The Greek paradoxes have reached us through a network of literature and oral tradition. Judging by how much nearly did not reach us and by the poor condition in which this material arrived, much must have been lost and much must molder in an unrecognized form on our library shelves. This chapter is about how paradoxes and our attitudes toward them have been shaped by their mode of transmission.
Paradoxes have not been handed down through the generations solely by virtue of their intrinsic interest. Often they hitch a ride on some weightier matter. For instance, the liar paradox owes some of its currency to the fact that Paul unwittingly packed it into the Bible.
And what appear to be mere accretions are sometimes the whole substance of the paradox. Many paradoxes of political philosophy and religion originated as incoherent compromises between vying factions.
The reverence or derision excited by a paradox is often an echo of the attitude first adopted toward those associated with the paradoxes. In Greece, dialectical struggle was generally a team effort. Philosophers had strong loyalty to their schools. Their competition for students and patronage was a matter of life and death—metaphorically for the group and literally for some members. Since the stakes were high, the tactics were more reminiscent of politics than of dispassionate inquiry. What could not be refuted was laughed down, stonewalled, or distorted.
The Megarian reputation for logic-chopping was consolidated by Euclides’ student and successor Eubulides. Diogenes Laertius describes Eubulides as “ . . . the author of many dialectical arguments in a question and answer form, namely, The Liar, The Disguised, The Electra, The Veiled Figure, The Sorites, The Horned One, and The Bald Head.” (1925 11, 108) Eubulides’ paradoxes are all discussed by Aristotle either directly or indirectly. But Aristotle’s discussion is spare and stiffly dismissive.
The only report of Eubulides writing a book is by Eusebius, a fourth-century bishop of Caesarea. Eusebius says Eubulides accuses Aristotle of being a spy for Philip of Macedon and further charges Aristotle with being disloyal to Plato. There may be some truth to both charges.
Diogenes says that Aristotle was disappointed when he was not chosen to succeed Plato as the head of Plato’s Academy. However, Aristotle was a foreigner from Macedonia and was forbidden to own land in Athens. Anti-Macedonian sentiment was growing because of Demosthenes’ warnings about the growing power of Philip. Demosthenes was Aristotle’s exact contemporary (both lived from 384–322 B.C.). Diogenes Laertius says that Demosthenes was probably a student of Eubulides. Perhaps Demosthenes was a channel of ill will between Eubulides and Aristotle. In any case, Aristotle left Athens and became the tutor of Philip’s son Alexander. When Athens and Thebes were defeated by Alexander, Thebes was razed and its citizens sold off as captives. Alexander offered generous terms to Athens to secure the cooperation of her navy in his plans for the conquest of Persia. The Athenians agreed to pay for a Macedonian garrison in their city and to exile Demosthenes for his role in rallying the Athenians against the Macedonians. Once Macedonian hegemony was established, Aristotle returned to Athens and founded the Lyceum. His well-provisioned school functioned as an intellectual counterweight to the Academy.
Aristotle appears to have been an informal ambassador from Macedonia. Ambassadors are often suspected of orchestrating espionage. Many tales about Aristotle were told. Biographers such as Hans Kelsen and Anton-Hermann Chroust pool these stories into a portrait of Aristotle as a kind of James Bond or Mata Hari.
Aristotle’s ambiguous status in Athenian society helps to resolve “Aristotle’s paradox of monarchy.” (Miller 1998) Aristotle says political justice and political community take place “among people naturally subject to law, . . . people who have an equal share in ruling and being ruled.” (Nichomachean Ethics, 1134b15) He writes that “legislation has to do with those who are equal both in kind and capacity.” (Politics, 1284b34–35) Yet there are also several passages in which Aristotle endorses kingship. A man of superlative virtue can surpass all others in his claim to rule. He is above the law and should rule permanently like the head of a household.
Scholars have tried to reconcile the conflict between Aristotle’s assertions that monarchy is the best form of government and that citizens should function as equals using their own rationality to make decisions. The contradiction in Aristotle’s writings arises from his need to appease the democrats in Athens and the need to appease Antipater, the regent of Macedon with whom he regularly corresponded. (Miller 1998) Aristotle also needed to appease Alexander himself. Although Alexander was thoughtful enough to send his teacher biological specimens, he also hanged Aristotle’s nephew, the historian Callisthenes, for refusing to worship him as a god. There is a suggestive fragment from one of Aristotle’s lost works that reads “Kings should not themselves be philosophers, but they should have philosophers as their advisors.” (Aristotle, 1955, 62)
When news of Alexander’s death reached the Athenians, they expelled the Macedonian garrison and recalled Demosthenes from exile. Aristotle, along with other Macedonian sympathizers, was charged by Demophilus and Eurymedon with impiety. Aristotle left for the city of Chalcis in Euboea where his mother’s estate was still under Macedonian protection. Aristotle said he did not want to see Athens sin twice against philosophy.
Aristotle died the same year from a stomach ailment. Demosthenes also died that year. He was driven to suicide when Antipater crushed the Athenian revolt.
After Macedon resumed its domination of Athens, Theophrastus, Aristotle’s designated successor, revived the Lyceum. The core of the school was Aristotle’s library. At about six hundred volumes, his library was one of the largest in the world. In addition to containing his own extensive works and notes, Aristotle’s library contained a wide range of Greek literature. He pioneered the practice of prefacing his own treatment of a topic with a survey of what had been written before. Students at the Lyceum emulated Aristotle’s methodology and his encyclopedic ambitions.
Theophrastus bequeathed the library to his pupil, Neleus of Skepsis. Theophrastus thought that Neleus would be his successor at the Lyceum and may have bequeathed the library to him to enhance the likelihood of his succession. However, the trustees of the school instead elected a younger man, Straton. Neleus, perhaps out of spite and perhaps in the hope of establishing his own school, carried the library of Aristotle and Theophrastus to the city of Skepsis. This contributed to the decline of the Lyceum. Neleus then bequeathed the library
to his heirs, ordinary people, who kept the books locked up and not even carefully stored. But when they heard how zealously the Attalic kings to whom the city was subject were searching for books to build up the library in Pergamon, they hid their books underground in a kind of trench. But much later, when the books had been damaged by moisture and moths, their descendants sold them to Apellikon of Teos for a large sum of money, both the books of Aristotle and those of Theophrastus. But Apellikon was a bibliophile rather than a philosopher; and therefore, seeking a restoration of the parts that had been eaten through, he made new copies of the text, filling up the gaps incorrectly, and published the books full of errors.
(Strabo 1929, 13.1.54)
Sulla seized Apellikon’s library and shipped it to Rome. Plutarch, in his biography of Sulla, says that copies of Aristotle’s writings were then made by the Greek philologist Tyrannion of Amisos (who had been in Rome since about 68 B.C.). Tyrannion gave them to Andronikos of Rhodes who edited them and compiled a comprehensive list of the works of Aristotle and Theophrastus. Although Andronikos’s commentary and bibliography were eventually lost, they led to a tradition of scholarship that preserved Aristotle’s work. What survives now is principally Aristotle’s lecture notes for specialists. Aristotle had also written more accessible works. The Roman orator Cicero (106–43 B.C.) praised Aristotle’s dialogues as composed in a “golden style” that he strove to emulate.
As Aristotle’s reputation reblossomed, the reputations of his adversaries withered. Cicero inaugurated a tradition of disparaging Eubulides. In Academic Questions, Cicero characterizes the Megarian paradoxes as “far-fetched and pointed sophisms.” The sorites is dismissed as a “very vicious and captious style of arguing.” Our principal source of information about Eubulides, Diogenes Laertius, quotes a comic poet: “Eubulides the Eristic, who propounded his quibbles about horns and confounded the orators with falsely pretentious arguments, is gone with all the braggadocio of a Demosthenes.” (1925, II, 108) The other commentators of antiquity also demonize Eubulides as a serpentine quibbler. Given this thin selection of uniformly negative “primary sources,” future historians had no textual basis to veer from Cicero’s verdict. Eubulides’ ignominy became self-perpetuating. Each generation’s dismissal expanded the basis for the next. As late as 1931, we find Eduard Zeller, in Outlines of the History of Greek Philosophy, characterizing Eubulides’ paradoxes as “clever but worthless fallacies.”
The emphasis on logic at the opening of the twentieth century elevated logical paradoxes to the status of instructive anomalies. Logicians lacked any historical grounds to challenge Cicero’s tradition. Yet, they began to feel toward Eubulides what Mark Twain felt toward another figure of antiquity:
I have no special regard for Satan; but, I can at least claim that I have no prejudice against him. It may even be that I lean a little his way, on account of his not having a fair show. All religions issue bibles against him, and say the most injurious things about him, but we never hear his side. We have none but the evidence for the prosecution, and yet we have rendered the verdict. To my mind, this is irregular. It is un-English; it is un-American; it is French.
(from “Concerning the Jews”)
In 1903, Gottlob Frege published his second volume of the Grundlagen which used a variation of the veiled figure to launch his theory of sense and reference. In 1905, Bertrand Russell published “On Denoting,” which deploys the paradoxes of identity as tests for his theory of definite descriptions. In “Vagueness,” Russell (1923) used the sorites to probe the applicability of classical logic to ordinary language. And Russell had previously used the liar paradox to model a refutation of naive set theory. In 1931, Kurt Godel was guided by the liar paradox in his construction of a proof of the incompleteness of arithmetic. And Alan Turing (1936) used the liar yet again in his derivation of the first uncomputable function (the halting problem). In 1950 Peter Strawson used the horned man paradox in “On Referring” to promote truth-value gaps. This led to a logic of truth-value gaps (“supervaluationism”).
When William and Martha Kneale published The Development of Logic in 1962, they were cognizant of the heights that were reached on the backs of these riddles. They boggled at the traditional dismissive treatment of the Megarian paradoxes: “All are interesting, and it is incredible that Eubulides produced them in an entirely pointless way, as the tradition suggests. He must surely have been trying to illustrate some theses of Megarian philosophy, though it may be impossible for us to reconstruct the debates in which he introduced them.” (1962, 114-15) Despite the meager historical record, the Kneales go on to make tentative suggestions as to how the paradoxes influenced Megarian logic and thereby Stoic logic through the efforts of Chrysippus.
There have been a spate of speculative reconstructions of Eubulides’ paradoxes and ancient efforts to solve them. They follow William and Martha Kneale’s suggestion that Eubulides should be understood in the same way as we understand Zeno (and Euclides): Eubulides’ paradoxes were a defense of Parmenides.
Recall that Parmenides infers that there is only one thing from the premise that concepts which employ negations do not apply to anything. Parmenides and especially Zeno seem sensitive to the self-refuting nature of this singular conclusion and the process of arguing for it. There can be an argument for Parmenides’ “All is one” only if there are premises that differ from the conclusion. But Parmenides’ conclusion implies that there are no differences between premises and conclusions. If Parmenides is right, there are no arguments!
When faced with inescapable self-refutation, a buoyant philosopher will modestly portray his arguments as dispensable tools. Once you cross the river, you no longer need the raft. To help others reach the other side, you send the raft back to the opposite shore.
Does the end, enlightenment, justify the means? If you do not personally accept the arguments composing the raft, then it seems wrong to propound them. Those who insincerely propound arguments are lying. They assert what they do not believe with the intention that their hearers will believe.
There are broader definitions of lying that associate it with nearly any kind of deceit. These fail to respect the moral asymmetry between lying and misleading. All lies are assertions. When I assert p, I invite you to take my word for it. Lying is graver than merely misleading a person because lying betrays trust.
Actually, there are two forms of argument in which the premise is supposed rather than asserted. In a conditional proof, one assumes a proposition p, deduces q, and then concludes “If p, then q.” In reductio ad absurdum, one assumes p, deduces a contradiction, and then concludes not p. Reductio is striking in that one assumes what one believes to be false. Reductio is frequently confused with modus tollens in which one argues: If p, then q; not q, therefore, not p. When q seems patently false (“absurd”), then there is some temptation to call the argument a reductio ad absurdum. But in modus tollens, two premises are asserted and q is merely some falsehood rather than a contradiction. Unlike the indirect forms of argument (conditional proof and reductio), modus tollens requires doctrinal commitments from the speaker.
Little can be proved without substantive premises. One loophole is to make one’s adversary assert the premises. Riddlers do not assert anything. They just ask questions. Diogenes Laertius is particularly fond of question-answer pairs in which a philosophical attitude is expressed in the answer:
[Thales] held there was no difference between life and death. “Why then,” said one, “do you not die?” “Because,” said he, “there is no difference.” To the question which is older, day or night, he replied: “Night is the older by one day.” Some one asked him whether a man could hide an evil deed from the gods: “No,” he replied, “nor yet an evil thought.” To the adulterer who inquired if he should deny the charge upon oath, he replied that perjury was no worse than adultery. Being asked what is difficult, he replied, “To know oneself.” “What is the most pleasant?” “Success.” “What is the divine?” “That which has neither beginning nor end.”
(1925 I, 34-36)
Some philosophical dialogues are just elaborations of this simple format. Others present the reasoning behind the answers.
Those who pose paradoxes are not asserting any of the propositions that comprise the paradox. They merely ask a question.
The Parmenidean master shies away from asserting “There are no negative truths” because that is itself a negative statement. But he can pose paradoxes that allow the student to attain the insight which is approximated by “There are no negative truths.”
As we shall see, this basic maneuver runs throughout the whole course of Western philosophy. It is also a steady favorite in Eastern philosophy. Some sects of Buddhism revel in the enigmas raised by their tenets. How can I aim for freedom from desire without desiring that freedom and thereby ensuring the frustration of my goal? How can everybody be reincarnated if there are more people now than there have ever been in the past? As if these anomalies were not enough, Zen Buddhists heap on extra puzzles in an effort to trigger enlightenment. Master Shuzan takes a bamboo stick and poses a dilemma: “If you call this a stick, you fall into the trap of words, but if you do not call it a stick, you oppose the fact. So what will you people call it?”
Paul warned Titus, his bishop on the isle of Crete: “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)
The prophet was Epimenides. Various poems have been attributed to Epimenides but none of his philosophical writings have survived. He is reported to have been born about 659 B.C. in Phaestus or perhaps Knossus, the capital city of Crete. The two most common dates of death reckon him as the most long-lived of philosophers: 157 years by one account, about 230 years by the other. Diogenes says that when Epimenides’ father sent him out to search for stray sheep, Epimenides lay down in a cave. He awoke fifty-seven years later. Epimenides returned to his fellow citizens with long hair and a flowing beard. He also had acquired superhuman knowledge of medicine and natural history. At his pleasure, his soul could leave his body and he could have intercourse with the gods—perhaps accounting for his gift of prophecy. His reputation as a seer led the Athenians to request his presence at rites of purification and propitiation to pave the way for Solon’s legislative reforms. The Cretans paid him divine honors upon his death. In Crete, there is still an important street named after him.
Epimenides’ remark “The Cretans always lie” was quoted for centuries because people realized that it is self-defeating for a Cretan to say “The Cretans always lie.” There is irony in self-defeat. But irony is not inconsistency. After all, some Cretan has at some time asserted something that was not a lie. Epimenides’ “The Cretans always lie” is just false. No paradox yet!
Eubulides may have poked through the ashes of Epimenides’ remark and discovered a live ember; it would be odd if Epimenides’ “The Cretans always lie” entails that some Cretan is not a liar. Sure, it is a historical fact that some Cretans sometimes tell the truth. But one should not be able to deduce this historical fact from logic alone. What if Epimenides were the only Cretan? Then we could not make “The Cretans always lie” come out false by finding a truthful Cretan. We would have a statement that must be neither true nor false!
The element of historical contingency and the vagaries of lying are both stripped away in the classic reformulation of the liar paradox: L: Statement L is false. If statement L is true, then it would be a true statement that says that L is false. Therefore, L is false. But if statement L is false, then it is correctly reporting its truth-value. If a statement says only what corresponds to reality, then it is true. Therefore, L is true if false and false if true!
A common first step toward a solution to the liar paradox is to maintain that “Statement L is false” is neither true nor false. One way to interpret this solution is as a repudiation of the law of bivalence. According to bivalence, every proposition has one of two truth-values: true or false. At this juncture, many philosophers claim a connection between the liar paradox and the paradox of the horned man: What you have not lost, you still have. You have not lost your horns. Therefore, you still have your horns. Deniers of bivalence go between the horns of the dilemma: “You have not lost your horns” presupposes that you had horns. A statement with a false presupposition is neither true nor false. A bachelor is not required to answer yes or no to “Are you still beating your wife?” Since none of the direct answers to this question are true, the bachelor must answer indirectly by correcting the false assumption that he is married. (Incidentally, “the horns of a dilemma” is derived from the name of the horned man paradox by means of the Latin argumentum cornutum.)
Samuel Wheeler (1983) conjectures that Eubulides solved the paradox by treating “your horns” as an empty name, like Pegasus. The second premise, You have not lost your horns, would then be meaningless rather than merely neither true nor false. Similarly, Wheeler suggests that Eubulides solved the liar paradox by denying that any statements are false. Falsehood concerns what is not, and what is not does not exist.
The Parmenidean approach could also be applied to the sorites paradox. If you have a heap of sand and subtract one grain, then you still have a heap of sand. One grain cannot make a difference between whether a collection of sand is a heap or not a heap. Given this principle is true, you will have a heap of sand regardless of how many grains of sand we subtract. But this leads to the absurd conclusion that one grain of sand is a heap!
In a commentary on Aristotle’s Nichomachean Ethics, Aspasius says that Eubulides used the sorites to criticize Aristotle’s theory of virtue. (Moline 1969, 396) Aristotle believed that virtues are dispositions that lie between an excess and a deficiency. For instance, courage lies between foolhardiness and cowardice. Generosity lies between liberality and stinginess. Aristotle concedes that the mean is not the same for everyone. The mean for a soldier’s courage is closer to foolhardiness than for a civilian. Perhaps the mean also shifts with one’s stage of life or circumstances. But even with this flexibility, Aristotle’s theory of virtue is vulnerable to a sorites argument. Suppose that in the case of Aristotle himself, a donation of one hundred drachmas to war widows would be generous. Donating ninety-nine drachmas would still be generous. A one drachma difference cannot make the crucial difference between a generous and nongenerous donation. Repeated applications of the principle leads to the conclusion that Aristotle would be generous if he donated a single drachma.
Aristotle frequently says that we should demand only as much precision as the subject matter allows. For instance, many factors of commerce depend on convention and fluctuating conditions. So a commentator on the economy must speak roughly and in outline rather than with the precision of mathematics or science. If Aristotle took these limitations about subject matter to be limits about the corresponding concepts, then he might have rejected Eubulides’ challenge to draw the line between generous and nongenerous donations. That is, he might have insisted there is a certain looseness in the concept of generosity that makes it illegitimate to ask which amount is the minimum generous donation. He might even have denied that there is any fact to be discovered. Textual evidence suggests that Aristotle demands precision from ethical concepts:
Similarly, too, we must state what quantity of money which he desires makes a man avaricious and what quality of pleasures which he desires makes a man incontinent . . . And similarly, in all cases of this kind; for the omission of any differentia whatever involves a failure to state the essence.
(Aristotle’s, Topics 146 b)
Aristotle’s ethical theory does seem to imply that there is a minimum generous donation (Moline 1969). One of his themes is that the judgment of a generous man sets the standard of generosity. When the generous man stops judging the donation as generous, the donation stops being generous.
Eubulides would have doubted that the generous man’s judgments are definite enough to support Aristotle’s solution. There is often no way to tell whether a difference of one drachma would alter the generous man’s opinion as to whether the donation was vague. Many generous men deny that there is such a thing as a minimum generous donation. If Aristotle lets the judgment of generous dissenters be the measure, his theory would be refuted. Their belief that there is no minimum would be enough to preclude a minimum.
The psychology of morality is as vague as morality itself. Sextus Empiricus liked to introduce the sorites paradox by first observing that it is not incest to touch your mother’s big toe. This is a slippery slope for “incest.” But it can also be pressed into service as a slippery slope for “judged to be incest by a virtuous man.”
A sorites argument can be raised for nearly everything. Nearly all of our words have borderline cases. “Table” is vague because borderline cases can be created by shaving off slivers of the table. Nearly everything can be whittled down to a doubtful case. Why pick on Aristotle?
This is a fair criticism of Eubulides. But it plays into Eubulides’ larger agenda. Parmenides reasoned that since nearly all of our concepts are concerned with differences between things, none of them applies to anything. Eubulides thought the sorites corroborated this sweeping nihilism. As followers of Parmenides, the Megarians would respond to all sorites arguments by denying the existence of the objects in question: there are no heaps or generous donations or incestuous acts. Ordinary things are illusions.