Paradoxes and Their Resolutions

A Thematic Compilation by Avi Sion

13. Protagoras vs. Euathlus

1.    An ancient paradox

One of the many alleged paradoxes that have come down to us from the Greeks is the dispute between Protagoras (of Abdera, ca. 480-410 BCE) and his student Euathlus (about whom nothing more is known). The story is told by Aulus Gellius (Roman, ca. 125-180 CE)[1], that Protagoras, a famous Sophist, and an expensive teacher, agreed with Euathlus to train him in rhetoric, a discipline essential at the time to argumentation in courts of law. The agreement was that Euathlus would not have to pay Protagoras the specified fee (or the unpaid portion of the fee, by some accounts) until he had been fully trained and went on to plead his first case and win it[2].

It is said that after Euathlus completed his course, he did not (for whatever reason) choose to use his newly acquired skills before any court of law, and so he never won or lost any case, and so was contractually not required to pay Protagoras anything. Nevertheless, Protagoras, with motives that we shall presently consider, sued him (in the court of Areopagus in Athens). Euathlus chose to personally defend himself. The following arguments were reportedly put before the court:

  1. Protagoras argued that he surely ought to be paid the fee, because (i) if the court ruled in his favor, he could on that basis demand payment; and (ii) if the court ruled in favor of the defendant, then the latter would have won his first case and therefore be contractually obliged to pay the fee anyway.
  2. Euathlus replied that he surely ought not to pay the fee, because (i) if the court ruled in favor of the plaintiff, then Euathlus would have lost his first case and therefore not be contractually obliged to pay the fee; and (ii) if the court ruled in favor of the defendant, then he would on that basis be exempt from payment anyway.

Thus, while the plaintiff argued, apparently convincingly, that he was certain to deserve payment however the court ruled, the defendant was in turn able to argue, apparently just as convincingly, that he was certain to be exempt from payment in either event. For this reason, this case is regarded as paradoxical. It is said that the court was so confused by these arguments and counter-arguments that it chose to adjourn sine die (or, some say, for 100 years) to avoid judgment.

The significance of this legal dispute for logic and philosophy is that it gives the impression that two people can argue dilemmatically and paradoxically in opposite directions and both be right. The enemies of human reason relish this kind of conclusion, since it would put in doubt the reliability and efficacy of human reason. But as we shall now show, the said impression is very superficial. There are, to my mind, at least two possible resolutions of this so-called Paradox of the Court. In fact, although I thought of the second before the first, the latter logically precedes the former. I later learned from Peter Suber’s survey of the literature on the subject[3] that the first resolution was long before me proposed by Aulus Gellius; the second resolution seems to be novel.

2.    First resolution

The simplest solution to this problem is to suppose that the wily Protagoras, seeing that Euathlus was taking his time getting to work, decided to speed things up. Protagoras trapped his pupil by using the above argument, knowing full well that he would lose a first trial, but win an eventual second trial. He knew he would lose a first trial, because the agreement between the parties only obligated Euathlus to pay the fee once he won his first case; it did not obligate him to practice law anytime soon, or even ever. Euathlus foolishly fell into the trap and personally argued the case in court. Had the court not adjourned sine die, it would have logically given him victory, thus making Euathlus win his first case. Thereafter, assuming that a second trial was legally permitted – and both the parties’ arguments above do make this assumption – Protagoras would have been the ultimate winner. Of course, no second trial would be necessary if Euathlus conceded that having won the first trial he was sure to lose the second, and settled the account forthwith.

In other words, Protagoras’s first argument (i) was mere camouflage; he was really relying on his second argument (ii). Euathlus let his vanity get the better of him and formulated two fancy counter-arguments, thinking to outdo his teacher. But Protagoras was more cunning than him. The only way Euathlus could have avoided being beaten was by hiring a lawyer[4]. If the lawyer won the first trial, as he could be logically expected to since the only condition for defeat here was not satisfied, Euathlus would not be considered as having personally won his first case in court, and Protagoras would not be able to win a second trial. Thus, the master was indeed superior to the pupil. There was no real paradox, since there was an actual way out of the apparent paradox.

Of course, one might add that Euathlus was in practice the winner, since through his counter-argument he managed to so bewilder the court that it gave up trying to judge the matter at all, and he was not forced to pay up. Maybe he hoped for that and he lucked out. But on a theoretical, logical level, in the present perspective, he proved to be not too intelligent. Not only did he foolishly not hire a lawyer to plead on his behalf, but he also wrongly assumed that his argumentation was effective in countering Protagoras’. He kidded himself into thinking that if he won the first trial, he would not have to face a second one. He should have examined his teacher’s argumentation more carefully.

Let us now look at the arguments in more formal terms, to clarify them. I shall introduce the following symbols: let P = Protagoras, Q = Euathlus; A = the agreed condition that Q wins his first case in order to be liable, and C = the agreed result that fee must be paid by Q to P. We know at the outset that if A then C, and if not-A then not-C: these are the terms of the agreement. The arguments are as follows (with my critical commentary in italics):

  1. According to P: (i) if court rules that P wins, then C is true (but objectively court cannot rule for P, since A not yet true, so this is a non-starter); whereas if (ii) court rules that Q wins (as it logically must), then not-C is true (at the conclusion of a first trial); but if Q thus wins, then A becomes true and C must (in a later trial) follow, in which event P finally wins.
  2. According to Q: (i) if court rules that P wins, then not-A is true (at the conclusion of a first trial); but if not-A is true then not-C must (in a later trial) follow, so Q finally wins (but objectively court cannot rule for P, since A not yet true, so this is a non-starter); whereas if (ii) court rules that Q wins (as it logically must), then not-C is true (but here Q fails to mention later consequences that P rightly pointed out).

In conclusion, P is logically the resultant winner; Q’s arguments are in fact insufficient to prevent this outcome. P pretends to seek to win immediately; but in truth his aim is longer term victory (in the second round). Q imagines he might lose the first round but win the second or that he might win the first round without having to face a second; but these are all fantasies. It is difficult to understand why the court found this case too confusing – the judge (or judges) can’t have been very bright fellows.

3.    Second resolution

A more complex solution to the problem is as follows. It is possible that Protagoras sued Euathlus by appealing to an unspoken clause of the agreement. The agreement contained only one explicit clause, viz. that Euathlus would have to pay Protagoras the fee if and only if he won his first case. If that was so, Protagoras would have no basis for requesting a trial, since that condition had obviously not been satisfied. But since he sued, he must have thought and argued that the agreement included a tacit (or perhaps implicit) understanding that Euathlus would practice law within a reasonable time lapse[5], at which time his new skills would be tested in a court and he would be expected to pay if he won. Protagoras couldn’t have imagined the judge would allow a trial to proceed, let alone would rule in his favor, without some good reason[6].

Clearly, what the judge was called upon to decide in this trial was (could only have been, in the present perspective) whether this claim by Protagoras, that there was a tacit clause to the agreement, was justified. He could well have justified it by considering that had Euathlus been allowed not to practice law at all or to practice it as late as he chose, the agreement would have surely specified the caveat. He could equally well have rejected it by considering that Protagoras took for granted something he should have explicitly obtained agreement on. So, the case hinges on a tacit issue, rather than exclusively on the explicit clause of the agreement; i.e. there was more to the story than is told.

Furthermore, it is evident from the arguments presented by both parties in this trial that each of them foresaw the possibility of a second trial in which the ruling of the first trial could be reversed. This is logically implied in the second argument of Protagoras and in the first argument of Euathlus. In these two eventualities, a second trial would be needed to finalize the judgment; unless, of course, the losing party freely concedes its inevitable result and settles the account in advance. The judge in the first trial could not decide in favor of either party and then against him in the same breath. A first judgment of win or loss would have to be established before a second judgment could be made, in view of the terms of the contract. So, there is a time factor to take into consideration in analyzing these arguments.

It should be noted that, whereas the first trial has an uncertain outcome, since it depends on the decision (the judgment call) of the judge regarding an alleged tacit clause to the agreement, the second trial has a foreseeable outcome, since it depends solely on the explicit clause of the agreement.

Clearly, as we shall now show, if we factor the above considerations into the arguments, the paradoxical appearance is easily dissolved. I shall here use the following symbols: let P = Protagoras; Q = Euathlus; A = the explicit condition that Q wins his first case; B = the alleged implicit condition that Q was required to practice within a reasonable amount of time; and C = the fee must be paid by Q to P. The arguments are as follows:

  1. According to P: (i) if B then C; and (ii) if not-B then not-C, but if not-C then (later) if A then C; therefore, C anyway.
  2. According to Q: (i) if B, then C, but if C then (later) if not-A then not-C; and (ii) if not-B, then not-C; therefore, not-C anyway.

From this we see that the two parties’ arguments are much alike, but each side has cunningly left out part of the consequences (shown in italics). P has truncated the consequences of (i) that Q points out, and Q has truncated the consequences of (ii) that P points out. For this reason, they seem to balance each other out. But if we take all the subsequent events (in a possible second trial) into account, we get the following more objective joint argument (c), which is clearly non-paradoxical:

  1. if B then C, but if C then (later) if not-A then not-C; and
  2. if not-B then not-C, but if not-C then (later) if A then C.

So, in (i) the final conclusion is not-C, while in (ii) it is C – which means that there is no paradox. This also shows that, while it cannot objectively be predicted whether P or Q will win the first trial, it can be said that (i) if P wins the first trial, he will lose the second, and if Q wins the first trial, he will lose the second. Obviously, P cannot argue that he has a right to a second trial (if he loses the first) but Q has not; likewise, Q cannot argue that he has a right to a second trial (if he loses the first) but P has not. So, we must take all later events into consideration to logically reconcile all the arguments. Note that if for some reason there is no second trial, there is also no paradox, since the conclusion will be either (i) C or (ii) not-C.

Whatever happens, there is no paradox because neither party can in fact, contrary to initial appearances, claim inevitable victory; victory does not come both ways for both parties, but only one way for each party.

Clearly, here, both parties were employing the common sophistical trick of hiding an inconvenient part of the unbiased argument from the court. Euathlus was a good apprentice of Protagoras’, since his counter-argument exactly mirrors the latter’s argument. That is, there was an element of dishonesty in both their arguments; both were intellectual frauds at heart, knowingly expounding half-truths. Therefore, this fake paradox presents no deep challenge to Logic, contrary to the claims of Relativists. In particular, Protagoras’ general claim that “there are two sides to every issue” (duo logoi) is shown to be spurious in the present context.

It is worth always keeping in mind that some people involved in philosophy and logic, as in life in general, are sometimes moved by the evil impulse; indeed, some much more than others. They may consciously lie, or subconsciously twist facts and arguments, for a large variety of motives. Usually, lusts for power, fame and fortune play some role. An academic may want the admiration of his peers or of his students; a husband may want to impress his wife; an unemployed may hope to get a job; and so on. It is wrong to look upon all philosophical statements as disinterested. Philosophers and logicians are not all pure scientists or saints.

I might add that the secret of success with finding solutions to philosophical and logical problems, and particularly to paradoxes, is the sincere desire to do so. Many philosophers and logicians approach problems with a negative attitude, not really wanting to solve them, but rather wishing to rationalize their antipathy to human reason through them. The honest researcher is moved by his better impulses; he is sincerely desirous to confirm the effectiveness of the human faculties of cognition – that’s precisely why he succeeds.

4.    Inadequate resolutions

As earlier mentioned, based on Suber’s account of the literature, the first resolution should be attributed to Aulus Gellius, but the second resolution seems to be original. Suber’s account shows that the court paradox has been discussed in a number of works over time, but more often apparently from a legal point of view than from a logical one. The legal issues involved are manifold, but most need not and should not be taken into consideration in a purely logical perspective. Why? Because the logician’s purpose here is not to decide the case, i.e. who should win or lose, but merely to explain and remove the appearance of paradox.

This remark can be illustrated with reference to the resolution (not mentioned by Suber) proposed centuries ago by Lorenzo Valla (Italy, ca. 1406-57). This attempt at resolution is not adequate because it relies on a thoughtless distinction between payment on account of the court’s verdict and that on account of the agreement. I quote an SEP article which describes it[7]:


“If Euathlus loses the case, he will have to pay the rest of the fee, on account of the verdict of the judges; but if Euathlus wins, he will also have to pay, this time on account of his agreement with Protagoras. Euathlus, however, cleverly converts the argument: in neither case will he have to pay, on account of the court's decision (if he wins), or on account of the agreement with Protagoras (if he loses).… Briefly put, Valla says that Euathlus cannot have it both ways and must choose one or the other alternative: he must comply either with his agreement with Protagoras or with the verdict passed by the judges. If they decide against Protagoras, he may try to reclaim his money in a second lawsuit.”


Let us analyze Valla’s proposed resolution using the following symbols: let P = Protagoras; Q = Euathlus; and C = the fee must be paid by Q to P.

  1. According to P: If Q loses (for whatever reason), then C (by verdict); but if Q wins, then C (by agreement).
  2. According to Q: If Q wins (for whatever reason), then not-C (by verdict); but if Q loses, then not-C (by agreement).

Valla’s conclusion, as here presented[8], is unclear. Apparently, he puts the onus on Euathlus in particular to “choose one or the other alternative” and comply with either the agreement or the verdict. This tells us nothing, since Euathlus’ argument shows he is willing to comply with either, except that he projects both as in his favor. Valla adds that if Protagoras loses a first lawsuit, he may win a second. But here again, this does not resolve the paradox, but only repeats one part of it.

From this we see that Valla has not thought the issues through: he does not consider on what grounds Q might lose or win ‘by the court’s verdict’; and he does not realize that the second argument by each party, where Q might alternatively win or lose ‘by the agreement’, in fact refers to a second lawsuit, since the agreement is evidently not about to be implemented voluntarily (Valla’s mention of a second lawsuit is placed beyond the four if-then statements, which themselves do not emphasize the temporal sequences involved). In the last analysis, then, Valla does not arrive at a ready resolution of the paradox.

The trouble with Valla’s approach is that it effectively takes the initial decision of the court to be arbitrary, i.e. unrelated to the agreement between the parties. It does not consider on what basis the court might judge that Euathlus can lose the first round. This may be acceptable legally, but it is not acceptable logically – and our concern here is with logic. Logically, the idea that Euathlus can lose the first round is a non-starter, if we go by the explicit clause of the agreement. He might lose the first round only if the court grants the supposition that there was a tacit clause to the agreement, such as that he had to practice law sometime soon.

In any case, the verdict for the first round cannot be arbitrary – i.e. irrational, unjustified - but must directly relate to the agreement. This is equally true for the second round. So, no disconnect between verdict and agreement is logically permissible.

Interestingly, my first reaction to the paradox a few years ago[9] was very similar to Valla’s. But as soon as I set about seriously considering the issues for the present essay, I realized its weakness and uselessness. The paradox appears neutralized if we insert a distinction between ‘payment following court verdict’ and ‘payment following contract terms’ and assign different symbols to these two consequences, say C1 and C2. These two terms may or may not be quantitatively identical; but they anyway refer to distinct events. This measure seems to nullify the paradox, because the consequences of the first and second if-then statements would be different for each party. Thus:

  1. According to P: If P wins, then C1; but if P loses, then C2.
  2. According to Q: If Q wins, then not-C1; but if Q loses, then not-C2.

It follows that if P wins and Q loses, then C1 and not-C2 are true; and if P loses and Q wins, then not-C1 and C2 are true. Since C1 and C2 are different terms, even if they happen to refer to the same monetary amount, the consequents in each party’s argument are not formally contradictory (since the defining motive is different), so there is no paradox. But, to repeat, this approach does not explain why the court would give a verdict inconsistent with the agreement, so it is artificial. For a genuinely logical resolution, we must focus attention on the agreement, and not admit arbitrary verdicts. Arbitrary verdicts just muddy the waters.

Other attempted resolutions I have seen are also flawed, either because they similarly conflate legal and logical issues, or because they do not follow through on all the consequences of all the suppositions. In any case, either of my proposed two resolutions suffices to unravel the paradox; but the two together take care of all eventualities. Note that, in the first resolution, we not only resolve the paradox, but also incidentally decide the case (in favor of Protagoras); whereas in the second resolution, we are content as logicians to resolve the paradox, leaving the task of legal decision to the judge.

To conclude: the paradox of the court is due to a number of factors, which must be untangled and taken into consideration if it is to be resolved. (a) The terms ‘win/lose’ cannot refer to arbitrary judgments by the court; if they do, the paradox may be perpetuated. (b) It must be realized that the arguments put forward by the two parties imply that the process of resolution has two phases: a first trial followed by a second trial (or by a ready concession and settlement without need of a second trial); the appearance of paradox is partly due to overlooking this time factor. (c) There is some vagueness in what is meant by ‘the agreement’, and all possible interpretations must be taken into account: if the agreement is taken to refer exclusively to the explicit clause, then the first trial concerns that only and is easily decided; but if the agreement is claimed to involve a tacit understanding, then the first trial aims at a decision regarding that tacit clause and the second round deals with the explicit clause.


From Topics (published only online so far).


[1]              Some 550-600 years after the fact, in his Attic Nights (ca. 150 C.E.).

[2]              The contract in question was presumably verbal, rather than written, in those days; but we may take it that both parties agreed on its stated clause.

[3]              In The Paradox of Self-Amendment. The section on the Protagoras v. Euathlus paradox can be read online; it is worth reading, including the notes, and not very long. See at:

[4]              Assuming this was possible in Athens in those days. This is a fair assumption (even if some commentators deny it) since, after all, Euathlus was apparently trained by Protagoras to be a lawyer himself.

[5]              Protagoras would also, of course, claim that the reasonable delay had expired. If the court agreed with the existence of a tacit clause but disagreed with the claim it was fulfilled, that would merely adjourn the case for a certain amount of time (of their estimate).

[6]              Needless to say, the present analysis is made entirely from a logical viewpoint, although a court of law might reason differently or even not reason at all (e.g. bribery, favoritism, pressure).

[7]              See Stanford Enc. of Phil., online at This issue is apparently treated in Repastinatio dialectice et philosophie. The author of the article is apparently Lodi Nauta (2013).

[8]              I have not read Valla’s work. I have to assume that the author of the SEP article on Valla correctly presented Valla’s reasoning. I suspect he or she may not have, as the details given in the article are rather vague and inconclusive; they do not clarify exactly what resolution of the paradox Valla had in mind. Not everyone is good at logic. (Note that I did write to the author, asking him or her to please verify the summary of Valla’s view given in the SEP article, but I got no reply. That the author did not deign to respond confirmed my estimate that this is not a very reliable source: intelligent people confidently welcome reasonable queries.)

[9]              In 2009, in an e-mail to someone.

Go Back


Blog Search

Blog Archive


There are currently no blog comments.