Buridan's bridge
Encyclopedia
Buridan's Bridge is described by Jean Buridan
, one of the most famous and influential philosophers
of the Late Middle Ages
, in his book Sophismata. Sophism 17 is a self-reference
d paradox
that involves a proposition
pronounced about an event that might or might not happen in the future.
Socrates
's response puts Plato
in a difficult situation. He could not throw Socrates into the water, because if he did he would violate his promise to let Socrates to cross the bridge if he speaks the truth. On the other hand, if Plato allows Socrates to cross the bridge it would mean that Socrates spoke untruth when he replied: "You are going to throw me into the water". In that case Socrates should have been thrown into the water. In other words, Socrates could be allowed to cross the bridge if and only if he could not be.
In response to the first question Buridan states that it is impossible to determine if Socrates' proposition is true or false. This is because the proposition "You are going to throw me into the water" is a future contingent that could be true or false depending on what Plato is going to do. Dr. Joseph W. Ulatowski says that Buridan apparently used Aristotle's
thesis about what "truth" is to come up with this response. Aristotle believed that a proposition is true if and only if it is verified by the state of things as they currently are. Contradicting the principle of bivalence
, Buridan implies a system of three-valued logic in which there are three truth values--true, false, and some indeterminate third value.
In determining the truth value of Plato's conditional promise, Buridan suggests that Plato's promise was false, and that because Plato gave his promise carelessly he is not obligated to fulfill the promise.
In discussing the third question, "What ought Plato to do to fulfill his promise", Buridan states that Plato should not have given a conditional promise in the first place. He also suggests that Plato could have made sure that the condition was formulated in such a way that it would not cause a contradiction; because Plato cannot fulfill his conditional promise without violating it, he is not obligated to fulfill the promise. Ulatowski points out that this is the contrapositive of the principle stated by Immanuel Kant
: "ought implies can".
applied the principle "nothing is true unless at this instant" ("nihil est verum nisi in hoc instanti") and concluded that "if a proposition is true it must be true now".
Dr. Dale Jacquette of the University of Bern says that "Plato can either permit Socrates to pass or have him seized and thrown into the river without violating his conditional vow". Jacquette argues that Plato's conditional promise was given only in regards of Socrates's proposition being clearly and unconditionally either true or false. To prove his point Jacquette asks, what would Plato have to do if Socrates had said nothing and was "as silent as a Sphinx
", or if he uttered something that could not be either proven or "undisproven", something like Goldbach's conjecture
. Jacquette concludes that Plato's conditional promise was true, and Socrates's proposition is "neither true simpliciter nor false simpliciter", and therefore Plato would be right regardless of the choice that he made.
In his book Paradoxes from A to Z Professor Michael Clark comes to the conclusion that if Plato is a honorable man, Socrates should not get wet under any circumstances. Clark argues that Socrates could say, "Either I am speaking falsely, and you will throw me in, or I am speaking truly, and you won't throw me in". Clark says that if this sentence is true, then it means that the first alternative "is ruled out", leaving us only with the second one. If this sentence is false, it means that both alternatives are false, and because Socrates spoke falsely "it will be false" to throw him into the river.
Dr. Joseph W. Ulatowski believes that since the truth value in Plato's conditional promise and even more so in Socrates's proposition is indeterminate, it means that Plato "ought to err on the side of caution with respect to the future contingency and allow Socrates to cross the bridge". In the same work Ulatowski offers a couple of humorous solutions to the paradox. Plato, Ulatowski says, could let Socrates to cross the bridge, and then throw him into water on the other side. Or both Plato and Socrates could combine their efforts and forcibly eject Buridan himself from Buridan's bridge.
in Don Quixote, when Sancho
was presented with the Buridan's bridge dilemma: A man who was going to cross the bridge was asked to respond truthfully where he was going or otherwise to face a death by hanging. The man "swore and said that by the oath he took he was going to die upon that gallows that stood there, and nothing else."
Sancho summarizes the situation by saying: "the man swears that he is going to die upon the gallows; but if he dies upon it, he has sworn the truth, and by the law enacted deserves to go free and pass over the bridge; but if they don't hang him, then he has sworn falsely, and by the same law deserves to be hanged". He then comes up with the solution, "that of this man they should let pass the part that has sworn truly, and hang the part that has lied; and in this way the conditions of the passage will be fully complied with". After Sancho makes this statement, the person who was asking for advice reasons with him: Sancho comes up with the moral solution:
Jean Buridan
Jean Buridan was a French priest who sowed the seeds of the Copernican revolution in Europe. Although he was one of the most famous and influential philosophers of the late Middle Ages, he is today among the least well known...
, one of the most famous and influential philosophers
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
of the Late Middle Ages
Late Middle Ages
The Late Middle Ages was the period of European history generally comprising the 14th to the 16th century . The Late Middle Ages followed the High Middle Ages and preceded the onset of the early modern era ....
, in his book Sophismata. Sophism 17 is a self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
d paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
that involves a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
pronounced about an event that might or might not happen in the future.
The sophism
The sophism is:Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
's response puts Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
in a difficult situation. He could not throw Socrates into the water, because if he did he would violate his promise to let Socrates to cross the bridge if he speaks the truth. On the other hand, if Plato allows Socrates to cross the bridge it would mean that Socrates spoke untruth when he replied: "You are going to throw me into the water". In that case Socrates should have been thrown into the water. In other words, Socrates could be allowed to cross the bridge if and only if he could not be.
Buridan's solution
In order to solve the paradox Buridan proposes three questions:- Is the proposition uttered by Socrates: "You are going to throw me into the water" true, or is it false?
- Is Plato's promise true or is it false?
- "What ought Plato to do to fulfill his promise?"
In response to the first question Buridan states that it is impossible to determine if Socrates' proposition is true or false. This is because the proposition "You are going to throw me into the water" is a future contingent that could be true or false depending on what Plato is going to do. Dr. Joseph W. Ulatowski says that Buridan apparently used Aristotle's
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
thesis about what "truth" is to come up with this response. Aristotle believed that a proposition is true if and only if it is verified by the state of things as they currently are. Contradicting the principle of bivalence
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
, Buridan implies a system of three-valued logic in which there are three truth values--true, false, and some indeterminate third value.
In determining the truth value of Plato's conditional promise, Buridan suggests that Plato's promise was false, and that because Plato gave his promise carelessly he is not obligated to fulfill the promise.
In discussing the third question, "What ought Plato to do to fulfill his promise", Buridan states that Plato should not have given a conditional promise in the first place. He also suggests that Plato could have made sure that the condition was formulated in such a way that it would not cause a contradiction; because Plato cannot fulfill his conditional promise without violating it, he is not obligated to fulfill the promise. Ulatowski points out that this is the contrapositive of the principle stated by Immanuel Kant
Immanuel Kant
Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
: "ought implies can".
Philosophers on the sophism and its solution
In his solution to the sophism, Walter BurleyWalter Burley
Walter Burley was a medieval English Scholastic philosopher and logician. He was a Master of Arts at Oxford in 1301, and a fellow of Merton College, Oxford until about 1310. He spent sixteen years at Paris until 1326, becoming a fellow of the Sorbonne by 1324. After that, he spent seventeen...
applied the principle "nothing is true unless at this instant" ("nihil est verum nisi in hoc instanti") and concluded that "if a proposition is true it must be true now".
Dr. Dale Jacquette of the University of Bern says that "Plato can either permit Socrates to pass or have him seized and thrown into the river without violating his conditional vow". Jacquette argues that Plato's conditional promise was given only in regards of Socrates's proposition being clearly and unconditionally either true or false. To prove his point Jacquette asks, what would Plato have to do if Socrates had said nothing and was "as silent as a Sphinx
Sphinx
A sphinx is a mythical creature with a lion's body and a human head or a cat head.The sphinx, in Greek tradition, has the haunches of a lion, the wings of a great bird, and the face of a woman. She is mythicised as treacherous and merciless...
", or if he uttered something that could not be either proven or "undisproven", something like Goldbach's conjecture
Goldbach's conjecture
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...
. Jacquette concludes that Plato's conditional promise was true, and Socrates's proposition is "neither true simpliciter nor false simpliciter", and therefore Plato would be right regardless of the choice that he made.
In his book Paradoxes from A to Z Professor Michael Clark comes to the conclusion that if Plato is a honorable man, Socrates should not get wet under any circumstances. Clark argues that Socrates could say, "Either I am speaking falsely, and you will throw me in, or I am speaking truly, and you won't throw me in". Clark says that if this sentence is true, then it means that the first alternative "is ruled out", leaving us only with the second one. If this sentence is false, it means that both alternatives are false, and because Socrates spoke falsely "it will be false" to throw him into the river.
Dr. Joseph W. Ulatowski believes that since the truth value in Plato's conditional promise and even more so in Socrates's proposition is indeterminate, it means that Plato "ought to err on the side of caution with respect to the future contingency and allow Socrates to cross the bridge". In the same work Ulatowski offers a couple of humorous solutions to the paradox. Plato, Ulatowski says, could let Socrates to cross the bridge, and then throw him into water on the other side. Or both Plato and Socrates could combine their efforts and forcibly eject Buridan himself from Buridan's bridge.
Use of Buridan's bridge in literature
Buridan's bridge sophism was used by Miguel de CervantesMiguel de Cervantes
Miguel de Cervantes Saavedra was a Spanish novelist, poet, and playwright. His magnum opus, Don Quixote, considered the first modern novel, is a classic of Western literature, and is regarded amongst the best works of fiction ever written...
in Don Quixote, when Sancho
Sancho Panza
Sancho Panza is a fictional character in the novel Don Quixote written by Spanish author Don Miguel de Cervantes Saavedra in 1605. Sancho acts as squire to Don Quixote, and provides comments throughout the novel, known as sanchismos, that are a combination of broad humour, ironic Spanish proverbs,...
was presented with the Buridan's bridge dilemma: A man who was going to cross the bridge was asked to respond truthfully where he was going or otherwise to face a death by hanging. The man "swore and said that by the oath he took he was going to die upon that gallows that stood there, and nothing else."
Sancho summarizes the situation by saying: "the man swears that he is going to die upon the gallows; but if he dies upon it, he has sworn the truth, and by the law enacted deserves to go free and pass over the bridge; but if they don't hang him, then he has sworn falsely, and by the same law deserves to be hanged". He then comes up with the solution, "that of this man they should let pass the part that has sworn truly, and hang the part that has lied; and in this way the conditions of the passage will be fully complied with". After Sancho makes this statement, the person who was asking for advice reasons with him: Sancho comes up with the moral solution: