Oxford University Press's
Academic Insights for the Thinking World

Eubulides and his paradoxes

Who was the greatest paradoxer in Ancient Western Philosophy? If one were to ask this question of a person who knows something of the history of logic and philosophy, they would probably say Zeno of Elea (c. 490-460 BCE). (If one were to ask the same question about Ancient Eastern Philosophy, the person might well say Hui Shi (c. 370-310 BCE). However, my story here is about the Western side of the Euphrates.)

According to Plato in the Parmenides, Zeno wrote a book in defence of Parmenides, containing many paradoxical arguments. Sadly, most of these paradoxes have not survived, with one notable exception: the famous paradoxes of motion, reported to us by Aristotle. With arguments such as Achilles and the Tortoise, and the Arrow, Zeno argued that motion was impossible. Zeno’s arguments have been much discussed through the history of Western philosophy, and — arguably — were finally laid to rest thanks to developments in mathematics in the 19th Century.

However, for my money, the answer would be wrong. The greatest paradoxer is not Zeno, but the Megarian philosopher Eubulides of Miletus (fl. 4c BCE). Again, nothing he may have written has survived; but fortunately we have commentators such as Diogenes Laertius who report his arguments. Eubulides is reputed to have invented seven paradoxes. Some of these seem to be variations of others, so that what we have is essentially four different paradoxes.

One of these is the Horns Paradox: if you have not lost something, you still have it. You have not lost horns. So you still have horns.

Most philosophers would now regard this as a simple sophism. Indeed, it is a standard fallacy, often called the ‘fallacy of many questions.’ To say that you have or have not lost something would normally presuppose that you had it to lose in the first place. So if you never had horns, the two premises both have false presuppositions. Whether that makes the premises themselves simply false, or neither true nor false, one might still argue about; but at least they are not true.

This one of Eubulides’ paradoxes is not, then, profound. However, the other three are a quite different matter.

The most famous of the paradoxes is the Liar Paradox: someone says ‘what I am now saying is false.’ Is this true or false? If it is true, it is false; and if it is false, it is true. So it is both true and false.

The paradox and its variations were discussed by Ancient philosophers, and have been subject to much discussion in both Medieval and modern logic. Indeed, those who have engaged with them in the 20th Century reads rather like a roll call of famous logicians of that period. But despite this attention, there is still no consensus as to how to solve such paradoxes. Solutions are legion; but the only thing that is generally agreed upon, is that all of them are problematic.

Zeno – portrait for a library by Szilas. Public domain via Wikimedia Commons.

Another of  Eubulides’ paradoxes is the Hooded Man Paradox: a man enters the room, wearing a hood. You do not know who the hooded man is. But the hooded man is your brother. So you do not know who your brother is.

This is a paradox of intentionality. Intentionality is a property of mental states that are directed towards some object or other — such as the hooded man. The topic of intentionality and its paradoxes was also discussed by Ancient and Medieval philosophers. However, in the 20th century, some influential logicians (such as W. V. O. Quine) dismissed intentional notions entirely as incoherent, largely because of this sort of problem. This is no longer the case, and various theories of intentionality have now been proposed. However, I think it is fair to say that there is still no general consensus as to how, exactly, to solve this sort of paradox.

The last  paradox is — at least for me — the hardest of all, the Heap Paradox: if you have one grain of sand it does not make a heap. If you add one grain of sand to grains that do not make a heap, they still do not make a heap. So, starting with one grain, and adding grain after grain, you will never have a heap, even when the sand is piled up to the height of a mountain.

The Greek for heap is soros. Hence, paradoxes of this kind are called Sorites paradoxes. Such paradoxes concern predicates that are vague in a certain sense, like ‘heap’, ‘drunk’, ‘child’, where very small changes do not affect their applicability. Indeed, virtually all of the predicates one commonly uses are like this. These paradoxes, too, have an interesting history. There was some discussion in Ancient philosophy; none whatsoever in Medieval logic (that I am aware of, anyway); and virtually none in modern logic until the 1960s, when the topic took off spectacularly. (Why this should have happened so suddenly after 2,000 years, I have no idea.) Since then, a very large number of solutions have been proposed. But, again, there is no consensus as to how to solve paradoxes of this kind. All proposed solutions face very obvious problems. Crucially, they all seem to impute more precision to matters than there really is.

What my story tells, then, is that Eubulides invented three profound logical puzzles. The puzzles problematise three very different, but very important, philosophical notions: truth, intentionality, and predication. And despite an enormous amount of effort by logicians over the last two and a half thousand years, there is still no agreed-upon solution to any of them. This is why I say, that of Zeno and Eubulides, it is Eubulides who is the greater paradoxer.

This may well not have been how Eubulides was seen in his own day, however. Diogenes Laertius reports that one of the comic poets of the time wrote: ‘Eubulides the Eristic, who propounded his quibbles about horns and confounded the orators with falsely pretentious arguments, is gone with all the braggadocio of a Demosthenes.’ Well, Eubulides himself may have gone, but his paradoxes remain to haunt us.

Featured image credit: Plato’s Symposium by Anselm Feuerbach from Google Cultural Institute. Public domain via Wikimedia Commons.

Recent Comments

  1. Karl Young

    Modesty no doubt precluded Professor Priest from making this recommendation but for studying the basis of one (or more accurately a set of) interesting logical approach(es) to examining paradox I’d highly recommend his “An Introduction to Non-classical logic”.

  2. Alex Stylianos

    The first ones are easy to deal with. The Heap Paradox probably has some answers out there. One of them is what artificial intelligence calls fuzzy logic. Human mind think this way a lot, although we do not use it in our educational system. If you read the basics about fuzzy logic you will see that this kind of thinking supports multiple categorisation of ranges. Like for example short men can be from 0.01m to 1.72 and men of average height start from 1.70. So 1.70 to 1.72 is a height range that belongs to two categories. The same way you can claim that a heap starts from an X amount of grains of sand and goes up while at the same time there is a range that is still questionable if it’s called a heap.

  3. kimber chai

    Well, I tend to think that the paradox is a fault of the way that language is used. The first that you call to be one of “truth” is dependent upon something called sense and reference as outlined by Frege. You can’t take the sentence in both ways at the same time. It invariably leads to a paradox. Often self-reference itself leads to paradox and this is what is happening with the liar’s paradox. Either we are focusing on the sentence in a sense as a mere sentence, or we are assessing its reference to some previous sentence that has been taken out to purposefully distort the situation.
    In the second paradox of intentionality, I think that there is a sort of linguistic sophism that puts the conclusion before the antecedent. The axioms are distorted.
    The third, of predication, is exactly that. These types of concepts involve a distinction between measurement. These are arbitrary labels to be sure, but sometimes, things are a bit arbitrary no matter how much philosophers hate that term. It’s like asking why do we call a rose a rose? Just my humble opinion. kimber chai

  4. Sean South

    The Heap Paradox, in my opinion, is sorely missing from Western thought as a defining characteristic of human experience and being in the world. I also believe it to be at the root of Buddhist thought and philosophy in terms of “emptiness” and “Ultimate Reality” vs “Conventional Reality”. All we experience during our day is part of one “heap” or another that can be whittled away, little by little, until we realize that in fact we have made our own illusion as to what something (or someone) is or is not. I believe it should take center stage again in Western thought.

  5. Marina Pavlova

    Thank you for an interesting article. What springs to mind first about the Heap Paradox is the arbitrariness of the notion of “heap”. At the core of the paradox there seems to be a clash of approaches. At the “micro-level”, when individual grains are added up, the approach is rather mathematical/abstract by nature while the evaluation of what makes (or does not make) a heap takes place at a “macro-level”, which in this case represents the “practical”, or “common sense” perspective according to which, say, four or five grains by no means make a heap. But supposing the observer evaluating the heap is something like a “mathematical point”. Or rather, let’s say the observer does have physical dimensions but they are smaller than an individual grain. According to Oxford Dictionary, a “heap” is “a collection of objects or material piled up so as to form a high (frequently conical) mass.” From the perspective of such a tiny observer as few as four grains could make a structure which would technically meet the linguistic definition of “heap.”

Leave a Comment

Your email address will not be published. Required fields are marked *