236:
4218:
4212:
4203:
4193:
4192:
4191:
4189:
4181:
4162:
4156:
4144:
4135:
4129:
4116:
4113:
4109:
4105:
4083:
4077:
4065:
4056:
4050:
4047:
4029:
4020:
4014:
3998:
3981:
3978:
3975:
3963:
3957:
3944:
3927:
3921:
3918:
3905:
3891:
3885:
3882:
3875:
3861:
3852:
3843:
3837:
3824:
3810:
3801:
3795:
3782:
3768:
3762:
3755:
3754:
3750:
3749:
3745:
3741:
3737:
3733:
3714:
3705:
3699:
3681:
3675:
3665:
3662:
3658:
3654:
3650:
3646:
3642:
3620:
3617:
3614:
3605:
3596:
3589:
3586:
3582:
3578:
3556:
3547:
3544:
3541:
3523:
3517:
3499:
3493:
3480:
3477:
3473:
3469:
3465:
3446:
3443:
3440:
3428:
3418:
3415:
3411:
3407:
3403:
3384:
3381:
3378:
3366:
3356:
3353:
3349:
3333:
3324:
3321:
3318:
3308:
3305:
3301:
3285:
3276:
3273:
3270:
3260:
3257:
3253:
3249:
3245:
3241:
3237:
3215:
3212:
3209:
3197:
3185:
3175:
3172:
3168:
3164:
3142:
3133:
3130:
3127:
3112:
3106:
3091:
3085:
3075:
3072:
3068:
3052:
3040:
3037:
3034:
3021:
3018:
3014:
2998:
2986:
2983:
2980:
2967:
2964:
2960:
2956:
2952:
2933:
2930:
2927:
2918:
2911:
2908:
2904:
2900:
2896:
2877:
2874:
2871:
2862:
2855:
2852:
2848:
2844:
2840:
2836:
2832:
2828:
2824:
2820:
2816:
2800:
2788:
2781:
2778:
2774:
2758:
2752:
2739:
2736:
2732:
2728:
2724:
2720:
2716:
2694:
2688:
2676:
2664:
2657:
2654:
2650:
2634:
2622:
2616:
2603:
2600:
2596:
2580:
2571:
2565:
2552:
2549:
2545:
2541:
2537:
2515:
2509:
2497:
2491:
2473:
2467:
2458:
2448:
2445:
2441:
2437:
2433:
2414:
2408:
2399:
2392:
2391:
2387:
2386:
2385:
2379:
2377:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2347:
2343:
2329:
2326:
2323:
2319:
2315:
2312:
2309:
2305:
2301:
2298:
2295:
2288:
2277:
2269:
2264:
2257:
2254:
2251:
2247:
2243:
2240:
2237:
2233:
2229:
2226:
2223:
2216:
2205:
2197:
2192:
2185:
2182:
2179:
2175:
2171:
2168:
2165:
2161:
2157:
2154:
2151:
2144:
2133:
2125:
2120:
2113:
2109:
2105:
2101:
2097:
2093:
2085:
2080:
2076:
2074:
2070:
2067:(true only),
2066:
2062:
2058:
2050:
2048:
2046:
2045:minimal logic
2042:
2038:
2034:
2029:
2025:
2022:
2018:
2010:
2006:
2000:
1996:
1992:
1987:
1983:
1978:
1959:
1950:
1947:
1940:
1939:
1938:
1921:
1918:
1915:
1902:
1901:
1900:
1883:
1880:
1874:
1871:
1864:
1863:
1862:
1845:
1839:
1836:
1833:
1826:
1825:
1824:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1776:
1772:
1768:
1763:
1761:
1757:
1753:
1749:
1745:
1741:
1738:cannot have (
1737:
1729:
1725:
1721:
1717:
1713:
1709:
1708:
1707:
1705:
1702:. A logic is
1701:
1693:
1691:
1689:
1684:
1682:
1678:
1677:
1672:
1668:
1664:
1660:
1656:
1651:
1649:
1645:
1641:
1621:
1618:
1615:
1609:
1585:
1582:
1579:
1573:
1549:
1546:
1543:
1537:
1516:
1470:
1450:
1447:
1437:
1436:
1435:
1433:
1425:
1422:is false and
1421:
1417:
1414:
1411:
1407:
1403:
1400:
1397:
1393:
1390:
1387:
1384:
1380:
1377:
1376:
1375:
1373:
1351:
1348:
1345:
1339:
1328:
1325:
1322:
1316:
1307:
1304:
1301:
1298:
1295:
1289:
1282:
1265:
1262:
1259:
1253:
1242:
1239:
1236:
1230:
1221:
1218:
1215:
1212:
1209:
1203:
1196:
1179:
1176:
1173:
1167:
1158:
1155:
1152:
1143:
1136:
1119:
1116:
1113:
1107:
1098:
1095:
1092:
1083:
1076:
1075:
1074:
1072:
1068:
1064:
1060:
1043:
1019:
1016:
1013:
1007:
1000:is true, and
986:
962:
959:
956:
950:
942:
938:
921:
913:
909:
904:
902:
898:
894:
890:
882:
880:
878:
862:
859:
856:
842:
838:
817:
811:
791:
785:
782:
779:
776:
768:
766:
763:
762:
759:
756:
754:
751:, as well as
750:
745:
743:
739:
735:
731:
727:
723:
719:
714:
712:
708:
704:
700:
699:commutativity
696:
695:associativity
692:
688:
684:
680:
675:
656:
653:
650:
644:
641:
638:
635:
628:
626:
623:
622:
607:
604:
601:
598:
595:
588:
586:
583:
582:
579:
573:
571:
567:
565:
561:
557:
553:
549:
545:
537:
535:
533:
529:
525:
521:
520:Graham Priest
517:
512:
509:
501:
499:
497:
493:
489:
488:Alfred Tarski
485:
484:metalanguages
481:
476:
472:
468:
465:
457:
455:
451:
449:
445:
441:
437:
433:
429:
425:
421:
417:
413:
409:
405:
401:
392:from 3 and 4
389:
386:
371:
361:
360:
354:
351:
336:
323:
322:
316:
313:
299:
296:
293:
283:
282:
276:
273:
258:
248:
247:
229:
223:
220:
210:
209:
206:
204:
200:
196:
192:
188:
184:
176:
174:
172:
168:
164:
161:
158:
154:
150:
145:
143:
139:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
7725:Three-valued
7675:
7666:Linear logic
7507:
7489:
7471:
7436:
7432:
7409:
7384:
7380:
7367:. Retrieved
7361:
7334:
7319:. Retrieved
7315:the original
7302:
7298:
7266:
7232:
7199:
7195:
7172:
7143:
7122:
7103:
7084:
7078:
7055:
7036:
7032:
7009:
6980:
6971:
6952:
6946:
6927:
6921:
6902:
6896:
6877:
6871:
6852:
6846:
6821:
6817:
6810:
6785:
6781:
6775:
6748:
6741:
6716:
6712:
6705:
6680:
6676:
6669:
6644:
6640:
6633:
6617:
6612:
6603:
6594:
6559:
6555:
6545:
6532:
6519:
6510:
6499:. Retrieved
6479:
6470:
6460:
6451:
6442:
6434:
6430:
6426:
6417:
6408:
6395:
6376:
6366:
6347:
6337:
6317:
6310:
6277:
6273:
6267:
6259:
6244:
6228:
6219:
6207:. Retrieved
6196:
6187:
6159:Formal logic
6113:B. H. Slater
6106:
6087:Val Plumwood
6076:
6073:Lorenzo Peña
6053:, 1878–1956)
5987:
5974:Bryson Brown
5942:, 1927-2013)
5920:
5900:
5897:Alternatives
5884:
5878:
5874:
5863:
5856:
5839:Hartry Field
5836:
5816:
5800:
5794:
5790:
5786:
5782:
5778:
5774:
5772:
5768:
5764:
5760:
5756:
5752:
5748:
5744:
5740:
5736:
5732:
5728:
5724:
5722:
5717:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5685:
5681:
5677:
5673:
5669:
5665:
5661:
5660:states that
5657:
5655:
5650:
5646:
5642:
5638:
5634:
5626:
5618:
5608:
5528:
5445:Epistemology
5412:
5409:Applications
5404:
5299:
5296:
5293:
5288:
5284:
5280:
5276:
5272:
5270:
5207:
5203:
5199:
5197:
5192:
5190:
5125:
5121:
5117:
5113:
5111:
5106:
5102:
5098:
5096:
5049:
5044:
5040:
5038:
4947:
4943:
4941:
4935:
4931:
4928:
4917:
4913:
4909:
4905:
4901:
4897:
4893:
4187:
4185:
4111:
4107:
4103:
3743:
3739:
3735:
3731:
3660:
3656:
3652:
3648:
3644:
3640:
3584:
3580:
3576:
3475:
3471:
3467:
3463:
3413:
3409:
3405:
3401:
3351:
3347:
3303:
3299:
3255:
3251:
3247:
3243:
3239:
3235:
3170:
3166:
3162:
3070:
3066:
3016:
3012:
2962:
2958:
2954:
2950:
2906:
2902:
2898:
2894:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2814:
2776:
2772:
2734:
2730:
2726:
2722:
2718:
2714:
2652:
2648:
2598:
2594:
2547:
2543:
2539:
2535:
2443:
2439:
2435:
2431:
2383:
2373:
2369:
2362:modus ponens
2357:
2353:
2349:
2345:
2341:
2339:
2072:
2068:
2064:
2060:
2054:
2030:
2026:
2020:
2016:
2008:
2004:
1994:
1990:
1985:
1981:
1976:
1974:
1936:
1898:
1860:
1816:
1812:
1808:
1804:
1796:
1792:
1788:
1784:
1779:
1764:
1759:
1755:
1751:
1747:
1743:
1739:
1733:
1723:
1719:
1715:
1711:
1703:
1697:
1687:
1685:
1674:
1652:
1644:modus ponens
1508:
1429:
1423:
1419:
1415:
1409:
1405:
1401:
1395:
1391:
1385:
1378:
1369:
1062:
1058:
905:
903:and others.
888:
886:
836:
834:
757:
753:linear logic
746:
741:
737:
733:
729:
725:
721:
715:
676:
673:
577:
568:
563:
559:
555:
551:
547:
543:
541:
513:
505:
479:
474:
466:
461:
452:
439:
435:
431:
427:
423:
419:
415:
411:
407:
403:
399:
397:
198:
185:(as well as
180:
166:
152:
146:
132:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
7765:Four-valued
7735:Łukasiewicz
7730:Four-valued
7717:Many-valued
7694:Description
7684:Dialetheism
6556:IEEE Access
6081:fuzzy logic
6020:Carl Hewitt
5970:(Australia)
5952:Nuel Belnap
5891:subcontrary
5881:David Lewis
5808:mathematics
5629:), and the
5584:Spintronics
5535:Electronics
3408:}&? = {
3302:}&? = {
2829:} (note: ~{
2647:** ?→{
1671:tautologies
1663:conjunction
1430:(Semantic)
1408:is true or
1370:(The other
1071:disjunction
1036:means that
979:means that
941:truth value
893:Argentinian
718:dialetheism
711:idempotence
693:as well as
516:dialetheism
508:information
171:dialetheism
160:philosopher
7830:Categories
7623:Fuzzy rule
7439:: 107342.
7321:2009-05-03
7311:10261/9714
7006:Dov Gabbay
6683:: 109927.
6626:1848901593
6501:2018-08-21
6209:1 December
5968:Ross Brady
5847:metatheory
5735:are true (
5568:Black hole
5497:metaethics
5436:Set theory
5191:So Γ→
5124:the value
1676:inferences
1530:such that
935:relates a
912:relational
908:functional
538:Philosophy
524:trivialism
502:Motivation
480:expressive
177:Definition
110:April 2018
80:newspapers
7777:IEEE 1164
7628:Fuzzy set
7453:0952-1976
7331:D. Gabbay
7305:: 28–56.
7293:(1996) .
7255:0812.4852
7169:D. Gabbay
6991:Resources
6838:264898768
6802:253064746
6733:264898768
6697:1568-4946
6661:264898768
6586:2169-3536
6238:Pittsburg
5948:(Belgium)
5940:Argentina
5817:Logician
5605:Criticism
5517:of large
5511:use cases
5418:Semantics
5386:Γ
5383:¬
5380:→
5377:Δ
5374:¬
5368:∧
5362:Δ
5359:¬
5356:→
5353:Γ
5350:¬
5344:∧
5338:Γ
5335:→
5332:Δ
5326:∧
5320:Δ
5317:→
5314:Γ
5253:→
5227:→
5221:¬
5163:→
5145:→
5142:Γ
5067:→
5018:∧
5000:∨
4982:→
4967:¬
4874:→
4865:→
4856:→
4841:→
4832:→
4823:→
4817:¬
4784:→
4778:¬
4772:→
4763:→
4754:→
4715:→
4706:→
4700:¬
4694:→
4685:→
4651:¬
4648:∧
4639:→
4630:¬
4627:∧
4594:→
4585:→
4576:→
4567:→
4561:¬
4528:→
4519:→
4510:¬
4507:→
4501:¬
4470:¬
4467:→
4461:¬
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648:¬
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149:Aristotle
7401:12125719
7369:June 17,
7224:10575304
6492:Archived
6302:10575304
6253:Archived
6203:Archived
6132:See also
6037:(Canada)
5976:(Canada)
5909:and the
5886:negation
5843:paradoxa
5804:theorems
5438:and the
5426:the Liar
4925:Strategy
4182:Excluded
3738:} with {
2837:} and ~{
2380:Included
2364:and the
1993:but not
1726:share a
1704:relevant
1646:for the
1505:is true.
1426:is false
1388:is false
1381:is true
1067:negation
1059:at least
244:Premise
157:Peruvian
7770:Verilog
7502:(ed.).
7484:(ed.).
7299:Sorites
7216:2274093
6628:. 2015.
6564:Bibcode
6465:(2007).
6294:2274093
5812:physics
5747:, both
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486:due to
448:trivial
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319:from 2
279:from 1
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101:JSTOR
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7449:ISSN
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5830:and
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1093:A
1087:(
1084:V
1044:A
1023:)
1020:0
1017:,
1014:A
1011:(
1008:V
987:A
966:)
963:1
960:,
957:A
954:(
951:V
922:V
863:A
857:A
843:(
818:A
792:B
783:B
777:A
738:A
730:B
726:A
657:B
651:A
645:,
642:B
636:A
608:B
602:A
596:A
564:p
560:p
556:p
552:p
548:p
544:p
440:A
436:P
432:P
428:A
424:P
420:A
416:P
412:A
408:P
404:P
400:P
372:A
337:P
300:A
294:P
259:P
230:P
221:P
201:(
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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