A paraconsistent logic system it is defined as

"a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic"

So, is there any paraconsistent logic system,or any other logic system, where illogical or impossible things would happen? Can they produce a solution to Russell's set paradox (in terms of naive set theory)?

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    "impossible - adjective - Not able to occur, exist, or be done" - source: Oxford English Dictionary. – Mawg says reinstate Monica Aug 24 '18 at 8:24
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    Paraconsistent logic does not allow the impossible to happen for the reason Mawg gives, It implies that impossible things could happen, for instance 'true contradictions', which is why it is not much used except by a handful of philosophers and even they do not so much use it as recommend its use. When applied to set theory it does not solve Russell's paradox but makes it evaporate. If we believe in true contradictions there's no paradox to solve. . – user20253 Aug 24 '18 at 12:03
  • Paraconsistent logic: Is that logic where you ignore inconsistencies? That could really, umm... simplify things! – Dan Christensen Aug 25 '18 at 2:08
  • @DanChristensen What do you mean? – bautzeman Aug 25 '18 at 12:10
  • Being facetious. I understand that PL allows inconsistencies. Seem weird, I know. Most people try to avoid logical inconsistencies. – Dan Christensen Aug 25 '18 at 12:20

Logic, paraconsistent or not, does not exactly make something happen, it is applied to reshuffle information already contained in a system. Paraconsistent logic does not even have to be applied to inconsistent systems, and even when it is, derivable contradictions do not have to be interpreted as "true".

What we need is not logic but semantics, although what kind of logic is used does impose some constraints on how the resulting system can be interpreted. Semantic interpretations that admit true contradictions, a.k.a. dialetheias, are called dialetheist. Priest and Routley, the founders of dialetheism, did draw inspiration for their interpretation of naive set theory from Wittgenstein’s remarks about the Russell’s paradox:

"Why should Russell’s contradiction not be conceived of as something supra-propositional, something that towers above the propositions and looks in both directions like a Janus head? The proposition that contradicts itself would stand like a monument (with a Janus head) over the propositions of logic".

This was developed into a body of inconsistent mathematics, by Meyer and others. The point is to obviate the negative conclusions of Gödel's incompleteness theorem by rejecting one of its premises, the assumption of consistency. Meyer's inconsistent arithmetic R# has no undecidable statements and he proved by finitary means that contradictions within it do not affect any numerical calculations. This is in a sense a realization of Hilbert's programme of proving consistency of arithmetic by finitary means, or at least as close as one can come.

Similarly, dialetheist interpretations were used to deal with the semantic paradoxes, like the famous Liar. If we admit true contradictions then a resolution of the Liar would be that the "I am false" sentence is just that. Hegel's philosophy, with its dialectic, and other non-dualist systems with their "unity of the opposites" (neoplatonism, Buddhism, Advaita, etc.) arguably affirm dialetheias, although this is debatable. Hegel does say that "one of the fundamental prejudices of logic as hitherto understood... [is that] the contradictory cannot be imagined or thought", but he uses "logic" in a different, old, sense, closer to today's "epistemology".

However, dialetheism is not the only, and not even the most common, way of admitting "impossible things". Pace Hume, who thought that things impossible can not be believed, or even conceived, things that turn out to be impossible are routinely conceived in reasoning provisionally, for example in reductio arguments. An ancient example is Euclid's proof which considers a rational number whose square is 2, and after a series of manipulations concludes that such a number does not exist after all, because a contradiction results. Russell's set is treated the same way in his paradox. To this day we do not know if an odd perfect number (equal to the sum of its proper divisors) is impossible or not, but mathematicians have been proving things about them for centuries. In other words, one need not believe in true contradictions to have a need to reason about the impossible.

This is handled by the epistemic logic, logic of what is known. Since the knower may not be smart enough to see through all the consequences of her assumptions she may well believe some hidden contradictions. Such belief systems are modeled using modal semantics that in addition to possible worlds admits impossible worlds. The sets of sentences describing them can imply contradictions, but derivations of contradictions have to be "long". The abridged descriptions are not closed under the logical consequence and hence avoid "overt" inconsistency. Other dialetheist and non-dialetheist interpretations are equally possible, as Priest points out:

"As far as I can see, any of the main theories concerning the nature of possible worlds can be applied equally to impossible worlds: they are existent nonactual entities; they are nonexistent objects; they are constructions out of properties and other universals; they are just certain sets of sentences."

Non-existent objects in ontology predate even modal logic, they were proposed by Meinong already in 19th century.

  • and with these philosophical approaches you quoted (like dialetheism) couldn't we conceive or describe the impossible? For example, once a philosopher told me: @Conifold – bautzeman Aug 25 '18 at 12:15
  • "This is a funny thing about logically impossible things. You can prove that they exist in any non-consistent or paraconsistent logic system. You might even be able to give a (nonsensical) description that satisfies some specific definition of said thing. But that still doesn't give you anything that makes sense. There, you are asking not only to prove that a very particular impossible thing exists, but you are asking for a detailed description of it to exist as well. I know of no method for doing that" (talking mainly about a solution that makes sense to Russell's set paradox) @Conifold – bautzeman Aug 25 '18 at 12:16
  • So is it there any logic/philosophy/method that would allow us to conceive/describe that? To conceive/describe the impossible? @Conifold – bautzeman Aug 26 '18 at 2:34
  • @bautzeman Yes and no. Impossible worlds can be as detailed as one wishes, as detailed as possible worlds are. But there is no way to describe even some possible things to everyone's satisfaction, as we know from Gödel's incompleteness. As for Russell's set, one does not even need impossible worlds, it suffices to call it a class rather than a set and the paradox goes away. We also easily "conceive" the Liar, the issue is how to fit it into our overall system. – Conifold Aug 26 '18 at 21:14
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    @bautzeman Whether something does the job depends on who you ask and what the job is. It is hard to tell from your remarks what your philosopher had in mind. Either dialetheism or epistemic logic work for most people, Hegel's dialectic is a more fanciful choice, assuming one needs any "method" at all. Transfinite cardinals, Lebesgue non-measurable sets and other such "possible" objects are much harder to "conceive" than self-referential trivialities like the Liar or the Russell's set, and yet they are handled by traditional mathematics. – Conifold Aug 26 '18 at 22:14

The most-commonly encountered example of a paraconsistent logic is a legal system, especially one that has both statutory and common-law components. Impossible things happen in the literal interpretation of legal status all the time. An explanation is given, or an authority intervenes, and the system heals around the contradiction.

Here is a real situation from my past. There is a statutory law against trespass -- you cannot walk across my lawn if it bothers me. There is also the common-law concept of easement -- traditional access routes cannot be blocked without giving notice. You are standing on my lawn because people in this neighborhood have used this as the shortest route from a park to the nearby junior high school for decades. I have just moved in here, I have never spoken to you before, but I am taken by surprise, and chase you off my lawn.

There is a contradiction. You have broken one law, and I have broken another but neither of us has done anything wrong. Until someone intervenes and explains what happened, it will remain a contradiction. It does not need to be resolved immediately, but we probably do need to decide who is right in the long run.

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