The Lair’s and other paradoxes.
Wim Dekkers, 5 April 2026
The question of logical paradoxes keeps many a philosopher busy. There are many different types of philosophical paradoxes, in this essay I will look into logical paradoxes, that seem to make no sense in the world of logical thinking, as their truth value seems to be impossible to establish. They are called antinomies, and are described as producing a self-contradiction by accepted ways of reasoning. In this category we have Bertrand Russell’s paradox about the barber who shaves all men in the village that do not shave themselves, and Russell’s famous paradox of the class of all classes that are not members of itself. Then there is the following logical paradox: is heterological heterological or autological (Grelling’s paradox) ? And a last famous example: all Cretans are liars, said by Epimenides the Cretan, the paradox of Epimenides, or the Liar’s paradox.
For philosophical logicians the antinomies paradoxes are deep logical contradictions, that no one understands. The question in this essay concerns the paradox of Epimenides specifically, a version of what is also called the Liar’s paradox. This paradox is especially confusing for logicians as the subject-matter is the concept of “truth”. Of course the “truth” is always involved in all the antinomies paradoxes, as logic defines these paradoxes as not being either true or false: it is impossible to define their truth value and that is why they are paradoxes in the first place. So, instead of being about men’s facial hair, mathematical classes, or adjectives, the Liar’s paradox is about the subject matter of “truth”, and as the other three paradoxes, the problem is that no truth value can be found. This is of course extra confusing, as the subject-matter and the reason for why it is a paradox both concern the “truth”, and now these two parts of the paradox, the subject-matter and the self-contradiction, are easily mingled in our human reasoning about it.
I will argue that all 4 of these paradoxes are simply erroneous reasoning, as a result of a wrong and a very confusing use of words. I will also argue that the problem about the Liar’s paradox has nothing more to do with the concept of “truth” as any of the other examples, and that the subject-matter, whether it is the “truth” or men’s facial hair or mathematical classes, has nothing to do with what the problem of confusion really creates. To do this I will analyse all four of our examples, to show the repetition of the same type of reasoning error made. The problem, that creates contradictions, lies in the meaning of the words, not in anything to do with formal logic or truth-value. I will start explaining all four mentioned logic paradoxes:
First, the barber shaves all men that do not shave themselves. Who shaves the barber ? If he does not shave himself, he belongs to the group of men that do not shave themselves, and so he has to shave himself, which is a contradiction. The problem is the definition of the barber, as someone who shaves the men of the island that do not shave themselves. This is a circular definition in the case that the barber is himself a man of the island that does not shave himself. Circular definitions are considered fallacious as they define a term in terms of themselves, and therefore do not define anything at all, but open an infinite circular reasoning. They are also begging the question. Now, on top of that, (and important for the creation of a contradiction), if you add a negative in the definition, this become immediately a contradiction. As in “this thing = not this thing”. In this case you would get something like “the man who is not shaving himself = not (the man who is not shaving himself)”. A man cannot at the same time and shave himself and not shave himself. So, the repetitive term in the circular definition, with an added negative, creates automatically a contradiction.
Next, is the adjective heterological heterological or autological? Heterological is the name for adjectives that do not describe themselves (like, German or monosyllabic), autological for adjectives that do describe themselves (like English or short). I think heterological and autological cannot be seen themselves as adjectives as they would then become circular definitions. They use already explicitly in their definition the concept of adjective, so you get the adjective is the adjective…which is circular. Added the negative of heterological and you get automatically a contradiction.
Then, the class of all classes that are not a member of itself. Is this class a member of itself? Again, as in the examples above: if you define a class with explicitly using itself in the definition, as happens already with classes that are defined as members of themselves, you have a circular definition (I can imagine that an infinite regress starts here in mathematics, but I am not sure, as I am not a mathematician). If you use the definition of classes that are not members of themselves, to define a new class, this definition of class is not only circular as it uses explicitly itself in it’s definition, but because you added the negative it herewith becomes a contradiction. This is how the paradox is born: a circular definition with an added negative in the definition.
Finally, all Cretans are liars, says the Cretan. Is the Cretan lying ? The circularity of the definition is clear and explicit: he, Epimenides, a talking Cretan that tells the truth, says all talking Cretans do not tell the truth. The problem is again the negative circularity in the definition. A talking Cretan (telling the truth, implicitly) defined as a talking Cretan not telling the truth. This is both a circular definition and, again, because of the negative in the definition (not telling the truth), it is also a contradiction.
So, here we have our paradoxes, which are all four circular definitions that do not define anything much but are fallacious as definitions and begging the question. But with an added negative in the definition they become contradictions. It is like defining a dog as being not a dog. Defining a dog as a dog is problematic, but defining a dog as not a dog is double problematic, and a contradiction. So that is the problem of the confusion of the 4 paradoxes, which is actually quite complicated erroneous use of language, that creates such confusion so that you do not see anymore what is actually the trouble: a double fallacy with the circular definition and a negative added that creates a contradiction.
So, we are here not arguing about self-reference, as is often done in discussions about these logical paradoxes, but about circular definition, which is a very special case of self-reference, with very special logical consequences. The added negative is finally responsible, in combination with the circular definition, to create the contradiction.
Now, if we go back to the second part of our question, whether one’s theory of truth makes a difference in whether the Liar’s paradox is a threat, I think it is now clear that that is not the case. That the Liar’s paradox has lying and telling the truth as it’s subject matter, just like the barber paradox has men’s care of facial hair as it’s subject matter, is not relevant for why it is a paradox. Like anyone’s ideas of men’s care of facial hair does not make the barber’s paradox more or less of a paradox. The problem with the paradoxes is not in the subject-matter but in the use of circular definition with a negative added to it. The fact that the subject-matter in the liar’s paradox is the concept of “truth”, and that the general problem of logical paradoxes is the impossibility of defining a truth value, just confuses even more the logical thinking about it, but it does not create the paradox.
Meaning exceeds classical logic.