25. Probation or Jail:
Buridan’s Bridge Paradox
‘Clever Alec’, a neighborhood thug, faces prosecution for a barroom fight. It is not quite clear, hoewever, who actually started the brawl, and Alec believes he can beat the rap by simply denying everything. He was just having a leisurly beer, he claims, when ‘Steve the Slugger’ entered the bar and started swinging.
The prosecutor is nevertheless confident that she can get Alec convicted. But in order to avoid a lenthy trial, she offers Alec a plea bargain: if he tells the truth, he will get only six weeks probation; if he lies, he will certainly have to do two months of hard time. While signing a plea bargain, a suspect has the right to say something in his own favor. He could claim mitigating circumstances, appeal to the prosecutor’s conscience, demand a better lawyer, and so on. Is there something Clever Alec can say and avoid both probation and jail?
Yes, there is.
Alec only needs to say “I’ll go to jail” … and go scot free. The prosecutor will have been duped: if she sends Alec to jail, Alec has told the truth. But if Alec says the truth, then, according to the deal, he should get probation.
The riddle has been known for several millennia. One account of it is traced back to Chrysippus of Sol, a stoic philosopher who is said to have died in 207 BCE in a fit of laughter. The best-known version of the conundrum was described by the French philosopher Jean Buridan (1300 – 1358): Plato, a landowner, has made a solemn vow that only those who speak the truth will be allowed across his bridge; those who lie will be thrown into the water.
When Socrates arrives at the river, he says “You will throw me in the water” and Plato is confronted with a dilemma. If he throws Socrates into the water, like Socrates said he would, Socrates told the truth. Hence, according to his vow, Plato should allow him to pass the bridge. But if Socrates is allowed to pass the bridge, he has uttered a lie. Hence, Plato should throw him into the water. What is Plato to do?
There are two questions that arise. First, is Socrates’ statement true or false? Second, what is Plato to do to keep his vow? The puzzle, recorded in Burdan’s collection of Insolubilia (Unsolvables), became known among the cognoscenti as Buridan’s Bridge Paradox.
Two centuries after Buridan, Miguel de Cervantes (1547–1616) took up the dilemma in his famous novel Don Quixote. The nobleman’s gofer Sancho Panza, when appointed governor of the virtual island Barataria, was called upon to settle a juridical question about just such a bridge. If the traveller uttered a lie, he would be hanged after crossing the bridge. “It happened one day that a certain traveller after being sworn, declared that by the oath he had taken, he was to die upon that Gallow.”
Same two questions…
The Buridan’s Bridge Paradox differs from the Liar’s Paradox (see Chapter ////) because, at the very time when Alec utters the sentence “I’ll go to jail”, he is not lying. But he is not telling the truth either. In contrast to Epimenedes’s claim (see Chapter ////) that “all Cretans lie…” which can be verified for its truthfulness right then and there, the veracity of Alec’s declaration and Socrates’s statement can only be established in the future,
That’s where the prosecutor in the above vignette was led up the garden path: she wanted Alec to tell the truth about the past, while Clever Alec said something about the future. Some philosophers maintain that the truth value of a proposition about the future cannot be determined. If a meteorologist predics a thunderstorm for the day after tomorrow, does he lie if the storm eventually does not materialize? Certainly, today he is not lying. And on the day after tomorrow….well, he may not have predicted the truth but we would be hard-pressed to make a case for lying.
So, in answer to the first question above – is Socrates’ statement true or false – we may say that it is neither. A proposition that depends on something whose veracity can only established in the future cannot be deemed, here and now, as either true or false.
Now to the question about keeping the vow. By making his vow, the philosopher seemingly placed himself in a dilemma in which he cannot act without violating his word. But does he have to act? Or may he refrain from acting while nevertheless adhering to his vow?
Plato has obliged himself to throw Socrates in the water if Socrates lies, and to let Socrates pass if Socrates speaks the truth. But now let us assume, for argument’s sake, that Socrates just walks silently over the bridge. Or that he pronounces the Riemann Conjecture while on the bridge, a mathematical conjecture that has neither been proven or disproven. Or that he says “in 2000 years, there will be no water under this bridge.” These scenarios can be said to be neither true nor false. So what is Plato to do?
Obviously, the vow does not cover all possible replies. As was just pointed out above, Socrates can do something other than utter a proposition that is either true or false. Therefore, under such circumstances Plato’s two-part conditional oath (“If you say A, I will do B, if you say C, I will do D”) does not commit him to doing anything.
Socrates' prediction “You will throw me in the water” is neither A nor C; it fails to satisfy either of the two conditions specified in the vow. Hence, Plato may abstain from any action without breaking his vow.
* * *
Poor Clever Alec. By the same argument that got Plato off the hook, the prosecutor is not bound by the plea bargain. She is free to ask the judge for jail time. Turns out, Alec was too clever by half.
Food for thought:
The above argument was put forth by Dale Jacquette in 1991. But is Plato, by tacitly letting Socrates traverse the bridge, not acting anyway? After all, he is letting Socrates cross the bridge. Let’s rephrase Plato’s vow: “I will throw Socrates in the water if he lies, I will let Socrates traverse if he says the truth.” In this phrasing, it is either one or the other and Plato remains bound by his vow. So, he’s back at square one.
 Dale Jacquette (///maybe omit///): “the problem of the bridge might not be resolved by the modern expedient of imposing either Whitehead and Russell's syntactical ramified theory of types, or Tarski's semantic hierarchy truth predications in languages and metalanguages…. Socrates' prediction of Plato's action on the basis of his conditional vow constitutes the fatal link implying semantic contradiction and ethical dilemma.”
Corrections, comments, observations: