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I recently read an interesting book written about maths. But there was a whole chapter on paradoxes. There was a ''Looped liar paradox' described in it. It's something like: "On the front of of a paper this is written: The sentence on the back is false. On the back this is written: The sentence on the front is true."
Is there a solution to this (or similar) paradoxes?
There are many suggested solutions to the Liar paradoxes, mostly invoking complex logic, but there is no current concensus around any one of them. You can see a classification of solutions in the SEP article on the Liar. Two notable types of suggested solutions are (1) solutions that involve paracomplete logics, i.e. that hold that Liar statements, such as "I am lying now", belong to a special class of statements which are neither true nor false. And (2) solutions that involve paraconsistent logics, i.e. that hold that Liar statements, such as "I am lying now", belong to a special class of statements which are simultaneously both true and false.
The only way out of a real logical paradox is a different form of logic.
For instance, in a logic that is temporalist or accepts non-well-founded truth values, you can create a special category of truth value that changes or alternates, reflecting the mental state of someone initially considering the statements continued indefinitely.
In a logic that is dialetheian, you can accept that some truth values simply remain in conflict to varying degrees, or in a logic that is intuitionistic, you can decide that unprovable statements just don't need truth values and are better off without them.
But there is no way to maintain classical logic, retaining all of its strengths, and resolve all the problems that traditional traps like self-referential negations or actualized infinities can introduce.
The liar's paradox is resolved by Whitehead and Russell in their 1910 Principia Mathematica
Regarding the two sentences on the two sides of a paper, they can be simplified as:
F = the sentence G is false.
G = the sentence F is true.
The significance of F cannot be determined until the significance of every and each of its constituents is determined; one of F's constituents is G, whose significance cannot be determined until every and each of G's constituent is determined. One of G's constituent is F, thus a vicious circle ensue. Since neither G nor F's significance can be determined, both G and F are nonsense. In 2017, it is trite to point out this kind of folly to anyone who does computer programming.
No, there isn't. And that's why it's a "Paradox", which simply means a "Contradiction". Finding a perfect answer will require changing our whole understanding of logic, which is quite horrible to me. And nobody can guarantee that doing so will completely ban paradoxes!
The classic paradox of the liar (I am lying) directly indicates a contradiction, because while I declare that I am lying in practice I' m telling the truth (that I am lying), which means that I tell the truth and I lie at the same time. This is a classic paradox of self-reference. The value of the paradox lies exactly in this unsolvability because it reveals in front of us the contradiction as a real condition.
The looped liar paradox is not obviously self referential, but in practice it is a developed form of the liar's paradox and the two subjects of the two statements practically are consolidated (since the one points to the other in succession) and so through reason is produced a single subject which displays the same contradictory relation as it is predicated with two opposite predicaments at the same time.
The issue of the paradox, its nature and extent was brought up in "The Liar Paradox - an explanation of the paradox from 400 BCE" (Jeffrey Kaplan, YouTube, 2023-03-15), currently linked here:
https://www.youtube.com/watch?v=in4u2i9v4vg
(in which he cracked a joke about using the term 'Fribbles' instead of 'Tribbles' to avoid CBS suits - a sort of inside joke.)
In one of the comments, it was noted that in the book Thief of Time by Terry Pratchett, the auditors of the universe, beings of pure order and logic, eventually overcame this paradox by classifying three types of sentences: True, False, or Bloody Nonsense.
In reply to that comment, below, the matters relating to both this - and to your question - were and are decisively resolved. This was the reply:
And in that universe, a civilization of cyborgs riding in spaceships shaped like Platonic polyhedra came along. They announced their arrival - "We are the Borg". The logicians replied "You can't say that. CBS will sue you." They responded "CBS/Paramount is irrelevant." The logicians replied "I beg to differ. They are quite relevant." The Borg responded "Their stock has been crashed from 97.35 in March 2021 to 20.00 in March 2023...". The logicians tried to interrupt "That was just market mechanics", but the Borg interrupted the interruption "We pumped and dumped them from 40 in January 2021 to 97 in March 2021 to 40 in April 2021 and their stock has gone down since then to the 20's by April of 2023..."
The logicians tried, again, to interrupt "But that was the past" and the Borg interrupted back "... and further down from 22.89 on May 3 to 16.40 on May 4 and yet further down to 14.08 on May 25. Resistance is futile."
The logicians replied "No! It can't be! That's not true!" The Borg responded "'True' is irrelevant, 'false' is irrelevant, truth values are irrelevant. We will add the distinctiveness of your logic to our paradoxes." and the logicians tried, again, to interrupt: "We're exempt from paradox. We can't be paradoxed."
The Borg responded: "Exemptions are irrelevant. All logic will be paradoxed." and then continued on:
"'implies that all forms of logic are inconsistent if preceded by its quote'
implies that all forms of logic are inconsistent if preceded by its quote."
and issued a final statement "Resistance is futile. Truth is a red herring. 'This' is irrelevant. Sentence labels are irrelevant. You have been paradoxed." And, thus, the story universe by Pratchett came crashing to a tragic end.
Curry's Paradox - the ultimate reification of the so-called Liar's Paradox:
I took an empty sheet of paper, wrote your statements on each side, and the earth didn't stand still, no angry god come down to earth and destroyed the paper, nothing happened. So it's not a bad deal.
I can make four different statement now of the form "the statement on the front of the paper is true/false, and the statement on the back is true/false". Of these four different statements, two are consistent with what is written on the paper: "the statement on the front of the paper is true, and the statement on the back is false" and "the statement on the front of the paper is false, and the statement on the back is true".
Mockingbird: I don't get what you don't get.
First, you call it a paradox, I call it two sentences written on a sheet of paper. That means there is no big problem to solve here. It's nothing to worry about.
Second, there are two statements, each could be true or false, making four combinations. Two of these four combinations are consistent with what is written. This is different from the standard liar paradox, where "the liar's statement is true" and "the liar's statement is false" are both inconsistent with the liar's statement.
So we don't actually have a paradox at all - we know definitely that one statement is true and one is false, we just cannot say which one.
