全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking
It then follows that the witness is false, i.e., what Epimenides said is false. However, this conclusion does not entail a contradiction. To see that, let s find out what that
conclusion does entail.
a. The witness is false, i.e., what Epimenides said is false.
b. Epimenides said, “The Cretans are always liars.”
c. Therefore, The Cretans are always liars is false.
d. Therefore, Some Cretans are not always liars is true.
[by Proof] [by (2)] [by (a) & (b)] A proposition is the corresponding O proposition. Or, in the language of predicate logic, ~( x)(…x…) is equivalent to ( x)~(…x…) (by the quantifier negation rule).
e. Therefore, there is at least one Cretan who does not always lie, i.e., who tells the truth on at least one occasion.
[by (d)] f. There are three ways for (e) to be true:
i. Epimenides told the truth on this occasion,
ii. He told the truth on at least one other occasion, or
iii. There is at least one other Cretan who sometimes tells the truth.
: I can t think of any other.
g. (i) is false.
h. We have no grounds for denying either (ii) or (iii).
[by (a)] : The fact that Epimenides lied on the relevant occasion alone implies nothing of his other assertions or other Cretans assertions.
i. Therefore, either (ii) or (iii) (or both) is true.
So, it follows from Epimenides having spoken falsely when he said that the Cretans are always liars that at least one Cretan spoke, or will speak, truly on at least one occasion. And there is certainly nothing contradictory about that. In fact, I m sure it s true (and you are too).
If the only thing any Cretan ever had said, or ever would say, were, “The Cretans are always liars,” then we d be in trouble; but since that isn t the case, we re not.
Thus, Paul s blunder isn t genuinely paradoxical. So, it s not really a version of the Liar, but I include it for its historical and religious interest.4
III. Reductio ad Absurdum?
4 I can t resist giving one more fascinating version of the liar paradox due to medieval logicians. So, for the sake of argument let s accept the medieval claim that God exists is necessarily true. Consider the following argument: God exists. Therefore, this argument is invalid. Is the argument valid? (Any necessarily true proposition may replace God exists. )