How/when and by what criteria is a paradox considered to be solved?


I understand that rejecting a premise and/or finding a flaw in reasoning can be wholly subjective. Considering the "this page is intentionally left blank" paradox, I could decide to reject the premise since there are words physically printed on the page but, it may still be argued that the page contains no useful information and one does not miss anything of import regarding the contents of the rest of the document...

But, it doesn't seem as though subjectivity would cut it when resolving a a paradox by any widespread consensus.

  • See Paradox : "By “paradox” one usually means a statement claiming something which goes beyond (or even against) ‘common opinion’ (what is usually believed or held)." Thus, one (possible) solution is to chabge 'common opinion'. – Mauro ALLEGRANZA Nov 22 '19 at 12:09
  • Would that make resolving a paradox something that is wholly subjective by nature? – Tim Burnett - Bassist Nov 22 '19 at 12:36
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    Why so ? "common opinion" is social, thus intersubjective. – Mauro ALLEGRANZA Nov 22 '19 at 12:49
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    What does it mean to solve a paradox? Does recognizing "This page is … blank" as deliberate ironic humour, solve (or even alter) the paradox? – Ray Butterworth Nov 22 '19 at 14:33
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    Presence of a flaw in a premise or reasoning can be up to subjective opinion, but then again, it can be an objective flaw. It can also be that there is no flaw, but only an appearance of inconsistency due to some hidden misconception. Finding an interpretation that makes such flaws or misconceptions explicit, and removes the paradox, counts as resolving it. (Credibly) rejecting the framework in which paradox is formulated is called dissolving it. But it is hard to say something more cogent in this sweeping generality. – Conifold Nov 22 '19 at 18:45

Technically speaking, a paradox isn't a problem to be solved in and of itself. A paradox points at a weakness, misconception, or internal contradiction of the philosophical/analytical structures we are using. When and if we fix the philosophical underpinnings, the paradox will simply become irrelevant.

Wittgenstein talked about this when he developed his 'therapeutic' approach to philosophy. He thought that intractable philosophical problems — which includes things like paradoxes — where nothing more than mistakes in language, where the philosopher misapplied the rules of philosophical grammar and created confusion. For Wittgenstein, trying to work through and solve such problems is a fruitless exercise; instead, we should work backwards to find where our language went astray. Once we discover where our language went wrong (assuming we ever do), the paradox will disappear in a puff of smoke and a great big Homer-esque "D'oh!".

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    I like it - the 'missing dollar' paradox comes to mind – Tim Burnett - Bassist Nov 22 '19 at 14:47
  • Wittgenstein was great. :-) – nomen Nov 22 '19 at 21:47
  • I have focused on Wittgenstein's notorious paragraph and believe that I have the analytical infrastructure to support his conclusion. I am responding to you because you are #1 ranked for the year and have no other way that I know that I can directly reach you. liarparadox.org/Wittgenstein.pdf – polcott May 8 '20 at 1:50
  • @polcott: Hunh. When did that happen? I'll rad over your thing; does it have contact information? – Ted Wrigley May 8 '20 at 3:33

The OP already mentioned 1 technique to resolve a paradox: reject a premise, a definition, identify a fallacy, etc. Another solution is a paradigm shift. I think perhaps looking at a historical example or two may help.

1) Russell's Paradox (aka Russell-Zermelo paradox)

Russell found a fundamental error in Frege's system of logic, which included naive set theory, and which he famously rephrased as a puzzle: There is a small town where the barber shaves those men, and only those men, who do not shave themselves. Does the barber shave himself? If you answer 'yes' then the barber can't shave himself, If you answer 'no' then the barber must shave himself.

This puzzle reveals a paradox in naive set theory: can the set of sets that are not members of themselves, be a member of itself? If you answer yes, it can't be a member of itself. If you answer 'no' then it must be.

There was no simple solution to this problem and indeed several solutions were developed to address it. One is to scrap the Comprehension Axiom and replace it with the Separation Axiom. That may not sound like a paradigm shift, but consider what happens when you add or remove axioms from Euclidean geometry.

There are other solutions- axiomatic set theories, natural set theories; Classes , Types, etc. But most new set theories required a shift in our understanding of what mathematics is, and thus constitute a paradigm shift within mathematics.

2) Achilles and the Tortoise

One of Zeno's paradoxes against motion involves Achilles chasing (or racing) a tortoise. The tortoise is a head of Achilles. For him to overtake it, he must first travel half the distance. And then half the distance again. And then half of the remaining distance, etc. The idea is that you can always divide the given distance in half, so in theory there is an infinite amount of distance to cover.

Clearly, Achilles can catch the tortoise, but how do you explain it? The problem seems to be with the ability to divide a distance ad infinitum. Ancient philosophers spent a considerable amount of time trying to solve this problem, including Aristotle who admitted that his proposed solutions were not convincing.

This paradox can be solved by considering the distance between Achilles and the tortoise as the sum of an infinite series, which in turn presupposes the concept of infinity. Since Calculus students learn this in high school this doesn't appear to be a big deal today, but infinity was controversial even when Leibniz (and Newton) invented Calculus, more than a thousand years after Zeno formulated his paradoxes. The point is that this required a new branch of mathematics to solve, and thus it required a paradigm shift.

There are numerous paradoxes in the history of philosophy, mathematics, and science. These examples certainly don't cover all of them, but hopefully they show how a paradigm shift can help resolve problems that were previously insurmountable.


Answers have discussed rejecting the premisses, identifying fallacies, but there is also this possibility: changing logic. First, yes, "to solve a paradox" is not imperative for philosophers. They were used in Antiquity as a kind of "reductio ad absurdum" arguments, which involved some premisse the philosopher would be willing to disavow. Also: dialetheists do not make sense of "to solve a paradox". But, assuming contradictions are problematic, I believe it is pretty understandable to mean by "to solve a paradox" not allowing the contradictory conclusion to follow from the premisses somehow. And, as I said, it can be arranged by changing the underlying logic of the argument. I can think of an example. There is a famous paradox of vagueness called Sorites paradox, which is:

(1) 10.000 grained collection is a heap;
(2) 1 grained collection is not a heap;
(3) A heap minus one grain is a heap (or: if k grained collection is a heap, then k-1 grained collection is a heap);
(4) Then, 9.999 grained collection is a heap;
(5) Then, 9.998 grained collection is a heap;
(10.002) Then, 1 grained collection is a heap.

Most proposals reject premisse 3, which is called "principle of tolerance". Yet, classical logic doesn't seem to differentiate much between the truth values of "10.000 grained collection is a heap" and "76 grained collection is a heap": they're either true or false. We though know that the former is truer than the latter. If we had a interval of truth values instead, such as [0,1], which includes 1 (true) and 0 (false), this could be done. In logic, we expect from an argument that it preserves truth, and we expect from an implication that its consequent is true when its antecedent is true. In this fuzzy logic we are using, all antecedents are truer than its consequents (as in: if 5453 grained collection is a heap, then 5452 grained collection is a heap), and this means they're not entirely true here: as the implication loses truth, the argument loses truth, until the conclusion is false.

If you have further interests, you may look at Sainsbury's Paradoxes and SEP's entry about Sorites. I hope this is helpful.

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