16 Oct It’s true if false, and false if true
Epimenides the Cretan said that all Cretans lie; did he tell the truth, or not? Let us assume, for the sake of argument, that every Cretan, except possibly Epimenides himself, was in fact a liar; but what then of Epimenides? In effect, he says he himself lies; but if he is lying, then he is telling the truth; and if he is telling the truth, then he is lying! Which then is it? The same conundrum arises from the following sentence: “This sentence is false”. That sentence is known as the “Liar Paradox”, or “pseudomenon”. The pseudomenon obeys this equation:
L =not L.
It’s true if false, and false if true. Which then is it?
That little jest is King of the Contradictions. They all seem to come back to that persistent riddle. If it is false then it is true, by its own deﬁnition; yet if it is true then it is false, for the exact same reason! So which is it, true or false? It seems to undermine dualistic reason itself. Dualists fear this paradox; they would banish it if they could. Since it is, so to speak, the leader of the Opposition Party, it naturally bears a nasty name; the “Liar” paradox. Don’t trust it, say the straight thinkers; and it agrees with them! They denigrate it, but it denigrates itself; it admits that it is a liar, and thus it is not quite a liar! It is straightforward in its deviation, accurate in its errors, and honest in its lies! Does that make sense to you, dear reader?
The name“Liar” paradox is nonetheless a gratuitous insult. The pseudomenon merely denies its truth, not its intentions. It may be false innocently, out of lack of ability or information. It may be contradicting itself, not bitterly, as the name “Liar” suggests, but in a milder tone.
Properly speaking, the Liar paradox goes:
“This statement is a lie”.
“I am lying”.
“I am a liar”.
But consider these statements:
“This statement is wrong”.
“I am mistaken”.
“I am a fool”.
This is the Paradox of the Fool; for the Fool is wise if and only if the Fool is foolish! The underlying logic is identical, and rightly so. For whom, after all, does the Liar fool best but the Liar? And whom else does the Fool deceive except the Fool? The Liar is nothing but a Fool, and vice versa!
“It can not be too strongly emphasized that the logical paradoxes are not idle or foolish tricks. They were not to make the reader laugh, unless it be at the limitations of logic. The paradoxes are like the fables of La Fontaine which were dressed up to look like innocent stories about fox and grapes, pebbles and frogs. For just as all ethical and moral concepts were skillfully woven into their fabric, so all of logic and mathematics, of philosophy and speculative thought, is interwoven with the fate of these little jokes.” —Kasner and Newman,“Paradox Lost and Paradox Regained” from volume 3, “The World of Mathematics”
A good read on Paradox: