# Is there any logical system/method where impossible/illogical/inconsistent things can exist (like a solution to Russell's paradox that makes sense)? [duplicate]

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Discussing with a philosopher about impossible things existing or being allowed within a particular logic system, he told me:

"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)

So ia it there any method/logic system or anything else where impossible/illogical/inconsistent things would be allowed? For example, If a solution to Russell's set paradox cannot exist and it is impossible to exist, is there any method/logic system or anything else where this solution could exist?

## marked as duplicate by Conifold, Frank Hubeny, Mauro ALLEGRANZA logic StackExchange.ready(function() { if (StackExchange.options.isMobile) return; \$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var \$hover = \$(this).addClass('hover-bound'), \$msg = \$hover.siblings('.dupe-hammer-message'); \$hover.hover( function() { \$hover.showInfoMessage('', { messageElement: \$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 30 '18 at 6:10

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• Talking with that philosopher I told you about about whether dialetheism was the right "method" to do what he was saying, he said "dialetheism is not unique in any way. It doesn't let you construct the impossible in a way that was not previously available to you, though it may change your mind about what is impossible, and it may change how you handle it." so I was wondering if someone knew an alternative "method" @Conifold – bautzeman Aug 29 '18 at 21:50
• There can not be any "objective" yes or no answer to the title question. Unfortunately, people can not mind read what would satisfy your philosopher, so you should decide for yourself if the "methods" described, dialetheism, epistemic logic, dialectic, etc., do it. And logic does not let you construct anything in a way not previously available to you, it can only reorganize what is already available. – Conifold Aug 29 '18 at 22:06
• Assigning a noun to any concept provides the basic tools you need to analyse the problem. Faster than light travel. Ok. Let's discuss it. In the programming language Java there is the concept of marker interfaces. I define the name grape and assign it to an object. But I can assign the name grape to a set of grape objects too. Mind bending stuff. It:s a kind of simplistic polymorphism. – Richard Aug 29 '18 at 22:48
• @Conifold "And logic does not let you construct anything in a way not previously available to you, it can only reorganize what is already available." but for example, in "standard" logic,impossible things like a solution to Russell's set paradox does no exist, but you said that in dialetheism for example it could exist. So something that was not available in one logic system is valid in another one – bautzeman Aug 29 '18 at 23:22
• Or in paraconsistent logic @Conifold – bautzeman Aug 29 '18 at 23:39

## 1 Answer

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)

Relax. Russell's Paradox was resolved over a century ago using what is now just ordinary logic and set theory. The problem was with the earliest axioms of set theory, those introduced by Cantor and Frege around 1900. They didn't work. The problem was resolved by introducing other axioms of set theory (ZFC being the most popular to date) from which it could be proven that the problematic set did not exist.

• Don't you think that common sense alone was enough to work out that the problematic set could not exist? I never grasped why this wasn't obvious to Russell right from the start. – PeterJ Aug 30 '18 at 9:29
• The non-existence of the Russell set could be proven using the rules of ordinary logic. The problem was that its existence could be formally proven using Cantor and Frege's axioms of set theory. For any formula F, they assumed that there existed a set S = {x | F(x) }. Seems reasonable even today, but it blows up for F(x) = x not in x.. Their axioms didn't work. They needed new axioms of set theory that avoided this problem. – Dan Christensen Aug 30 '18 at 14:58
• Thanks Dan. This is a useful comment. I wonder why they didn't simply accept that this set doesn't exist. I've never quite grasped this. It may not matter in maths which approach we take but it sure does in metaphysics. – PeterJ Aug 30 '18 at 16:04
• @PeterJ The problem was the above proposed axiom of set theory. It led to an inconsistency with basic logic. Some (the intuitionists) blamed the system of logic and banned proofs by contradiction. That, too, got rid of Russell's Paradox because it relied on this method to prove the non-existence of the Russell set. That was throwing the baby out with the bathwater IMHO. – Dan Christensen Aug 30 '18 at 16:32
• Thanks again. I'd agree with the your last sentence. This 'paradox' would be the central problem of metaphysics for me and you've explained why so few can solve it and why so many prefer to throw out the baby and allow contradictions. Fascinating issue but off-topic here. – PeterJ Aug 30 '18 at 17:09