Jean Buridan (1295-1356) practiced what William of Ockham preached. He abstained from theology and concentrated on semantics, optics, and mechanics. In the case of insolubles, his secular focus on details led to an understanding of the “paradoxes of self-reference” that was only matched in the twentieth century.
Buridan’s career follows the pattern encouraged by Ockham. He enrolled in the University of Paris, which was then the most prestigious school in Europe. He was later hired as an instructor and rose through the ranks. Buridan did not pursue the usual master of arts degree and so was not licensed to teach theology. He was a “secular cleric,” a priest who did not affiliate with an order. Consequently, his work was not promoted in the way Dominicans perpetuate Thomas Aquinas and Franciscans advance Duns Scotus. Buridan’s fame and influence came from research and administrative service. Buridan served as university rector in 1328 and again in 1340. In 1345 he was chosen to defend the interests of the University of Paris before Philip of Valois at Rome. To quell strife, the pragmaticBuridan banned some of Ockham’s nominalist writings (which, logically enough, eventually led to some of Buridan’s own writings being posthumously banned).
As a side effect of the medieval educational system, sophisms had become standard classroom tools. A sophism is a sentence that poses an instructive analytical difficulty. Usually the problem is an embarrassment of riches: there is an argument in favor of the sentence’s truth and an argument against it. Albert of Saxony’s eleventh sophism in his Sophismata is:
A. All men are donkeys or men and donkeys are donkeys.
Here is the argument that A is true: A conjunction is an “and” statement and so is true when all of its conjuncts are true. “All men are donkeys or men” is true and “Donkeys are donkeys” is true. Therefore, A is a true conjunction. Here is the argument that A is false: A disjunction is an “or” statement and so false when all of its disjuncts are false. “All men are donkeys” is false and “Men and donkeys are donkeys” is false. Therefore A is a false disjunction. The quandary is that A is both true and false. This completes Albert’s exposition of the sophism.
Now comes Albert’s solution. Statement A is ambiguous:
A1. (All men are donkeys or men) and (donkeys are donkeys).
A2. (All men are donkeys) or (men and donkeys are donkeys).
The sophism prompts students to distinguish between the main connective of a sentence and subordinate connectives. If “and” is the main connective, then A means A1. If “or” is the main connective, A means A2.
In addition to logical sophisms, there are grammatical sophisms. Linguistics began with puzzle sentences such as “Love is a verb.” Twentieth-century linguists continued the practice with counterexamples to increasingly sophisticated grammatical generalizations. For instance, one natural theory of how pronouns work is that they borrow the reference of an earlier referring phrase (unless the meaning is supplied from outside the sentence, as when we point). Thus, “Francis touched the beggar and cured him” is solved as “Francis touched the beggar and cured the beggar.” In 1967, Emmon Bach and Stanley Peters pointed out that this theory leads to an infinite regress when applied to cross-referential sentences such as “The pilot that shot at it hit the Mig that chased him.” In the Mig sentence “it” means “the Mig that chased him” and “him” means “the pilot that shot at it.” Substituting one phrase for the pronoun always leaves the other pronoun. Since we are finite beings, we cannot go round and round substituting forever. Do we understand the sentence by leaving some pronoun ungrounded? Or is the Mig sentence meaningless? Buridan would have loved the Bach-Peters paradox.
As sophisms became more challenging, their solutions became controversial. Writers of logic manuals would then review past solutions, present their own, and finally show the advantages of their proposal. The insolubles lie at the extreme end of this continuum of difficulty.
But how did the medievals come by the insolubles? In the previous chapter, I argued that they were unable to recover them from the fossil liars embedded in the Bible, Cicero, Aristotle, and sources now unknown to us. The insolubles were not simply imported to medieval Europe from the Greeks. Nor do the Islamic commentators play a role. The medievals reconstructed the insolubles from a pedagogical practice of their own design—the obligational dispute.
At the University of Paris, virtually all students were in the Arts Faculty. All of these students were required to spend the first two years studying logic. As is still true, logic is taught by frequent assignments and tests. But since the costs of writing were much greater, far more of the course work was oral. Beginning students were obliged to engage in formal debating games. More advanced students participated in the contentful debates recounted by Peter Abelard and Thomas Aquinas. But I trace the recovery of the insolubles to the introductory-level debates.
The scholastic obligational dispute is a skeletalized descendant of the debating game Aristotle sets out to codify. He states this purpose in the opening of the Topics: “Our treatise proposes to find a line of inquiry whereby we shall be able to reason from reputable opinions about any subject presented to us, and also shall ourselves, when putting forward an argument, avoid saying anything contrary to it.” The topics indicated by Aristotle are similar to the ones still debated in high school and college debating competitions. Whereas modern debaters are judged principally on rhetorical criteria, the measure for Aristotle’s debaters was logical consistency. Since the point of the debate is to check for contradictions, Aristotle shapes the debate to make internal conflicts easier to spot and prove:
With regard to the giving of answers, we must first define what is the business of a good answerer, and of a good questioner. The business of the questioner is to develop the argument as to make the answerer utter the most implausible of the necessary consequences of his thesis; while that of the answerer is to make it appear that it is not he who is responsible for the impossibility or paradox, but only his thesis; for one may, no doubt, distinguish between the mistake of taking up a wrong thesis to start with, and that of not maintaining it properly, when once taken up.
(Topics VIII 4)
Aristotle is picturing a cooperative exchange between knowledgeable, mature individuals. The point of the dispute is to create a specimen for postgame analysis.
Medieval obligational disputes were an adaptation for a rowdy, naive crowd. The format does not presuppose any knowledge. Obligational disputes appeal to the male appetite for mock combat (which is intensified in isolated, male-only institutions such as medieval universities). Yet obligational disputes have a surprisingly pure logical structure. An obligational dispute resembles the children’s game king of the hill, in which the defender on the hill wins unless he is dislodged. But here the king’s opponent chooses the hill. Specifically, the opponent in an obligational disputation posits a proposition. If the proposition is consistent, then the respondent is obliged to consistently defend the proposition against the opponent’s cross-examination. The respondent has a limited range of answers. In the early history of the game, the only responses are “I grant it” and “I deny it.” Ockham describes a later version in which the defensive repertoire is enriched with “I doubt it” and even “I distinguish it.” If the opponent extracts contradictory answers from the respondent, then the respondent loses. After all, a pair of conflicting answers is a sure sign that the respondent committed a logical error: a consistent proposition cannot have inconsistent consequences.
Opponents usually saddled the respondent with a patently false posit. This strategy is shrewd. Since the respondent does not believe what he must defend, his background beliefs conflict with the thesis in the foreground. If the respondent fails to censor his real beliefs, a later answer will be inconsistent with earlier answers. This mental leakage explains why prevaricators have trouble sustaining consistency.
The length and pace of the debate needs to be regulated because the players have opposite strategic preferences about how much is said by the respondent. The opponent’s chance of detecting a contradiction increases with the number of replies made by the respondent. Thus, the opponent wants to maximize the number of answers while the respondent wants to minimize this exposure.
The opponent has the exciting role. He is motivated to keep the pace fast. He has a far wider range of possible moves than the respondent’s meager stock of “I grant it” and “I deny it.” When not diverted by compassion, people prefer to identify with those who have the more pleasant perspective. Since human beings also prefer offense to defense, audiences have a tendency to root for the opponent in an obligational dispute. And the rooting was sometimes boisterous. There were university statutes excluding students who demonstrated by “clamoring, hissing, making noise, stone-throwing by themselves or by their servants and accomplices, or in any other way.” (Thorndike 1944, 237)
Devious opponents eventually noticed that respondents can be disarmed with pragmatic paradoxes. These posits make false comments about the respondent. If you posit “You do not exist,” then I must defend this proposition because it is consistent. If you follow up with “You are a handsome fellow,” then what am I to reply? If I say “I grant it,” my answer implies that I exist and so I have contradicted my earlier position that I do not exist. If I say “I deny it,” my answer also implies that I exist and so I have still contradicted my earlier position that I do not exist. You never gave me a chance!
Instructors tried to rescue respondents by outlawing “You do not exist” as a posit. But clever students deployed variations such as “You are asleep” (which was not always false; courses ran from 4 A.M. to 7 P.M. with hardly a break). Here the problem is that an obvious truth about the act of disputing conflicts with the content of what is said in the dispute.
Some instructors reacted by rescinding the opponent’s freedom to choose the starting position themselves. However, as Jean Buridan points out, at any stage of a yes-no obligational dispute, the respondent can be trapped with “Your reply will be negative.” If a respondent replies yes, he is affirming that he is not affirming. If he replies no, then he is denying that he is denying.
Buridan repackages “Your reply will be negative” as a stepping stone to a more famous sophism: Suppose Plato is a bridge keeper. Plato is enraged and tells Socrates, “If what you say is true, then I will let you cross the bridge; and if what you say is false, then I will throw you into the water.” Socrates replies, “You will throw me into the water.” What Socrates said is true if and only if it is false.
Although Buridan is sometimes credited with being the inventor of the bridge paradox (Jacquette 1991), the puzzle probably goes back to Chrysippus. In “The Auction of Philosophers,” Lucian (ca. 115–ca. 200) depicts a slave market with Zeus in charge and Hermes as auctioneer. The offerings include Pythagoras, Diogenes, Aristippus, Democritus, Heraclitus, Socrates, Epicurus, Chrysippus, Aristotle, and Pyrrhon. The philosophers are put through their paces. As a fatalist, Chrysippus is resigned to being sold as a slave. He illustrates his value as a logician with an exhibition of paradoxes: the Reaper, the Electra, and the Sorites. Finally, he rescues a customer’s child from a hypothetical dilemma:
(Lucian 1901, 413)
Chrysippus does not answer but the grammarian Aphthonius is on the record recommending that the crocodile be told “You do not intend to restore it.”
In the second century A.D., the Roman Pausanias (1971) pioneered the literary genre that we call the travelogue. To convey local history, he recounts legends associated with particular sites and artworks. Pausanias’s section on Eleia contains a story about how Anaximenes pleaded with Alexander the Great. The king had just conquered Anaximenes’ native city of Lampsacus. Since the citizens had sided with the Persian king, they feared that the enraged Alexander would enslave them and destroy their city. They sent the respected philosopher Anaximenes to petition for mercy. Before Anaximenes could speak, Alexander interrupted and
swore by the gods of Greece, whom he named, that he would verily do the opposite of what Anaximenes asked. Thereupon Anaximenes said, “Grant me, O king, this favour. Enslave the women and children of the people of Lampsacus, raze the whole city even to the ground, and burn the sanctuaries of their gods.” Such were his words; and Alexander, finding no way to counter the trick, and bound by the compulsion of his oath, unwillingly pardoned the people of Lampsacus.
(1971 6.18.2–4)
Since Anaximenes lived two centuries before Alexander, this anecdote is chronologically impossible. Even so, the tale shows that the topsy-turvy aspect of countersuggestibility was already well known in the second century.
Storytellers continue the Greek tradition of incorporating paradoxes. Miguel de Cervantes used Buridan’s bridge in Don Quixote. This made the bridge an element of the western literary canon.
Buridan may have adapted the bridge from the story related by Pausanias. Although Pausanias received almost no attention for hundreds of years (his papyrus rolls were impractically bulky for travelers), lovers of antiquities eventually realized that Pausanias’s travelogue had aged into an atlas of treasure maps. This kept the ancient legends in circulation.
To salvage the obligational disputes, the logic teachers needed to formulate a general rule about how the game should start. One reform was to prohibit personal posits that described the respondent. But students would side-step this rule with impersonal posits such as “No one exists.” If reference to any individual was prohibited, Buridan suggested that opponents choose “No proposition is negative.” The medievals regarded “No proposition is negative” as a contingent proposition because they took propositions to be actual assertions. This magnified the importance of pragmatic paradoxes.
William of Ockham proposed that the respondent’s commitments not include facts about the game itself. This requires a distinction between the content of what is said and what follows from the fact that it was said. If I assert “The Black Death was caused by fleas,” I invite the inference that I believe the Black Death was caused by fleas. This goes some way in explaining why it is difficult to assert “The Black Death was caused by fleas but I do not believe it.” The sentence is consistent but I cannot consistently believe it. This limit to consistent belief is interesting because we tend to assume that any consistent proposition can be consistently believed.
Buridan’s sophisms can be traced to loopholes in obligational disputes. They are dialectical variants of pragmatic paradoxes and the liar paradox.
Buridan’s first sophism in his chapter on the insolubles is “Every proposition is affirmative, so none is negative.” Is the argument valid? Pro: the premise implies the conclusion because it is an instance of the valid argument from “All Fs are Gs, therefore, no Fs are non-Gs.” Con: a contingent proposition cannot imply a necessarily false conclusion. “No proposition is negative” is false whenever it is uttered.
Buridan’s solution is to insist that a proposition can be possible even if it cannot be a true utterance. It is good enough if the facts could be as the proposition says. For instance, “No sentence on this page is nine words long” expresses a possibility because there could be an absence of nine-word sentences on this page. But the sentence cannot be both true and inscribed on this page because it is itself nine words long. Under this conception of possibility, the argument is valid.
Buridan’s second sophism is “No proposition is negative, therefore some proposition is negative.” The argument seems invalid because the conclusion contradicts its contingent premise. Yet the argument also seems valid because the conclusion is made true by the premise. The premise is a negative proposition and so the conclusion is confirmed by the premise. However, Buridan thinks this is the wrong kind of support. Given the conception of possibility and validity he promotes in his first sophism, Buridan must reject this second sophism as an invalid argument. For what counts is whether the conjunction of the premises and the negation of the conclusion could be made true by a fact. If such a fact is possible, the argument is invalid. If no propositions were uttered, then the premise of the second sophism would be true and the conclusion false.
Buridan’s discussion accumulates insights and constraints that guide the development of subsequent sophisms. As a prelude to the direct liar paradox (“What I am saying is false”), he reviews indirect liar paradoxes. The simplest specimen, sophism nine, consists of Plato saying “What Socrates says is true” and Socrates replying “What Plato says is false.” If Plato’s statement is true, then it is false. If Plato’s statement is false, then Socrates’ statement is also false—which means that Plato’s statement is true after all.
This looped liar shows that no direct self-reference is needed for the liar paradox. It also shows that the paradoxical nature of an utterance need not be an intrinsic property of the sentence itself. The paradoxical aspect of Plato’s “What Socrates says is true” depends on another utterance. If Socrates had instead said “My father was Sophroniscus,” then Plato’s remark would have been unparadoxical.
Although there is no direct evidence that the Greeks were aware of looped liars, some of their humor shows they understood that the paradoxical nature of a statement can rest on other statements. In his essay “On False Modesty,” Plutarch relates an incident involving Menedemus, a student of the Megarian Stilpo, who had a reputation for teasing others with paradoxes: “When he heard that Alexinus often praised him, he said ‘And I’m always chiding Alexinus; so Alexinus must be a bad man, since he either praises a bad man, or is chided by a good one.’” Alexinus’s otherwise innocent remark has been dragooned into a paradox.
These contingent paradoxes refute subjective definitions of “paradox” that require any paradoxical statement to seem absurd to someone. Plato could have drawn a random statement from an urn, declared it true without reading it, and then cast the unread message into the sea. If that unread statement was “What Plato says is false,” then Plato’s original remark was paradoxical even if it never seemed absurd to anyone.
Paradoxes are as objective as diseases. My subjective sense of disorder is evidence of a disorder but is not itself a disorder. I can be sick without feeling sick and without the possibility of a physician being able to detect any illness. Just as there are diseases that will never be discovered, there are paradoxes that will be forever unknown.
Many of these paradoxes will never be known because of a real disease: the Black Death. In addition to killing Buridan, Ockham, and a third of Europe, this plague lowered the prestige of the Church and its satellite institutions. Insights into the liar paradox were packaged in an esoteric terminology and format that received blanket condemnation by disaffected survivors. As intellectual life reconstituted, thinkers turned toward accessible writers such as Cicero and Augustine. Contempt for the liar paradox enjoyed a renaissance.
Obviously, the Church was still an important institution. Scholastic philosophy lingered for centuries. However, the intellectual superstructure of Christianity was increasingly marginalized by humanists. The future belonged to skeptics and scoffers such as Erasmus (1466–1536) and Montaigne (1533–1592). These men were fideists who denied that complicated reasoning could improve on simple faith. They lampooned the scholastics’ efforts to draw positive morals from philosophical paradoxes. Paradoxes were either dismissed as piffles or deployed negatively to humble the pretensions of reason.
People were more receptive to the “new” paradoxes emerging from empirical discoveries. In 1522, all of Europe was astounded by the circumnavigator’s paradox. When Ferdinand Magellan’s ship sailed around the world, a whole day apparently was lost. One of the eighteen survivors of the original 270-odd crew relates the incident:
On Wednesday, the ninth of July, we arrived at one these islands named Santiago, where we immediately sent the boat ashore to obtain provisions. . . . And we charged our men in the boat that, when they were ashore, they should ask what day it was. They were answered that to the Portuguese it was Thursday, at which they were much amazed, for to us it was Wednesday, and we knew not how we had fallen into error. For every day I, being always in health, had written down each day without any intermission. But, as we were told since, there had been no mistake, for we had always made our voyage westward and had returned to the same place of departure as the sun, wherefore the long voyage had brought the gain of twenty-four hours, as is clearly seen.
(Pigafetta 1969, I, 147–48)
It turns out that one of Buridan’s young colleagues, Nicole Oresme, wrote extensively on this paradox. (Lutz 1975, 70) (Oresme may have picked up the paradox from Syrian geographers.) In “Traitié de l’espére,” Oresme describes two imaginary travelers Jehan and Pierre who go around the equator in opposite directions and rendezvous simultaneously at their point of departure. Each covers 30 degrees of longitude per 24-hour day. Jehan, who goes west, reports that his journey took eleven days and nights. Pierre, who goes east, says that it lasted thirteen days and nights. As a control, there is a third man, Robert, who remains at the starting point. Robert says that only twelve days and nights had elapsed since both travelers had set out. Oresme realized if you travel in the same direction that the sun appears to move, you will lengthen the interval to the next sunset or sunrise. After a complete circuit, the increases will add up to a whole day.
Lewis Carroll (1850, 31-33) embellishes the circumnavigator’s paradox by imagining a strip of land circling the earth in which everyone speaks English. You embark Tuesday from London at 9 A.M. and travel quickly enough to keep the sun in the same position in the sky. As you go along, you check the time by asking the locals, “What time is it?” They always answer, “9 A.M.” Indeed, that is the answer when, 24 hours later, you return to London. But the Londoners also report the day as Wednesday rather than Tuesday. So where did Wednesday begin?
The circumnavigator’s paradox was rendered obsolete in 1878 when the International Date Line was declared at 180 degrees east from Greenwich, England. It is a credit to scholasticism that the need for such a convention was noted five hundred years earlier by Nicole Oresme. He thought through these issues in an entirely hypothetical manner. In deference to the condemnation of 1277, Oresme denied that strict demonstrative proofs were possible in the physical sciences. First, he would argue that the earth goes round the sun. Then, he would turn around and argue that the sun goes round the earth. Oresme invites us to conclude that neither reason nor experience can settle the issue. He adopts the traditional view, that the earth does not move, on faith.
Was the circumnavigator’s paradox a theoretical paradox when debated in the sixteenth century? On the one hand, a widely acknowledged expert, Nicole Oresme, had definitively solved the problem in the fourteenth century. So it was no longer a paradox to those attending Oresme’s lectures; there was no longer any conflict between observation and calendar theory. However, in the sixteenth century, Renaissance men were opting out of the old system of intellectual division of labor. Their experts did not include fourteenth-century philosophy professors. The medieval syllogisms of Buridan and Oresme was likened to spiders’ webs; hard for any man to precisely imitate but only strong enough to ensnare the feeble. By disavowing the past, Renaissance men created a new environment for the circumnavigator’s paradox.