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More or less the question title.

My final-year logic course a few years back covered a number of non-classical logics (deontic, Kleene/Lukasiewicz multi-valued, etc.), however dialetheism was left as little more than a footnote in spite of a somewhat important name in the area being from the same university. I would like to read a little more on the topic to actually understand what is going on, and I'm looking for the first work that introduced it as a serious, defend-able position under the assumption that it is likely to have a reasonable explanation of the fundamental argument in the face of the obvious initial objections.

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    It may help to see Dialetheism. – Mauro ALLEGRANZA Aug 31 '15 at 6:00
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    Every treatise written on Common Law is, in fact, about a working dialethian system. Rights conflict, outright, by nature. So the abstract concept of rights upon which the better half of all legal contention proceeds is an artificial dialethian model with direct application. The weird failure of philosophy to take that seriously just makes it seem like there is an unsolveable problem where there is really a very important set of obvious solutions in practice currently. One could claim that the ethics of negotiation and legislaion basically is the specialist study of dialethian logic. – jobermark Aug 31 '15 at 14:04
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    George Melhuish's book 'The Paradoxical Universe' pre-dates the work of Priest and Routley and seems to form its basis. Priest's Dialethism is logically absurd (that's the point of it) and I see no reason to endorse it, but it's a valuable study since it introduces us to the crucial logical problem that the Perennial philosophy solves where other approaches become enmired in problems and have to start speaking of paradoxical universes. I mention Priest as the main exponent of Dialethism but this is not en endorsement of his wild misunderstandings of Buddhist philosophy. . – PeterJ Aug 13 '18 at 11:32
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According to Graham Priest, although it was Kant, in his Critique of Pure Reason, who first noted the natural occurrence of contradiction when we attempt to apply our notions outside of their natural range of application, it was in fact Hegel, in his Logic, who first accepted the legitimacy of contradictory concepts - i.e., that there are true contradictions. In fact, again according to Priest, Hegel held that all of our concepts are contradictory.

Kant considered the application of concepts beyond the bounds of our experience to be illegitimate. While Hegel agreed with Kant that such arguments that end in contradiction proceed by perfectly legitimate reasoning, he found no reason for declaring the applications of concepts in such arguments to be illegitimate. Hegel argued that the distinction between objects of our experience and objects of our thought had no particular ontological significance.

Thus, according to Hegel, if correct reasoning using legitimate applications of certain concepts leads to contradiction, then the concepts are contradictory. And, according to Hegel, since a sound argument must have a true conclusion, there must be contradictions which are true.

Quoting Priest, from his book In Contradiction :

The point I wish to isolate and highlight is Hegel’s contention that our concepts are contradictory, and there are true contradictions.

EDIT

Conifold's excellent answer is probably the answer you are looking for. However, I note that Priest, who defines dialetheia as true contradictions, identifies Hegel's Logic as the first to accept their legitimacy.

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    Priest fails to see that these contradictions are not 'true'. They merely show the contradictory nature of our concepts. Nobody has ever discovered a true contradiction. – PeterJ Aug 13 '18 at 11:34
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According to Priest, Routley and Norman's book Applications of Paraconsistent Logic the term "dialetheism" was coined by Graham Priest and Richard Routley in 1981. They cite Wittgenstein as inspiration who in Remarks on the Foundations of Mathematics describes ‘This sentence is not true’ as a Janus-headed figure facing both truth and falsity. Di-aletheia is literally double-truth, it refers to sentences that are both true and false. There is an obvious affinity to a much older term "dialectic", literally double-speak, which goes back to pre-Socratics, especially Heraclitus ("we do and do not step into the same rivers, we are and we are not"), and was revived in more recent times by Hegel (“Something moves, not because at one moment it is here and another there, but because at one and the same moment it is here and not here, because in this ‘here’, it at once is and is not”).

Dialetheism is not a property of a logical system but of its interpretation, any paraconsistent logic can accomodate dialetheism non-trivially by limiting explosion of contradictions. However a logician does not have to allow that A and ¬A are both true just because A∧¬A obtains. Kleene/Lukasiewicz three valued logic can be interpreted dialetheically if one chooses to think of the third truth value as "both true and false", but one can equally well interpret it as "neither true nor false" or "undefined".

In addition to SEP Dialetheism entry I recommend ones on Inconsistent Mathematics and Impossible Worlds. For instance Meyer constructed inconsistent arithmetic R#, where "it was demonstrable by simple finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert's goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable". In any consistent arithmetic this is unachievable due to Godel incompleteness, there are also extensions to analysis and beyond.

Impossible worlds extend semantics used to interpret modal logic, and go back to non-normal worlds of Kripke. They are literally dialetheic, some sentences are both true and false in them. Among other things, they are applied to analyzing counterpossible reasoning by fallible agents, who may admit premises that are, unbeknownst to them, inconsistent. Proofs by contradiction in mathematics are an example since assumed premise turns out to be necessarily false, and therefore uninterpretable in possible worlds. "Jago constructs an epistemic accessibility relation between worlds which is structured by rules of deduction: a world is ruled out as incompatible with the agent's cognitive state when it makes true an impossibility that the agent can recognize as such by applying a limited amount of logical reasoning. Some absolute impossibilities, though, may be too difficult to be detected, because their negation follows from basic principles of logic or mathematics only via long and complex chains of deduction".

  • Nice answer. I read the question as more about the concept than the term. Still, nice answer. – Nick R Aug 31 '15 at 19:12
  • I still like this answer, but upon re-reading it, I'm not sure you have correctly expressed what a dialetheia is. The example "This sentence is false" is both true and false. But this statement is not the dialetheia. If P = "This sentence is false", then it is the statement "P and not(P)" is the dialethia, and this statement is true (according to dialethism). – Nick R Aug 31 '15 at 20:05
  • This is an excellent answer. I'm going to accept Nick R's answer instead because it's a bit closer to what I was asking in terms of literary references about the concept, but I really am grateful that you took the time to write this and it has been quite useful. Thanks! – DTR Aug 31 '15 at 20:10
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    @Nick R "This sentence is not true" is not the same as "This sentence is false" outside of classical logic, it is a variant of "strengthened Liar". The option of making it both true and false is seen as an advantage of dialetheism since other approaches have difficulties with it, so it is a dialetheia. See Priest's analysis plato.stanford.edu/entries/dialetheism/#3.2" – Conifold Aug 31 '15 at 20:38
  • My use of "false", rather than "not true", was probably due to my haste. The linked SEP article appears to be contrary to Priest's definition in his book In Contradiction. Regardless, the SEP article says that both A and not(A) are true. It does not say both true and false. Whatever the case, I'm no expert and I learned a lot from your answer. I've probably misunderstood Priest's def of dialetheia vs dialetheism. – Nick R Aug 31 '15 at 20:47
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I don't know the answer to the specific question; however, Graham Priest is a well-known defender of this position. He suggests they arise on the borders of expressibility.

For example, Kant's antinomies arise in exactly this way and, much earlier, so do those of Parmenides and Zeno. But whereas Kant took this as a point demonstrating unintelligibility, Priest, possibly inspired by Hegel, still assumes intelligibity: the question is how? For Hegel, his orientation is on the unity of opposites (for Aristotle - a contrary).

note 1: The early work of Priest was in Buddhist logic. Dialethism is a strong tradition there (and likely a stronger tradition than it is in the West). It's plausible that it gained its impetus from Buddhist metaphysics.

note 2: Jobermark's comment on the ethics and process of dialogue as a process of contra-dicta or dia-logos is very useful here, too. Truth is a process, not at a moment, possibly formalisable in some paraconsistent logic, and as a sinn in some paraconsistent logos.

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