16

I may need to refine this question, since I am mostly grappling with a murky intuition and haven't yet done the real work.

When I encounter many of the well-known paradoxes, such as Zeno's dichotomy, Russell's set containing itself, problems of "zero" and "one," the concept of the infinitesimal in calculus or a "point" in time, Hegel's "identity of identity and difference," even aspects of quantum indeterminacy, they all seem tantalizingly similar. They all seem to arrive at a point-position, like the "origin" in Cartesian coordinates, that must be defined as both "inside" and "outside" the totality under consideration. Or something like that.

Is there some reduction of these and other paradoxes that is "more fundamental" or best clarifies what they have in common, some sort of meta-paradox? As I say, this may be either too wooly or something everyone with a background in logic already knows, but I thought I'd toss it out there.

  • 5
    There is an implicit instability in negation, so these share that. But not all paradoxes involve negation. Some involve mutual dependence, e.g. the Prisoner's Dilemma, or infinity, e.g. the Trolley-Car paradox that presumes the infinite value of life, instead. – jobermark Sep 21 '15 at 19:41
  • Thanks. If you have a short answer to what you mean by "instability of negation" I'd be glad to hear it. And you are correct, I was not trying to encompass "all" paradoxes, such as game theoretic or ethical. I don't really have a good way to classify the types of paradox. – Nelson Alexander Sep 21 '15 at 20:26
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    Description "defined as both "inside" and "outside" the totality under consideration" sounds like use of impredicative definitions, which do indeed produce paradoxes of self-reference en.wikipedia.org/wiki/Impredicativity I do not follow how Zeno's dichotomy, infinitesimals or quantum indeterminacy fall under this though since they rely on spatial intuition, or classical physical intuitions more broadly, to produce the paradoxality, rather than on logical constructions. – Conifold Sep 21 '15 at 20:49
  • Thanks, I'll look that up. I am netting some useful terms I didn't know. – Nelson Alexander Sep 21 '15 at 21:11
9

You might find Graham Priests book The Limits of Thought helpful in refining your question. Priest argues that

thought runs into true contradictions when it runs up against its own limits

This was noted by Kant - his famous antinomies - which motivated his critical project; however Priest credits Hegel for deciding the contradictions are unavoidable, and their underlying structure.

The book runs through a history of limits; and argues for a typology; limits of:

  • expression (Plato on something unchanging as a precondition for meaning, Aristotle on Prime Matter, Cusanus on the ineffability of God)

  • iteration (infinite regresses in Zeno, Aristotle and Liebniz)

  • cognition (Sextus's sceptism and Protagoras's relativism)

  • conception (Anselms ontological argument and Berkeleys inconcievability argument for Idealism).

In a different direction, it's worth noting that some of the mathematical Paradoxes of Cantor, Russell's, Tarski, Turing and Godel have been noted to rely on a similar argument - diagonalisation - which was later marshalled into formal argument (under the auspices of category theory).

  • I've deleted all the comments... If @jobermark wants to make 1 comment expressing his objections, I think we can allow that. But please avoid arguing in the comments. – virmaior Sep 26 '15 at 2:40
3

Paradoxes can generally be categorised by their traits, but there cannot exist a paradox that describes the fundamental components to all other paradoxes. I will perform a proof by contradiction.

Let's pretend there does exist a fundamental paradox that describes all others. That would entail that there exists a paradox that describes both binary paradoxes and infinite paradoxes. This on it's own is a binary paradox, meaning that the fundamental paradox would have to describe itself as well. But if it describes itself that means that it is not a fundamental paradox because it itself is a paradox. You can continue this pattern on and on until you just give up.

I hope this helps and please comment if I made a mistake in communication or in my arguments.

  • Hmmm, your "proof by contradiction" sounds like it might be an answer. I'm unfamiliar with the classification of "binary" and "infinite" paradoxes, though it makes sense, so I'll do some further reading and ponder. Thanks. – Nelson Alexander Sep 21 '15 at 20:21
  • @NelsonAlexander just for clarification, binary paradoxes are what I use to describe a paradox that struggles with two possible outcomes. Infinite paradoxes are those which continue on repeating the same process over and over without any definite answer. I glee that helps a bit. – Teagen Dix Sep 21 '15 at 20:23
  • So, roughly same as "antinomies" versus "infinite regress" or "undecidability"? – Nelson Alexander Sep 22 '15 at 13:19
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    There is no good reason the fundamental paradox could not be a binary paradox that describes itself. Description is a reference, not some kind of incorporating inclusion, so there is not necessarily any infinite regress here. – jobermark Sep 22 '15 at 20:42
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    Also, your definition of an infinite paradox includes things like 'Find the largest integer', which is not very paradoxical, just impossible. Recursion is not in-and-of-itself a paradox. – jobermark Sep 22 '15 at 20:45
3

One incredibly fundamental issue that arises with paradoxes is Tarski's undefinability theorem. It is a limit of formal languages. It states that any formal language meeting a particular criteria cannot define its own semantics. That criteria is rather broad: any formal language which can describe arithmetic and has the negation operator cannot define its own semantics! The "famous" paradoxes are typically ones which have received strict mathematical attention, so tend to be in comparatively formal form already.

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    Thanks! That sounds like exactly the sort of fundamental take I am looking for. Will definitely follow up. – Nelson Alexander Sep 23 '15 at 16:02
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    Glad it might help! I only discovered it recently, and I have found it quite useful. It also points to an intriguing alternate solution to paradoxes: as the language becomes more and more informal, it becomes possible for one individual's semantics to lead to a paradox, and another's semantics to show no conflict. In my experience debating paradoxes (for fun), I've noticed that often people don't see any particular conflict in a paradox, you have to teach them to see the conflict by imposing a semantics, and then they see it as a paradox. – Cort Ammon Sep 23 '15 at 16:12
3

According to a research paper referenced below, "many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme".

The examples brought out in the paper:

  1. Cantor’s theorem that N ℘(N)
  2. Russell’s paradox
  3. The non-definability of satisfiability
  4. Tarski’s non-definability of truth and
  5. Gödel’s first incompleteness theorem.

See A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.

2

One way to describe a paradox is that it's a situation where two of your intuitions conflict. You have lots of intuitions, and they can conflict about all sorts of things, so there's no reason to expect that all paradoxes can be related to each other.

In particular, a situation feeling paradoxical is as much a statement about your state of mind as a statement about the situation. You might discard some intuitions and refine others and later find that something that used to feel paradoxical doesn't anymore.

  • 1
    An interesting application, but it sounds more like ambivalence. Paradox is usually thought of as a conflict in reasoning, thus apparent of "all rational" thinkers and having little if anything to do with conflicted feelings. Though I might agree that in some sense "paradox" goes deep into the human condition, as deep as subjectivity. – Nelson Alexander Sep 22 '15 at 3:31
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    @Nelson: I think this is just incorrect. For example, I think Zeno's paradoxes were very clearly in hindsight about conflict between intuitions people had about motion and intuitions people had about infinity. A modern mathematician has more refined intuitions about infinity and with these refined intuitions there's nothing particularly paradoxical going on. – Qiaochu Yuan Sep 22 '15 at 4:19
  • Whatever reasoning is, it's in particular something that happens in human brains; let's not forget that. – Qiaochu Yuan Sep 22 '15 at 4:20
  • @yuan: I think there is more to say Zenos Paradoxes than is usually admitted; the debate then was about the possibility of change, ie motion. – Mozibur Ullah Sep 22 '15 at 10:59
  • I guess I do not understand how you are using "intuition," which is usually a rationalist term thought of as "a priori," and thus not determined by any subjective states or even materialist "brains." But in either case I don't see how adding "intuition" instead of, say, "thinking" really advances the question. And I do agree with Mozibur U., Hegel, Russell, and others that Zeno is still meaningful. My original question arose because I felt I was seeing Zeno redux everywhere. – Nelson Alexander Sep 22 '15 at 13:26
2

The particular paradoxes that you want to tie up together do seem to share an aspect from a given logical point of view.

From a 'hard Intuitionist' point of view, our linguistic construct of negation is incomplete. Brouwer traces them to our difficulty combining two different kinds of distinction related to time. And they don't fit together properly.

One is related to counting, time's successiveness, and another is related to continuity, time's 'even flow'. From the one related to counting we get the Law of the Excluded Middle, everything is either before or after this point. And from the one related to continuity we get the notion of completeness, that you can always look closer and see more stuff, but what you see won't contradict what you have already seen from a less complete viewpoint. These inherently contradict, but we can't see that clearly, and we see paradoxes when we have to choose just one of them.

Zeno's paradox is the clearest example of how they fail to fit together, we count off subdivisions of space, as our intuition of space naturally fills up all the gaps, and we are left wondering whether the process should or should not be allowed to finish. The problems with the unrestrained use of infinitesimals are just Zeno's Paradox generalized. We have to accept continuity over iteration, and that does not sit well with us. Cantor gives us some idea why. But we can't really keep it straight in our heads.

Russell's paradox is more about looking outside than inside, but from an intuitionistic point of view, the fact we can always find a set to wrap around any given set is splitting infinity in the same way Zeno is splitting a finite distance. It creates a boundlessly multiplying world, which like the continuum, 'presumably fills in' all gaps made by any distinction. We then attempt to apply the Law of the Excluded Middle to every distinction, all at once. And we fail.

Our basic notion of implication also intrinsically involves negation. We like the notion that A -> B <=> B v ~A. But this notion leads us into paradoxes like "If this statement is true, then Santa Claus said it first." Our wish to have LEM conflicts with our wish to have every statement be true or false, and this one is either both, or vacuous.

Etc.

But negation is not the only source of confusion in classical logic. There are other equally weak concepts that incompletely capture other intuitions.

Our grasp on what can and cannot be compared, and why, is somewhat weak. We have a bias toward the idea that any partial order easily yields a total order. This makes us imagine the Prisoner's Dilemma should have the wrong answer, and to balk at the Game of Intransitive Dice and the related oddness in the way probabilities behave.

We can see how the first boring number would not be boring, because it would elucidate the nature of boringness, which would make it less boring. We have a definite partial order which cannot base a total order. And that is annoying.

Infinity makes this situation worse, netting us various ethical problems which involve multiple lives and attempts to compare different losses of life, which basically challenges our intuition to come up with consistent extra rules to order ratios of infinities. And the Axiom of Choice, which is equivalent to this notion that order is either there or not, via Zorn's Lemma, leads straight into the Banach-Tarski paradox about measurability, size and the nature of solidity.

Putting our problems with probability and divisibility together, statistics, which is infinitely divisible probability, creates its own host of paradoxes.

So I would claim there may be large classes of paradoxes traceable to any one specific weak intuition, but that there are at least a few unrelated sources of paradox, and beyond that different paradoxes arise by combining different sets of weak intuitions.

  • Thank you, an excellent clarification, though a few of the terms escape me. This may be about as definitive an answer as I can get here. Of course, it leaves much to mull over and I suppose it simply takes practice, practice, practice to find applications and to reduce to the simplest models. My present lack of formal training in logic forces me into "picturing" things, but this may not be all bad. – Nelson Alexander Sep 22 '15 at 16:53
  • Don't take it as too definitive. Brouwer's ideas about conflicting intuitions, the weakness of negation, etc. are actually quite unpopular. But I hold this particular point of view. From the viewpoint of Intuitionism, we need to put mathematics or logic that relies upon strong negation, actual infinities, the Axiom of Choice (particularly in the form of Zorn's Lemma), and a few other obscure axioms into a "provisional" category, in order to avoid instances of paradoxes. This does not make for fans. – jobermark Sep 22 '15 at 19:00
0

A paradox is a proof that your way of thinking about things is too naive or really bad, and should be changed if you want to make sense of the previous things. There are some special kind of paradoxes, called "diagonal arguments", that plays a special role.

Diagonal arguments are basically happening when one is stating a sentence that "ask too much", and involves some universal quantification. It is based on the fact that if "everything is something" is true, then it is true that "everything is something" is also something. I invite you to replace the previous something by subjective, meaningless, relative, false, or whatever you like. Basically, the idea of diagonal argument is to apply an idea to itself, and it's actually a pretty good check for consistency. In this respect, subjectivism or nihilism are clearly not diagonally admissible, while the naive classical truth one is.

Now, not all paradoxes are like that. For example, one should definitely make a difference between an antinomy (what is both true and false at the same time) and a paradox (what goes against intuition). In any case, one shouldn't think that paradoxes are things one cannot do anything against. For example, linear logic is a logic of quantity/information. There, some truth are temporal such that what is true at some point is not necessarily true at the other. In particular, linear implication A -> B has to be understood as "given A, it transforms into B" : you start with A, and you end up with B but you lost A in the process. In this framework, the liar paradox becomes a valid statement, but it is cyclic. Indeed, if A = "He is lying when he says that he is lying", and B = "He says the truth when he says that he is lying", then A -> B because if he is lying when he claims to lie, then he actually says the truth. Yet, if he says the truth when he is lying, then clearly he is actually lying, thus B -> A. The paradox is now a cycle of length 2, and is perfectly meaningful.

The same would happen if one was able to have a formal logic with memory (thus non commutative as quantum mechanics) for the prisoner paradox: you just stop the reasoning at such a point, but the conclusion would change if you redo it knowing such information, and it'll never end.

Note that algebraically, one is always free to interpret a paradox as a fixed point of the negation, but I don't find it enlightening at all.

  • Not sure why the down vote, but I voted back up. Thanks for the concepts to pursue, I will look into the "diagonal." Lots of reference information here. However, I do agree with down vote that it is not a very clear or concise answer to the question, though part of that may be my unfamiliarity some of the terms. – Nelson Alexander Sep 22 '15 at 13:34

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