Comment by skissane | Hacker News Reader

Comment by skissane | Hacker News Reader

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> one axiom is the existence of existence, which cannot be denied, because any question, doubt, skepticism, reason, point, or imagined counterargument is something that exists and is within existence

You argue that the denial of the axiom of "existence of existence" is self-refuting. But don't all arguments that something is self-refuting depend on the acceptance of the law of non-contradiction ("for all A, not (A and not-A)"?) If your argument for the axiom of "existence of existence" depends on the axiom of non-contradiction, is the axiom of "existence of existence" truly axiomatic any more?

What about the axiom of non-contradiction? Do we have to believe that? Well, dialetheism [1] denies that the axiom of non-contradiction holds (in the most general case), and yet the whole system doesn't fall apart. The secret is to allow contradictions, but limit their consequences, by rejecting the classical principle of ex contradictione quodlibet (ECQ, from a contradiction anything follows, aka the principle of explosion). An unlimited allowance for self-contradiction destroys all possibility of non-trivial reasoning, whereas a more limited allowance for self-contradiction leaves open the possibility of non-trivial thought. So, if we have a choice whether to accept the axiom of non-contradiction, maybe your attempt to escape Münchhausen's trilemma has not succeeded.

[1] https://plato.stanford.edu/entries/dialetheism/

Fair point, but how do you determine when to stop following the consequences of the contradictions? Furthermore, from the point at which you stop, you are then re-enforcing the principle of non-contradiction, so why not enforce it from the start and call it a day? To me, dialetheism has always seemed very arbitrary, but maybe that's exactly the point of it. If that's true, then it's just another flavor of extreme relativism where anything can be true or false and any reasoning is meaningless.

On top of that, it seems to me that even in denying the principle of explosion, one must follow it if they want to be consistent in the denial. Otherwise the denial itself won't hold, because if you allow contradictions, by definition you must also allow their opposite, which is the principle of non-contradiction. And as far as I'm aware there is no proof in logic that allows for an exception to ECQ. Because once you allow any contradiction, you implicitly allow anything, which makes thought impossible.

Well, one approach is to start with an assumption that the law of non-contradiction is true in most cases, and then look at specific cases in which it might be worthwhile to make exceptions to that generalisation - for example, the liar sentence (This sentence is false), Gödel sentences, Russell’s paradox (“this set both is and isn’t a member of itself”), etc. Dialetheism in which every contradiction is accepted is trivial. In non-trivial dialetheism, you argue that there are good reasons for accepting certain contradictions that do not apply in the case of others.

My questions still remain unanswered. Regarding the examples you raised: the liar paradox can be solved without breaking explosion by observing that all assertions carry the implicit ending "and this sentence is true". The liar sentence then becomes "This sentence is false and this sentence is true", which is simply false. As for Russell's paradox, as far as I know this has been solved in ZF theory, although admittedly with a few extra axioms needed. Not sure about Gödel's sentences.

It just doesn't seem worth it to me to suspend the fundamentals of thinking just to solve some paradoxes that can anyway be solved within our current model of logic.

> Regarding the examples you raised: the liar paradox can be solved

There are various proposed solutions, but they all have drawbacks. You have to compare the drawbacks of the different options. (From my limited memory, your proposed solution is not one of the standard solutions proposed in the literature.)

> As for Russell's paradox, as far as I know this has been solved in ZF theory

ZF theory pays a price – the restriction of the axiom of comprehension. The point is you don't solve Russell's paradox for free, every solution has its price, every solution involves giving up some component of naïve set theory; ZF chooses to partially give up the axiom of comprehension; inconsistent set theory (part of inconsistent mathematics [1]) chooses to partially give up the axiom of non-contradiction instead. If we have to give something up, how do we decide which part to give up? You think that giving up the axiom of comprehension is a smaller price to pay than giving up the axiom of non-contradiction – but is that a subjective value judgement? Or, can it be objectively justified? (And if so, how?)

A good argument for paraconsistent logic is that relevant implication is a more accurate model of natural language than material or strict implication, and relevant implication is paraconsistent. (That said, dialetheism goes beyond mere paraconsistency.)

I'd suggest (if you have the time/inclination) reading Graham Priest's book In Contradiction. It explains the arguments for all this far better than I can from memory. (I read that book 15 years ago.)

[1] https://plato.stanford.edu/entries/mathematics-inconsistent/

I was referring to Arthur Prior's solution to the paradox.

As for dialetheism being a better model of natural language, I'm familiar with the claim but I think its main argument fails to grasp that contradictions in natural language do not happen "in the same sense and at the same time". For instance, the sentence "This sentence is false" is true and false in natural language, but in succession and not at the same time. Moreover, dialetheism does not seem to be able to explain hierarchies and cause-effect relationships, which are key constructs of language.

I think that when it comes to logical systems such as mathematical theories, the point is not necessarily to come up with a theory that is true (that would ultimately be impossible due to Agrippa's trilemma), but to have a theory that has predictive power and is self-consistent, and this is what ZF is. Self-consistency seems to be the minimum requirement for any theory, lest it falls into triviality.

The little I know about paraconsistent logic is that it suspends consistency in a few select cases. But generally, consistency still holds, otherwise, as you said, the system would become trivial. I guess my question was: would paraconsistent logic claim to be true? Or, at least, identical to itself? If yes, then if would itself be subject to consistency and identity. And if it chooses to temporarily suspend those, then in that time it can no longer claim to be true. But if in general it complies with consistency (lest it becomes trivial), then it must reject the subsets that are not true, otherwise it could no longer be self-consistent.

I guess my point (and I could be wrong) is that paraconsistent logic is entirely non-consistent, unless it chooses to redefine what consistency means, in which case, we're back to triviality.

Thanks for the book suggestion, I'll definitely read it and hopefully find some answers.

> If your argument for the axiom of "existence of existence" depends on the axiom of non-contradiction, is the axiom of "existence of existence" truly axiomatic any more?

Obviously since there's no need to rely on precisely one axiom.

My point being, if you use A to demonstrate B, then B is not axiomatic in the sense of Münchhausen's trilemma. In the sense of Münchhausen's trilemma, an axiom is a belief accepted as true without being justified on the basis of some more basic belief. When you use A to demonstrate B, B ceases to be a belief accepted as true without being justified on the basis of some more basic belief, since you are justifying it on the basis of A.

I see what you meant and that's actually interesting regarding LNC. What I mean is if you can use A and B to prove C, it doesn't matter that you may be able to use A and something else to prove B. Relative to C, A and B may be considered axioms.

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