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This question is exercise 1 from Manuel Bremer's An Introduction to Paraconsistent Logics.

The question

Often the sentence given as the Liar example is "All Cretans are liars." said by a Cretan.

Why does this not work as an antinomy if there is more than one Cretan?

I really don't understand why if there are more than one Cretans this isn't a paradox, can you please explain to me?

Thanks!

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  • I’m voting to close this question because the Liars paradox is game theory, not philosophy.
    – Swami Vishwananda
    Commented Jan 18, 2021 at 4:09
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    @SwamiVishwananda Liar's paradox is about logic, and logic is part of philosophy.
    – MathematicalPhysicist
    Commented Jan 18, 2021 at 10:45

2 Answers 2

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There are, in fact, two problems with the "All Cretans are liars" paradox.

The first is that a liar is not a person who tells nothing but lies, it's a person who tells (some) lies. If all Cretans are liars, one could honestly tell you so. (But if you did redefine it to mean "all the time", a Cretan can lie and tell you that all Cretans are liars and not even be one himself -- he tells the truth some of the time, just not now.)

The second, the one you are referring to, is that the opposite of "All Cretans are liars" is not "No Cretans are liars" but "Some Cretans are not liars." A lying Cretan could claim that they all are, and thus make false claims about some other Cretans, without forming a paradox. He's a liar, and he's lying when he says all Cretans are.

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  • OK, thanks. You are indeed correct, if he were referring to himself only then that would be the old classical paradox.
    – MathematicalPhysicist
    Commented Jan 17, 2021 at 19:32
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    I've always interpreted it as simplified language in which "liar" does mean someone who tells nothing but lies.
    – Barmar
    Commented Jan 18, 2021 at 1:10
  • @Barmar If it's a person who tells nothing but lies, a person who claims to be a liar could still be lying because he occasionally tells the truth.
    – Mary
    Commented Jan 18, 2021 at 13:52
  • For the purpose of resolving this as a non-paradox it's not necessary to make the assumption from the middle para. The reasoning from your last para works just as well if the def taken for liar is that they lie all on every utterance. Basically if there are two Cretans X and Y, X lies all the time, and X says "X and Y are liars", X is true to its nature and just said a lie in the case/model where Y is not a liar.
    – got trolled too much this week
    Commented Apr 4, 2021 at 13:01
  • Only if you want to consider statements like "1+1=2 and 1+2=4" as "partly false" (instead of just false) does it make sense to bother with a weaker notion of liar (i.e. someone who lies some of time). Basically, with a classical valuation of the conjunction (and likewise for the universal quantifier "all") and only two truth values for sentences, the "liar lies all time" def works fine.
    – got trolled too much this week
    Commented Apr 4, 2021 at 13:13
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Lets call the Cretan Ari. If his statement is true then all Cretans always tell untruths. But then Ari's statement is itself untrue. So, if Ari's statement is true then it is not true, which proves that Ari's statement is not true. Since it is not true, then at least one statement of at least one Cretan is true (note that this is certainly not Ari's statement, as we have proved that it is not true). This is not a logical contradiction.

However, we managed to prove that at least one statement of at least one Cretan is true without any insight into the statements of the Cretans. I think we can consider that a paradox.

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