Socrates thought that paradoxes are best approached through free inquiry. This is not an option if you adhere to a sacred text. These creeds channel inquiry. As a rule, the effect is stifling. This chapter reviews an exception to the rule.
Nine years before William of Ockham (ca. 1285–1349) was born, a logician was elected pope. Peter of Spain, who became Pope John XXI, is the author of three logical treatises. A few scholars are not sure the logician Peter was the pope. But then it would be a miracle that Peter’s Summulae Logicales became the most popular logic text with 166 editions printed over the next three hundred years.
Peter of Spain drew a highly influential distinction between two ways of parsing sentences that use the word “infinite.” Under the mild interpretation, “The number of dead men is infinite” means that for every natural number n, there is a stage of history at which the number of dead men is greater than n. Under the more problematic reading, “The number of dead men is infinite” means that there is a stage of history such that for every natural number n, the number of dead men is greater than n. Peter suggests that the paradoxes of infinity can be conjugated away by choosing the milder reading. Peter’s proposal is a linguistic replacement of Aristotle’s metaphysical distinction between potential and actual infinity. By keeping the distinction at the level of grammar, Peter avoids Aristotle’s problem of explaining why the potentially infinite cannot become actual.
Like twentieth-century linguistic philosophers, Peter preferred to cast issues in terms of words rather than things. He was especially suspicious of pagan cosmology. Peter preferred Augustine over Aristotle. To continue his research, the pope had a private chamber added to the papal palace at Viterbo. But while he was using his new room, the ceiling collapsed. The pope died of his injuries less than a week later.
Five months before the ceiling fell in, Pope John XXI commissioned the bishop of Paris, Etienne Tempier, to investigate whether the radical Aristotelians at the University of Paris were heretics. In addition to the Thomists, there were characters such as Siger of Brabant. Whereas Aquinas thought that the Christians had argued the Greeks to a draw (at least) on all key issues, Siger thought that the Islamic commentators on Aristotle had shown that reason favors a universe with an infinite past. Siger therefore maintained that faith must sometimes run contrary to reason (not merely beyond reason). After Bishop Tempier issued his unexpectedly detailed condemnation of 219 propositions of “Latin Averroism,” Siger had to flee Paris. He was rumored to have been murdered. Aristotelians worried that Rome would sin twice against philosophy.
The bishop of Paris had exceeded his commission but Pope John XXI went along with the downgrading of Aristotle. Negatively, the condemnation of 1277 created a tradition of Aristotle-bashing that continued long after the Aristotelians reasserted themselves. In 1536, the French logician Peter Ramus presented a master of arts thesis reputedly entitled “Whatever Aristotle Has Said Is False.” Peter Ramus continued to peck away at Aristotle for thirty years. When this French logician was murdered on the third day of the Massacre of St. Bartholomew, there was a report that the assassins had been hired by an academic opponent.
The condemnation of 1277 strongly reaffirms the all-powerful nature of God: anything that is logically possible can be done by God. Aristotle’s natural necessities do not circumscribe the divine sphere of action. Aristotle says nature abhors a vacuum, but since God can create empty space, Christian physicists were obliged to take vacuums seriously.
The Christian elite viewed purges much as a forest ranger views controlled burns. Yes, there is the immediate horror of destruction. But clearing the old, dead wood makes way for fresh, safer growth. The condemnation of 1277 relieved many Christian scholars from the stifling obligation to respond to Aristotle. There was a wave of fresh thought experiments in physics: What would happen if a feather and a stone were dropped in a glass cylinder free of air? Would they reach the bottom simultaneously? Could one see into the vacuum or does one need a medium to see? How would heavenly bodies look if the speed of light were infinite?
At the same time, logic entered a golden age. As an unexpected bonus, a scientific revolution was conceived—only to be tragically aborted by the Black Death.
Researchers on motion and perception were free to question the common sense that Aristotle so ably fortified. Instead of having to take on this intellectual giant with infant theories, the post-1277 scholar was entitled to ignore him.
These liberated theorists could also ignore most theology. In times of intellectual persecution, prudent thinkers seek safety in specialization. They become newly appreciative of the intrinsic interest of their esoteric subject matter. They claim that their research is irrelevant to the controversies of religion and politics. For extra protection, these professionalized Christians claimed their research was irrelevant to neighboring fields of research. Like-minded researchers clustered together, achieving critical mass within and uncritical mass without. Each group operated with the understanding that they would not interfere with those outside their specialty. Good fences make good neighbors.
Purges also create academic vacancies. Inoffensive specialists can hope to swiftly climb institutional ladders that were recently thick with senior faculty. Technically oriented logicians were in an especially good position to move up. Since logic is only concerned with what follows from what, it is topic neutral. By being about no doctrine in particular, logic is heresy-proof (at least with respect to consistent dogma). Yet this topic neutrality also makes it applicable to all arguments whatsoever.
The condemnation of 1277 served as a garden trellis for Ockham’s reasoning. He had faith in the absolute power of God. He adopted Augustine’s interior perspective. Ockham emphasizes that what matters morally is the intention behind the act, not the act or its consequences. A man who attempts adultery and fails is just as guilty as a man caught in the act. A man who throws himself off a cliff in suicidal despair is forgiven if he manages to repent halfway down.
Each agent has intuitive knowledge that his will is free. No proof of freewill is possible. Indeed, very little is provable. Ockham felt that reverence for Greek philosophy led Aristotelian Christians to overintellectualize Christianity. Little of positive theology can withstand careful logical scrutiny. Since theology is not the queen of the sciences, it should step aside and let other areas of study develop on their own.
Part of wisdom is recognizing one’s limits. In the case of paradoxes, this means a willingness to concede that one is unable to reconcile some apparent inconsistencies. For instance, in his Commentary on the Sentences, William of Ockham declared, “It is impossible for any [created] intellect, in this life, to explain or evidently know how God knows all future contingent events.” (d. 38, q.1) Yet he denies that this irresolvable appearance of inconsistency compels him to weaken his belief in God’s foreknowledge or to hedge his belief in human freedom. Ockham is not merely asking for patience. He thinks we will never figure out the solution because we cannot figure it out.
This impotence is embarrassing. To avoid the appearance of irrationality, most theologians fell into the reality of irrationality by worshiping in houses of cards. Their systems are rationalizations for propositions that must stand on faith. The light of logic exposes cracks in the foundations and reveals how much of Christianity rests on the grace of God.
Intellectual pride prejudices theologians into accepting flawed formulations of the problems they “solve.” For instance, Ockham thought Aquinas merely exploited formulations of the problem of foreknowledge that make time important. The real challenge lies in the fact that God’s absolute power is determining our actions to the same degree that it determines any other event.
God’s omnipotence is not entirely bad news on the problem-solving front. Ockham thinks God’s absolute power explains why God is not to blame for the evil in the world (since God knowingly and voluntarily created the world). Ockham accepts the divine command theory of ethics: An action owes its rightness solely to the fact that God approves of it. An act owes its wrongness solely to the fact that God disapproves of it. Since God does exactly what he wills, he cannot do anything blameworthy. If God made a world worse than the actual world, that gratuitous inferiority could not be the basis for reproaching him.
Although the divine command theory is so popular with laymen that it tends to be presupposed rather than asserted, theologians are impressed with an objection Socrates musters in the Euthyphro. Is an act pious because it pleases the gods or are the gods pleased because it is pious? If the gods approved of cruelty would that make cruelty right? The divine command theorist cannot answer by saying that God would never approve of cruelty because cruelty is wrong. For under their theory, God’s approval of something makes it right.
Instead of rushing to answer the Euthyphro dilemma, Ockham calmly pushes the dilemma to a logical extreme. Suppose God commands a man to disobey him. The man is obliged to disobey God (because anything commanded by God is obligatory). Yet the man is also obliged not to disobey God. “The man disobeys God” is a consistent proposition. If God is free to will any consistent state of affairs, then he is free to will that.
Ockham rode the condemnation of 1277 into a protoscientific dawn. By restraining theology, he encouraged a rigorous, piecemeal approach to theoretical issues. Little improvements could accumulate without external meddling. He was making the academic world safer and more productive.
Ironically, the autonomy he promoted for puzzle specialists came at the price of high-level disruption of his own career. Ockham’s influential attacks on the theological establishment alarmed John Lutterell, the chancellor of Oxford University. He prevented Ockham from obtaining a masters of arts degree at Oxford, thereby denying him a license to teach. Lutterell also denounced Ockham to church authorities. With no more official status than an undergraduate philosophy major, Ockham was summoned to papal court in Avignon to meet charges of being a heretic. During the four-year process, the general of the Franciscan order (to which Ockham belonged) maintained Pope John XXII was mistakenly opposed to the doctrine of apostolic poverty. This is the issue Umberto Eco dramatizes in the “Fifth Day” chapter of The Name of the Rose. Eco’s monks raucously debate the vexed issue whether Jesus owned the clothes he wore. If the clergy should imitate Christ and Jesus failed to own even his loin cloth, they should own nothing as individuals. If the religious groups should imitate Christ, then even collective ownership is forbidden. Ockham was asked to investigate the history and doctrinal aspects of apostolic poverty.
Ockham concluded that the pope was a heretic. In 1327 it became evident that Pope John XXII was going to enforce his rejection of apostolic poverty. Ockham and his companions fled from Avignon and sought the protection of Emperor Louis of Bavaria. They were immediately excommunicated. For the next twenty years, Ockham devoted himself to writing treatises on issues of papal authority and civil sovereignty. His polemics angered the pope so much that he threatened to burn down the city of Tournai if the citizens did not capture Ockham and turn him over. After the death of his protector Louis, Ockham realized his antipapal crusade was a lost cause and may have attempted a reconciliation. He died in 1347, probably from the Black Death.
The medievals called the liar paradox an “insoluble.” Just as sailors do not imply absolute invisibility when they describe sandbars as invisible, Ockham does not imply that the liar paradox is absolutely recalcitrant. Ockham thinks the liar paradox is merely difficult for us to solve.
The insolubles encompass what current philosophers call the “paradoxes of self-reference.” Twentieth-century philosophers rediscovered most of them independently, but some have been directly mined from medieval literature. For instance, Stephen Read (1979) blew the dust off of a paradox discussed by Pseudo-Scotus (so named because he was long confused with Duns Scotus). Under the traditional definition, an argument is valid if and only if it is impossible for the premises to be true and the conclusion false. In a commentary on Aristotle’s Prior Analytics, Pseudo-Scotus presents an apparent counterexample: “God exists, therefore, this argument is invalid.” Pseudo-Scotus is presupposing that “God exists” is a necessary truth. Less-devout readers are free to substitute any necessary truth, say, “All equilateral triangles are equiangular.” If the argument is valid, then since the premise is true, the conclusion must be true. But then the argument is both valid and invalid. Contradiction. Therefore, the argument is not valid. But wait! If the argument is invalid, then it is possible for the premise to be true and conclusion false. Since the premise is a necessary truth, the argument is invalid only if the conclusion is false. But then the argument is valid! Contradiction again!
Pseudo-Scotus suggests that the counterexample can be avoided by adding an extra requirement for validity: the conclusion must not deny its own validity. Here he is applying Ockham’s proposal that the insolubility be stopped by banning self-reference.
Ockham realized that a ban on all self-reference would mean the loss of innocent sentences such as “This sentence is in English.” He was willing to accept such casualties as collateral damage. When a gardener poisons aphids, he foresees that he will poison innocent insects along with the pests. There might be more specialized poisons but the gardener does not feel under an obligation to avoid all side-destruction. The medievals are interested in eliminating the glitch but are not expecting the solution to be deeply illuminating. They just want to de-bug their logical systems.
This engineering attitude toward the liar paradox contrasts with the pure theoretical concerns of the ancient Greeks. Diogenes lists Chrysippus as devoting at least six works to the subject. Athenaeus reports that the poet and grammarian Philetas lost much sleep over the liar paradox. He also lost his appetite. Philetas became so thin that friends had to attach lead weights to his feet. They feared the emaciated insomniac would be blown over by the strong winds of Cos (the beautiful Aegean island where Philetus lived). His epitaph: O Stranger: Philetas of Cos am I. ’Twas the Liar who made me die, And the bad nights caused thereby. Most medievals only lost sleep over the liar paradox in the way a bookkeeper loses sleep over ledgers that will not tally. The bookkeeper does not think the inconsistency has dark implications for the foundations of arithmetic.
I have suggested that the technical attitude toward paradox was sharpened by the condemnation of 1277. However, this merely amplified a preexisting tendency of late-medieval thinking. The full story only emerges when we examine the strange saga of the Christian liar paradox.
Ockham’s insights emerged from an educational system (much still intact) that was guilded together after the European economy began to revive around 1100. Prior to this period, schools clustered around ecclesiastical centers. Scholars were not required to take higher orders. But most did, partly out of religious devotion and partly because of practical advantages: clerical status offered scholars independence and protection against the brutality of the locals.
With the decline of the Roman empire, paradoxes that seemed irrelevant to religious issues lapsed into desuetude. They were either forgotten or demoted to the status of intellectual diversions. The liar paradox is a particularly striking example (Spade 1973). One might expect it to be kept steadily before the Christian eye because it is repeated in the Bible: “One of themselves, even a prophet of their own, said, The Cretans are always liars, evil beasts, slow bellies. This witness is true.” (Epistles 1: 12–13). But Christians had the same attitude toward the liar paradox as they did toward a mosquito that is preserved in a reliquary; they were curious as to how the pest became ensconced but were not curious about the creature itself. Augustine cites the verse only to raise the issue of why sacred Scripture should quote pagan sources. Many Christian scholars had access to Cicero’s remarks about the liar paradox in his Academica. It is also explained in Aristotle’s Sophistic Elenchi (25, 180a27-b7) which appeared in the Latin West around 1130. But this led to no competent commentary.
Is the liar paradox so hard to understand? In the twentieth century, the liar paradox became part of popular culture. In the 1967 Star Trek episode “I, Mudd,” the android leader Norman short-circuits when he hears the following exchange:
| Captain Kirk: | Everything Harry tells you is a lie. Remember that—everything Harry tells you is a lie. |
| Harry Mudd: | I am lying. |
The scriptwriters could safely presuppose that most of their audience could follow Norman’s oscillations: If Mudd is lying, then Mudd is telling the truth—and if Mudd is telling the truth, then he is lying.
One might suggest that the liar is only easy in hindsight. Medieval pictures of construction sites depict wheels and hand barrows but never depict wheelbarrows. The wheelbarrow is an obvious combination once you see it, but it requires insight to discover.
This analogy is marred by the fact that the medievals had access to the specimen liar paradoxes—plus intelligent commentary about them. The real problem was that the medievals were at a false summit. They had enough sophistication to escape sloppy formulations of the liar paradox but not enough sophistication to strengthen the paradox in a way that exposes pseudosolutions.
The first hundred years of commentators on Sophistic Elenchi simply accept Aristotle’s cursory “solution.” Aristotle says the liar paradox commits the fallacy of secundum quid et simpliciter: treating a statement that is true in one respect as if it were true absolutely: for instance, “Ethiopians are white with respect to their teeth, therefore, Ethiopians are white.” Consider a man who swears he will break his oath to go to Athens. If he breaks his oath to go to Athens, has he sworn truthfully? On the one hand, he has done what he swore he would do (break the oath to go Athens). On the other hand, he has not done what he swore to do (go to Athens). If we forget to relativize “true” and “false” to the distinct oaths indicated in parentheses, then “The perjurer kept his oath” seems both true and false. Similarly, says Aristotle, we will appear to fall into contradiction if we do not relativize for the liar paradox. Aristotle leaves it to the reader to fill in the details.
The medievals do not express dissatisfaction with the sketchiness of Aristotle’s answers. For centuries, most scholastics mention the liar paradox only as an example of a fallacious argument. They actually show more interest in Aristotle’s case of the perjurer who swears to break his oath. For instance, Giles of Rome became interested in the moral implications of the perjurer. If you swear to break an oath to go to Athens, should you break your oath to go to Athens? Giles concludes that you ought not. Although it is bad to swear to break an oath, it is worse to fulfill the tainted oath.
As time passed, commentators gravitated toward leaner formulations of the liar paradox. In De fallaciis Thomas Aquinas writes: “Likewise here, ‘The Liar speaks the truth in saying that he speaks falsely. Therefore, he speaks the truth.’ It does not follow. For to speak the truth is opposed to what it is to speak falsely, and conversely.” Perhaps inadvertently, Aquinas undermines the analogy with the perjurer’s statement by putting the liar paradox in the present tense. This frees the liar paradox from an association with the thorny issue of future contingent propositions. Albert the Great tends to formulate the perjurer example in the present tense to preserve the analogy.
Sadly, the analogy with the perjurer’s oath became a way of defeating non-Aristotelian solutions to the liar paradox. For instance, those who suggested that the liar statement “says nothing” were ridiculed because the perjurer’s oath is obviously meaningful.
The historical puzzle about the liar paradox has three parts. The first subproblem is to explain a thousand years of Christian incomprehension of the liar paradox. The second task is to explain how these “dunces” could go on to rapidly reach a new pinnacle of understanding. The third stage of the historical puzzle is to explain why understanding of the liar paradox then declines again into a four-hundred-year plateau of facile complacency. In the next chapter I will focus on these three tasks.