Buridan's Bridge (also known as Sophism 17) is described by Jean Buridan, one of the most famous and influential philosophers of the Late Middle Ages, in his book Sophismata. It is a self-referential paradox that involves a proposition pronounced about an event that might or might not happen in the future.
The sophism is:
Socrates wants to cross a river and comes to a bridge guarded by Plato. The two speak as follows:
Plato: "Socrates, if in the first proposition which you utter, you speak the truth, I will permit you to cross. But surely, if you speak falsely, I shall throw you into the water."
Socrates: "You will throw me into the water." [1]
Socrates' 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 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. [2]
In order to solve the paradox Buridan proposes three questions:
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. [3]
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. [3]
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 interpretation is the contrapositive of a principle of Immanuel Kant: "ought implies can". [3]
In his solution to the sophism, Walter Burley (d. 1344/1345) 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". [4]
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 regard to 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. [3]
In his book Paradoxes from A to Z Professor Michael Clark comes to the conclusion that if Plato is an 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. [5]
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". [3] 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. [3]
Buridan's bridge sophism was used by Miguel de Cervantes in Don Quixote , [5] 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." [6]
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". [6] After Sancho makes this statement, the person who was asking for advice reasons with him:
"But then, señor governor," replied the querist, "the man will have to be divided into two parts; and if he is divided of course he will die; and so none of the requirements of the law will be carried out, and it is absolutely necessary to comply with it." [6]
Sancho comes up with the moral solution:
...there is the same reason for this passenger dying as for his living and passing over the bridge; for if the truth saves him the falsehood equally condemns him; and that being the case it is my opinion you should say to the gentlemen who sent you to me that as the arguments for condemning him and for absolving him are exactly balanced, they should let him pass freely, as it is always more praiseworthy to do good than to do evil. [6]
In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "p is the case" and "p is not the case" are mutually exclusive. Formally this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case" holds.
In logic, the law of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws.
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
Chrysippus of Soli was a Greco-Phoenician Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of the Stoic philosopher Cleanthes. When Cleanthes died, around 230 BC, Chrysippus became the third head of the Stoic school. A prolific writer, Chrysippus expanded the fundamental doctrines of Cleanthes' mentor Zeno of Citium, the founder and first head of the school, which earned him the title of the Second Founder of Stoicism.
A sophist was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught arete – "virtue" or "excellence" – predominantly to young statesmen and nobility.
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."
Zeno of Elea was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Plato and Aristotle called him the inventor of the dialectic. He is best known for his paradoxes.
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
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Gorgias is a Socratic dialogue written by Plato around 380 BC. The dialogue depicts a conversation between Socrates and a small group of sophists at a dinner gathering. Socrates debates with the sophist seeking the true definition of rhetoric, attempting to pinpoint the essence of rhetoric and unveil the flaws of the sophistic oratory popular in Athens at the time. The art of persuasion was widely considered necessary for political and legal advantage in classical Athens, and rhetoricians promoted themselves as teachers of this fundamental skill. Some, like Gorgias, were foreigners attracted to Athens because of its reputation for intellectual and cultural sophistication. Socrates suggests that he is one of the few Athenians to practice true politics (521d).
Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses OA to mean it is obligatory that A, and PA to mean it is permitted that A, which is defined as .
Hippias Minor, or On Lying, is thought to be one of Plato's early works. Socrates matches wits with an arrogant polymath, who is also a smug literary critic. Hippias believes that Homer can be taken at face value, and he also thinks that Achilles may be believed when he says he hates liars, whereas Odysseus' resourceful (πολύτροπος) behavior stems from his ability to lie well (365b). Socrates argues that Achilles is a cunning liar who throws people off the scent of his own deceptions and that cunning liars are actually the "best" liars. Consequently, Odysseus was equally false and true and so was Achilles (369b). Socrates proposes, possibly for the sheer dialectical fun of it, that it is better to do evil voluntarily than involuntarily. His case rests largely on the analogy with athletic skills, such as running and wrestling. He says that a runner or wrestler who deliberately sandbags is better than the one who plods along because he can do no better.
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In Plato's The Republic, a noble lie is a myth or a lie knowingly propagated by an elite to maintain social harmony. Plato presented the noble lie in the fictional tale known as the myth or parable of the metals in Book III. In it, Socrates provides the origin of the three social classes who compose the republic proposed by Plato. Socrates speaks of a socially stratified society as a metaphor for the soul, wherein the populace are told "a sort of Phoenician tale":
...the earth, as being their mother, delivered them, and now, as if their land were their mother and their nurse, they ought to take thought for her and defend her against any attack and regard the other citizens as their brothers and children of the self-same earth...While all of you, in the city, are brothers, we will say in our tale, yet god, in fashioning those of you who are fitted to hold rule, mingled gold in their generation, for which reason they are the most precious—but in the helpers, silver, and iron and brass in the farmers and other craftsmen. And, as you are all akin, though, for the most part, you will breed after your kinds, it may sometimes happen that a golden father would beget a silver son, and that a golden offspring would come from a silver sire, and that the rest would, in like manner, be born of one another. So that the first and chief injunction that the god lays upon the rulers is that of nothing else are they to be such careful guardians, and so intently observant as of the intermixture of these metals in the souls of their offspring, and if sons are born to them with an infusion of brass or iron they shall by no means give way to pity in their treatment of them, but shall assign to each the status due to his nature and thrust them out among the artisans or the farmers. And again, if from these there is born a son with unexpected gold or silver in his composition they shall honor such and bid them go up higher, some to the office of guardian, some to the assistanceship, alleging that there is an oracle that the city shall then be overthrown when the man of iron or brass is its guardian.
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