I have read many times that some problems or logical propositions do not have solutions or are outright impossible. These are three examples of such problems:

  1. [The Russell's paradox] which is deemed as contradictory with no possible solutions

  2. [The Liar's paradox] which is similar to Russell's paradox in the sense that it is also contradictory and no logical solution can be found out of it

  3. Some mathematical impossibilities like trying to find a positive integer number to divide a prime number (e.g. 2, 7, 53, 181...) other than 1 and itself

However, is it really true that no solutions exist for these situations? Could we solve these problems using alternative axioms of logic and mathematics? Or perhaps our brains need to develop or evolve more to conceive such solutions?

  • 1
    Russell's Paradox has been solved with Axiomatic Set Theory and similar. Dec 1, 2022 at 11:39
  • 1
    There are many interesting approaches to the Liar Paradox but not commonly agreed solution still. Dec 1, 2022 at 11:40
  • There are still many Unsolved problems in mathematics, but some of previous unsoved ones have been solved. Dec 1, 2022 at 11:41
  • 5
    3. Isn't really impossible it's the definition of a prime number...
    – haxor789
    Dec 1, 2022 at 14:25
  • (1) Russell's Paradox: Caused by faulty axioms of set theory. Resolved in the early 1900's. (2) Liar's Paradox: A non-issue in mathematics AFAIK. If formalized as S => ~S, then it is not a paradox at all. By contradiction, S is simply false. (May ruffle some feathers.) (3) Your prime number "paradox": No such thing. See any definition of the prime numbers. Dec 16, 2022 at 17:40

4 Answers 4


These are issues of logic. When you discover a paradox you have the following choices. Accept the paradox indicates a flaw in your logic. Or. Change some part of your logic.

  1. Birds must be able to fly.
  2. Ostrich is a bird.
  3. Ostrich cannot fly.

To rescue this, you must change one (or more) of them.

  1. Maybe not all birds fly.
  2. Maybe ostrich and bird share a common ancestor that is not a bird, maybe a "proto bird" that might or might not fly.
  3. Maybe ostrich can fly. Such as when the zoo ships their prize flock to another zoo for a loan, and sends them by air freight.
  • The example is not a paradox: they are three statements two of which are false. And the purported solution is not a "change in logic": it is a proposed correction to the wrong statements. Dec 1, 2022 at 13:46
  • 1
    @MauroALLEGRANZA You think a paradox can be based on true statements?
    – Boba Fit
    Dec 1, 2022 at 13:47

One alternative-logic option for (1) is plural quantification. This allows us to refer to all and only well-founded sets without referring to a set of all of these.

Or, then, one might stipulate that the qualifier "and only" be disallowed: so you could have a set X that has all well-founded sets as elements, but X has parafounded elements, too (itself, say). Broaching the subject in these terms leads us to issues with unrestricted universal quantification.

As far as the liar paradox goes, the argument to the paradox relies on a subtle equivocation in that the "is not true" of the liar sentence L is not the same kind of "is not true" that is attached to the sentence when L is externally adjudged. Consider:

  1. This sentence is unknown to be true. Formally, S: ∃S(~kS) (where "k" is a predicate).
  2. It is known or unknown that (1). Formally, K/~K(1).

In (1), knowledge is a predicate; in (2), it's a sentential operator. Likewise, when the liar paradox is derived, we conflate the antitruth-predicate by virtue of which L is defined with a prosentential antitruth operator. In effect, the problem with the derivation of the liar paradox is akin to the problem with Anselm's ontological argument, where a concept's term is put in a predicate position that it either doesn't belong in, or which, even if it does, is such that the concept doesn't equate to the term that we construe as generating the paradox. I.e., T/~T(L) is no more contradictory than, "The existent X doesn't exist," is.

Famously or infamously, all that goes to the question of truth being a predicate vs. truth being an object, or then also the question of truth functionality overall, and a myriad options besides (truth-maps, truth-sets, truth-elements, truth-what-have-you's...).


As per Aristotlean Logic, which basically make foundation of all logic:

"X and Not-X cannot both exist at same time" or simply "A thing either exist or not (nothing can half exist)".

No, liar's paradox has no solution. Its designed to have no solution.

The thing is, logic is a tool, to understand this universe. Logic works because this universe works on "X and Not-X cannot exist at same time". No observation ever found a violation of that.

So when you encounter paradoxes just disregard them as rubbish. Thats what they truely are. You gain nothing contemplating on them.

An actual always-liar can never say "I always lie" while being always-liar. Cannot happen in this universe.

You can apply the same dont-pay-attention attitude to other paradoxes as well.

Look, some questions are indeed wrong (your school teacher is wrong). The question "When did your mother release from jail?" is wrong if the said woman never went to jail. There is no right answer to this question (yes, no, null are all wrong answers).

Any question that has an assumption thats not true is wrong. Ignore such questions.

  • half existing can be, but nothing named something. When glass is full by half and empty by half, half from both sides - that is half existing of something. you can say - he lied. So, lie is existing. Jan 27 at 6:59

I not understand 3#

And as i understand 1#=2#

So only liar paradox

Liar said: im liar

or little easy

lie is existing ...

as Gödel sad you can't fix paradox in it's own space, so you need a protospace.obviously.

Okey, lets talk about what is a lie. You have the nut in a core(a core in the shell). You don't know what is inside. But it is the nut. It is probably good, or probably wormy. You need a good one - crack! It is good - true! Another one - crack! Wormy - lire baker sold wormy nut, give my money back!


You need wormy one - crack! The best one - lie, lier-backer sold good nut to me! That means, lie depends on your waiting. Waiting is the predictable result.

If the prediction is same as the result - true.

If the prediction is not same as the result - lie.

But what is the prediction - it is default the TRUTH.

a LIE can't be a predicted, it is not a predicted result.

The TRUTH is exist as the prediction, a LIE is not exist as the prediction.

lie is existing - incorrect input (we have no full parts). Except lie=/=LIE or it is not a prediction.

if lie=/= LIE then lie it is local true mean, and true is local lie mean. local means confusion.

if it is not The prediction, then it is a result. If this it ll mirror image but back view. Where lie is exist and true is not exist.

okey, another nut - crack! Empty one...

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