A logical paradox is any piece of reasoning, often offered in the form of an argument, whose premises appear to be plausible but whose conclusion seems absurd or at least counterintuitive. A large collection of paradoxes has been collected throughout the history of philosophy. Along with Zeno of Elea, the ancient Greek philosopher Eubulides of Miletus (fourth century B.C.) is perhaps the most re-known discoverer of logical paradoxes and for this reason deserves a special place in ancient philosophy alongside with Plato, Aristotle, and just a few more. To him are attributed four of the most discussed arguments: the liar paradox, the sorites paradox, the paradox of the hooded man, the paradox of the horned man. The study of the paradoxes played a key role in the development of twentieth-century logic, epistemology, and metaphysics.
The Liar Paradox
The liar paradox deals with certain puzzling consequences of predicates expressing the truth or falsity of a sentence. Let’s make an example. The sentence you are reading is false. Is the sentence you just read true? If it is, then it is false, as that’s not what it says; if instead the sentence is false, then it is true, because that’s what it says. Or, consider this other case made up of two sentences. The next sentence is true. The previous sentence is false. Suppose that the first sentence is true; then the second sentence is also true; but this implies that the first sentence is false, contrary to the initial assumption. Suppose instead that the first sentence is false; then the second sentence is also false; but this implies that the first sentence is true, contrary to the initial assumption. Alfred Tarsky and Saul Kripke gave important contributions towards the solution of these sort of paradoxical reasoning; different solutions were offered upon the development of non-classical logics, such as dialetheism.
The Sorites Paradox
The sorites paradox exploits the indeterminacy of some terms in our language. The Madison Square Garden with just one person inside is not crowded; and if exactly twenty thousand people are in it, it is definitely crowded. But, at which exact number does it stop being not crowded and becomes crowded? Or, consider a man slowly loosing hair, one at a time: at which point does he become bold? Or consider building a heap of corn by adding one kernel at a time: at which point do you have a heap? In all those cases, it seems that one item more or less cannot make a difference; and yet it must be that some item or other makes a difference! The paradox has been at the center of a vast debate in logic, epistemology (theory of knowledge), philosophy of language, and metaphysics. It is indeed unclear what is responsible for the paradoxical character of these lines of reasoning: is it the bad logic of the arguments; or rather our ignorance of some matters of fact; or is the world indeterminate in some respects?
The Paradox of the Hooded Man
The paradox of the hooded man exploits the different senses that terms such as “to know” can acquire even within the same piece of reasoning. You know your philosophy professor; a person walks into the room with a hood on the head and you fail to recognize that it is your philosophy professor; so, you do not know your philosophy professor. Analogous paradoxes can be run for belief. You believe that Superman is a great guy; you believe that your colleague is Clark Kent; but you fail to believe that Clark Kent is a great guy. Solutions to this paradox will explain the nuances of knowledge or belief attribution.
The Paradox of the Horned Man
The paradox of the horned man plays with other contextual ambiguities, including different understandings of negation. What you have not lost, you still have. You have not lost horns, so you still have horns. On the one hand, you can be said not to have lost what you have, intending that you could not be said not to have lost what you never had. But, on the other hand, there is also a sense in which you can be said not to have lost what you never had.