# What is the solution to Protagoras' Wager?

Asked by Smashley (4110) November 15th, 2010

Perhaps there is no solution but I thought I’d give it a go:

Protagoras teaches law (well, “sophistry” but you get the point). Eulathus is one of his students but has no money to pay tuition. The two make an arrangement: Eulathus will pay the tuition after he wins his first case. Indeed Protagoras guarantees that Eulathus will win his first case. Upon completing his training however, Eulathus doesn’t argue any cases and never pays. Annoyed, Protagoras sues his former pupil.

“Either I win and you must pay me, or you win, and thus having won your first case, you must pay me.”

“No,” says Eulathus. “If I win I am not required to pay, and if I lose, I have lost my first case, and am not required to pay you.”

Is there a logical error in this paradox? Who is right? Can they both be right, or are they both wrong?

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## 17 Answers

It seems they’re working with two opposite standards: those of the trial, which say Eulathus has to pay if he loses, and those of their wager, which say he has to pay if he wins. Both of them try to use half of the rules from the trial, and half of the rules from the wager, to turn the situation into a win/win situation for them.
They should be fair, and choose either trial rules or wager rules, by the flip of a coin if they have to. They clearly can’t use both at the same time.

Fyrius (14525)

In order for the student to be required to pay, there has to be a contract. In order for there to be a contract, then there needs to be an offer, acceptance, consideration, and according to the statute of frauds for contracts lasting more than one year, a reduction to writing. The terms must also be sufficiently definite so that a court of law could determine the terms of the contract.

Of course, is this a logical or legal problem? If it’s just a logic problem, I doubt we’ll get to the answer.

iamthemob (17135)

P.S.
But yes, it is a paradox. Protagoras’ suing his student has put them in a logical stalemate. They can resolve it by agreeing to disregard one set of rules.

Fyrius (14525)

@Fyrius – I presume that since there is no specific requirement that they change the rules, so neither would agree to one, as a concession from either one might injure their logical constructions. A coin flip might be “fair” but who would agree to it when they believe they are guaranteed to win anyway? It’s certainly a case of “which rules do we follow?” so what’s the answer? Is it just “might makes right”? That is, if the court or Protagoras has the power to force Eulathus to pay, then Protagoras is right, and if neither does, then Eulathus is right?

Smashley (4110)

@iamthemob – the contract, an oral one, seems binding enough to both parties, and they are both attempting to stand behind it, so there is a mutual acceptance of it.

The terms could be described as: A teaches B. B will not pay A for the service until such time as he argues and wins his first legal case. Should B not win his first case, he is not required to pay.”

The suit would then follow: “B has deceived A, choosing not to argue any cases, and thus A will never receive payment for the services given. Is their agreement now invalid, and must B pay A immediately?”

I suppose it’s a logical paradox, but only a legal dilemma. Ultimately, one side is going to win, and will be declared “right.”

Smashley (4110)

@Smashley – the issue, of course, is what constitutes a “win” in this case. We could resort to trade terms in this case. If we’re going to do that, then the contract isn’t going to be enforceable at this point because a “win” would mean the litigation of a case of another in the court of law and winning such case for a client. The contingency on which the contract becomes enforceable is the legal win.

If we don’t, and it’s unclear what a “win” is, then the problem is this ambiguity goes to the heart of the contract. Therefore, it is unclear when the contingency occurs. Additionally, the contingency does not clearly specify a time upon which the payment must occur – it’s simply “after.”

There’s also the issue of whether this is law school. If it is, then there is no way that the contract can be enforceable if not in writing as law school is three years (at least in the U.S.). If it takes anything equal to or longer than a year to complete training, then the contract cannot be completed within a year, and it has to be reduced to writing.

And if the meaning of win allows for a case within the time of training, then there really isn’t any consideration, meaning that the process was meant to delay payment until the profession had started. If the contract is written in a manner where payment can be demanded at this point when the student is still a student and still has no money, then the student has received nothing but has to pay for it. ;-)

There just doesn’t seem to be any contract.

iamthemob (17135)

I think that idiot teacher should have sued the pupil for something else and lost intentionally, but noooo, he had to create a paradox. What a fool.

ragingloli (40218)

I side with the student’s logic, it beats that of his teacher.

I do not believe it is a stalemate. Rather a moral choice.
If he should lose in court he has no choice but to pay or face a lien of property which would be enforceable,
However should he win he only has a moral obligation which is only enforcable in your own mind. i.e. “I feel guilty” kind of thing.

@ragingloli – That’s an amazingly simply and good point.

@iamthemob – I agree that there is an implication in the contract that the “win” must be done for a paying client, and that is perhaps part of the absurdity of the question. The assumption is that the “contract” is considered valid by both parties, as they both stand behind it to support their own argument, and are willing to abide by it to the letter. But there is a second part to the problem: both men consider the judgment of a court to be as valid as their agreement. They are simultaneously agreeing to enforce their “contract” and abide by the outcome of the suit. Are these declarations mutually exclusive? Is this where the paradox is? Since the contract and the legal decision will necessarily contradict each other, mustn’t one take absolute priority over the other?

@thekoukoureport – but if the student wins (and is thus responsible for paying, based upon his “win”) wouldn’t that be a separate suit altogether, with fewer paradoxes, regardless of his moral obligation? Whereas the first suit is more ambiguous, the second one would simply be “He won a case and he hasn’t paid. Make him pay!”

@Simone_De_Beauvoir – That was my initial thought, too, but could you unpack that thought a bit?

Smashley (4110)

@Smashley It was my initial reaction to your details. When the teacher said “If I win, you must pay me”, he was referring to, imo, a new situation, not the one they made a contract about. The student in his response was following the initial contract’s rules and therefore his logic, to me, is more sound.

@Smashley
I think in this situation it would be in either party’s best interests to resolve the stalemate. As it is, they just don’t agree about the results of winning or losing the trial. Protagoras wants to be paid. Eulathus doesn’t want to pay, but he also wants Protagoras to stop bugging him about it. Unless they can agree on a way to settle the disagreement, neither of them is going to get what he wants.
If they decide to forego the wager and just follow whatever is the outcome of the trial (because otherwise Eulathus could cheat by being a crappy defendant on purpose), then Protagoras gets a realistic chance of getting paid, and Eulathus gets a realistic chance of Protagoras getting off his case and leaving him alone.

Maybe I’m going about this the wrong way. It’s a logical thought experiment, but I’m approaching it more like a problem in game theory.

@Simone_De_Beauvoir
By those standards, only half of his logic is sound; the other half – about what happens if the trial is won – abandons the initial contract and goes with the rules of the trial instead of just paying the heck up. The same goes for the teacher’s logic, but reversed.

Fyrius (14525)

@Fyrius Ah, I see what you’re saying.

@Smashley – Really interesting points! Thinking about it…though…contracts are only enforced through legal means when one party has breached the contract in some way, shape or form. Otherwise, the parties abide by the terms of the contract, and the courts are never involved. In essence, the contract is an assurance that if one party backs out, the other can use the law to enforce the obligation to them in some way, but usually to get the damages associated with the value they lost.

Therefore, since there is a disagreement about the terms, the court will look at the contract. In order for it to do so, there must be terms that it can interpret. And in general, it will not look outside the contract unless necessary to interpret the contract. If that court cannot do so, and apply the concrete terms presented to it, it will declare that there was no enforceable agreement. Therefore, both parties can agree that they agreed this will happen, but the law will not be invoked to enforce the agreement on either party.

This is often the case, not only with ambiguous contracts, but clear contracts that violate public policy or those that are unconscionable. You could sign an agreement for a loan that would charge you 10% compounded daily to be paid back with interest in 30 days. Will any court in the U.S. enforce that? No…and this is despite the fact that both parties agreed to the terms and they were clear. You can contract for someone to kill your husband, but the court won’t require compensation to the party who paid for the hit, but the husband is still alive.

Therefore, regardless, the court has to agree that the contract is there, and what the contract is.

iamthemob (17135)

Great points everyone! Thanks for the input!

@iamthemob – So both parties are acknowledging that the courts are the ultimate authority? It’s a little hard to apply modern legal standards to a pre-Socratic thought experiment, but I agree that a contract, or “wager” or what have you, can only exist if all parties acknowledge the existence of a higher authority that can arbitrate upon any disagreement.

Smashley (4110)

@Smashley – I agree, which is why I was hesitant to go the legal route. However, we must assume some higher authority as you mentioned, because…well…the person is being trained in the law, so there’s a body administering it.

If either party has the ability to enforce the agreement against the other party without any third party stepping in, then the thought experiment becomes kind of meaningless.

iamthemob (17135)

The solution, in my opinion, is this: the court should find in Eulathus’ favor since he has not yet won his first case, at which point Protagoras can either demand payment or file a second suit that he will win (now that Eulathus has won his first case). As such, Protagoras is correct—though in a slightly more roundabout way than his claim might suggest. Eulathus’ mistake is in thinking that the court finding that he does not have to pay yet is the same thing as saying he does not have to pay period.

Perhaps Eulathus should then sue Protagoras for not doing a better job of teaching him. Then again, there are two rules for success: (1) Never tell everything you know.

SavoirFaire (22919)

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