I don't think all axioms are "merely asserted rather than defended". For example, 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.
Another is consciousness. To grasp that existence exists, or make any attempt to reason either way about the proof or truth of anything, is itself an expression of consciousness.
Existence exists, and so does consciousness capable of conceiving that. Start there and work down.
> 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.
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.)
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.
You seem to be summarizing Descartes' "Cogito Ergo Sum" - "I think therefore I am". It does indeed prove the existence of something, but it doesn't actually prove anything about the nature of that existence. Are you an actual human being on a planet, the way it looks like, or are you a mind trapped in the Matrix? Does one thought cause the next, or are your thoughts just recordings on a tape of some sort, played sequentially, so the apparent causality is just an illusion?
My point is, you really don't get very far before you need to believe in some axiom, e.g. "I am a sensing being whose subjective experience of reality resembles the objective reality around me" (I know this sounds very basic, but try listening to someone who thinks they can slow Earth's rotation by meditating, and you'll realize that not everybody starts with an axiom where subjective and objective reality are separate, or where subjective reality is a derivative of objective reality and not the other way around)
By century-old arguments due to Russell, this "existence of existence" is incoherent. For example, as an imagined counterargument, imagine that there is a set of all things which exist, are sets, and do not contain themselves. By your axiom, this set exists. By definition, this set does not contain itself. Therefore, and by definition, this set contains itself.
Formal proofs are syntactic. As a result, they do not need consciousness to be verified; verification is algorithmic and can be done by a machine. Proofs can also be searched for by machines, although such proof search does not run at practical speeds on existing hardware yet. Nonetheless, to take your claim at face value is to claim that computing machines are conscious; are you prepared to embrace this result? (I don't personally have a problem with it, but many people do.)
This sort of word salad, more generally, is why Objectivism has such a poor reputation: It doesn't make sense when examined with even basic critical thinking.
Russell's paradox deals with self-referencing logically unconstructible objects, not with the impossibility of proving anything as existing.
>Formal proofs are syntactic. […]
This confuses means of formal proof with meanings attached to them. A proof, whatever the means used to perform it, only matter to conscious beings. A computer can help to solve a logical problem, but so far none have been catched wondering which logical problem would be interesting to solve as its next challenge – as far as I know.
In mathematics, we do actually take logically self-defeating objects to be non-existent, as a matter of classical reasoning: If they don't exist, then they must not-exist; otherwise, they would exist, and immediately defeat themselves in a puff of logic [5]. Such classical reasoning can reject a large number of similarly-shaped objects [2], and Russell's antimony is actually only the tip of the iceberg of a bundle of formal results, notably Tarski's [3] and Lawvere's [4].
The Gödel Machine algorithm [0][1] is only a few years old, and it cannot yet be tested due to practical hardware constraints, but it provably implements a system which wonders, specifically, about which direction of proof search will speed up its ability to search for proofs, to consider directions, or to wonder.
Yes, this is the lumine naturali of Decartes! The undoubtable doubt was presented and the rest is history. Needless to say it has been contended against, but the heritage of this line of reasoning as a foundation for epistemology cannot be disputed.
Using just an axiom of existence, you cannot get very far. Notice that, by the end of this post, you have already introduced a second axiom, concerning consciousness. Working down without additional axioms is the difficult part.
With regard to "the existence of existence", you appear to have started on your way down the "regressive argument" branch.
I'm not sure! There is the idea of the "stolen concept fallacy", which is to reason about something while denying that you are able to reason as the result of sensory perception, which if you accept, might mean there's some naturally resulting axioms about the senses and what a consciousness is capable of knowing.
These aren't my ideas. At the risk of having this and my first comment downvoted into oblivion, I got them from Ayn Rand. She has a short (80-page) nonfiction book about just these things which I think is remarkable, called Introduction to Objectivist Epistemology. If you are not predisposed to despising the author, it's a quick, very interesting read that I recommend.
Your starting point is taken up by Descartes in his Meditations on First Philosophy.
The more modern expansion of these ideas can be found in Phenomenology by people like Husserl or Heidegger.
There are a lot of contemporary philosophers dealing directly with philosophy of mind and consciousness which is more of an offshoot of these questions once you sideline questions of grounding.
And there is also more contemporary philosophy of science/math that deals directly with questions of consistency and the grounding of logical or rationalistic methods.
Basically there are a lot of resources if you are interested in this stuff, and I would heavily recommend reading and understanding what others have thought about before trying to re-invent the wheel as your starting point. That takes too long and there's a long way to go.
Not trying to reinvent anything, just sharing things that I've found appealing. Not claiming this author was first or last or original or not, only an influence on myself. Have you read the book? If not, it may contain perspectives you haven't encountered yet. Thanks for the recs!
Nope, the law of identity is incoherent and doesn't exist in physical reality. I'm extremely frustrated that Western Philosophy has reasoned about this extremely poorly and the assumptions made from antiquity seem to continue to be unquestioned.
The Cogito, or other ontological foundations of existence which rely on appearances are completely wrong. "I think therefor I am?" How do you know that you are thinking? How do you know that appearances of thinking are actually in existence? It's possible that you did not think! and that what you believe to be existence is not existence at all.
At best, you can say "it appears to me that I think therefor it appears that I exist". You cannot just escape radical doubt because of appearances.
Also, the no-cloning (and the other no-go) theorems and bells theorem (no determinism) seem to imply that Quantum Theory has concluded that the law of identity doesn't hold up in reality. Objects are NOT equal to themselves. We do not know that a fundamental minimal quantized unit of time exists, meaning that we cannot say that "an object is equal to itself at a particular time". If we accept the Kantian thesis that time and space are the "vessel" or "canvas" of reality, than it's not possible to confirm that an object exists in the same point in time, and the same point of space with itself without a fundamental minimal quantized unit to measure those in.
You should instead conclude that the metaphysical constitution of objects is ultimately in-determinant
The law of in-determinancy would say that "objects can be infinitesimally close to identical with themselves"
A =/= A. The left-side A exists in a different point in time and space as the right-side A. You cannot even think of these in your own mind at the same time, in the same part of your "mind-space". There is no verifiable measure of the "present" so we cannot constitute an object in exactly one point of time and space at a particular "moment"
All of what I've just said is basically a more modern version of Nietzsche's revulsion to the exact same "A = A" crap that was being thrown around in his own time.
Another is consciousness. To grasp that existence exists, or make any attempt to reason either way about the proof or truth of anything, is itself an expression of consciousness.
Existence exists, and so does consciousness capable of conceiving that. Start there and work down.