Fig. 8.1
The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato
—Alfred North Whitehead
Most philosophers think that a paradox involves an argument that moves from acceptable premises to an unacceptable conclusion via acceptable reasoning. I have already objected that this definition is too narrow; it precludes the perceptual paradoxes of chapter one and the meaningless paradoxes of chapter three. I now want to criticize this reigning view as also being too broad. My counterexamples exploit the preface paradox and the “paradoxes of strict implication.”
Plato’s real name was Aristocles. “Plato” was a nickname. It means “the broad-shouldered one.” The name was apt because of Plato’s powerful build as a wrestler.
But was Plato’s original name even more apt? Someone is well named if the predicate corresponding to his name describes him. Professor Sober is well named because he is sober. Professor Grim is ill named because he is not grim. Aristocles means well-named. So the issue is whether Well-Named is well named.
We would run into a dilemma if Plato had instead been named Ill-Named (Geach 1948). Either Ill-Named is well named or ill named. If Ill-Named is ill named, then the predicate corresponding to his name accurately describes him. So if he is ill named, then he is well named. But if Ill-Named is well named, then the predicate corresponding to his name does not apply to him.
The ill-named dilemma is a version of the liar paradox: L: Statement L is false. If L is true, then since it says that it is false, it must be false. But if statement L is false, then things are just as L says—in which case L is true after all. L is paradoxical because there is no consistent assignment of truth-values to it.
The opposite problem makes the truth-teller sentence paradoxical: R: Statement R is true. R is paradoxical because we can consistently assign either truth-value to R. Since there is no further basis for assigning a truth-value, R would be true or false without anything making it true or false.
Most logicians react by saying that R is neither true nor false. This is based on the feeling that the sentence is empty. But if R is neither true nor false, then R seems to falsely claim to be true.
“Well-Named is well named” is a version of the truth-teller paradox. We can consistently say that “Well-Named is well named” is true and we can consistently say that it is false. If we conclude that it lacks a truth-value, then “Well-Named is well named” seems false because well named fails to truthfully describe Well-Named. But if well named does accurately describe Well-Named, what is the property it so accurately describes? The very property of being well named?
Is Aristocles Aristocles? This philosophical question was missed by Plato. What a lost opportunity! Philosophy could have become a series of footnotes to Aristocles.
Was Plato aware of the liar paradox? The legend of Epimenides had been in circulation for hundreds of years before Plato’s birth. Plato has Socrates raise an objection to Protagoras that tacks near the shores of the liar paradox. Protagoras’s slogan “Man is the measure” is presented as implying that all beliefs are true: What seems to a man, is to him. To refute Protagoras, Socrates needs a criticism that will work within a single individual’s belief system. He sees an opening in the fact that each person has beliefs about his beliefs:
. . . there is no one in the world who doesn’t believe that in some matters he is wiser than other men; while in other matters, they are wiser than he. In emergencies—if at no other time—you see this belief. When they are in distress, on the battlefield, or in sickness or in a storm at sea, all men turn to their leaders in each spheres as to God, and look to them for salvation because they are superior in precisely this one thing—knowledge. And wherever human life and work goes on you find everywhere men seeking teachers and masters, for themselves and for the other living creatures and for the direction of all human works.
(Theaetetus 170 B–C)
Consider someone who believes that at least one of his own beliefs is false. Protagoras’s principle that all beliefs are true implies that “At least one of my beliefs is not true” is true for this modest individual. Thus “All beliefs are true for the believer” when applied to the modest believer yields “Not all beliefs are true for the believer.” Since Protagorean relativism implies its own falsity, Protagorean relativism is false.
Socrates’ objection to Protagoras resembles D. C. Mackinson’s (1965) preface paradox. In the preface of the book you are now reading, I apologize for the errors that are sure to be in the text. This acknowledgment of my fallibility is good common sense. Yet, it does make it impossible for all of my beliefs to be true. If the belief expressed in the preface is true, then one of the beliefs in the text is false. If all the beliefs in the text are true, then the belief in the preface is false. Either way, I have a false belief.
Although the preface paradox damages Protagoras’s relativism, it also undermines Plato’s assumption that rationality implies consistency. Since it is impossible for all of my beliefs about this book to be all true, they are jointly inconsistent. Yet, the belief I express in the preface is rational.
Indeed, this degree of intellectual humility is intellectually mandatory. Know thyself! If I failed to believe that some of the beliefs expressed in this text are false, then I would be a vain scholar.
In addition to being modest about whether my beliefs are true, I should be modest about whether all my beliefs are consistent. The more I say, the more opportunities I have to contradict myself. I say very much in this book and so believe that the assertions in this text (even apart from those in the preface) are jointly inconsistent.
Take the first 10,001 assertions I make in this book. I believe that any conjunction of 10,000 of them is inconsistent. Now consider any argument that takes 10,000 of my 10,001 assertions as the premises and takes the negation of the remaining assertion as the conclusion. This jumble argument would fit R. M. Sainsbury’s definition of a paradox:
This is what I understand by a paradox: an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable. So, generally, we have a choice: either the conclusion is not really unacceptable, or else the starting point, or the reasoning, has some non-obvious flaw.
(Sainsbury 1995, 1).
The conclusion of the jumble argument is unacceptable to me because I sincerely assert its negation in my book. Each premise of the jumble is acceptable because I sincerely assert it in my book. The reasoning in the jumble argument is acceptable to me because I think the argument is deductively valid: Any argument with jointly inconsistent premises is automatically valid. This principle follows from the definition of valid: an argument is valid if there is no possibility that all its premises are true while its conclusion is false.
An argument is sound if it is both valid and has true premises. Since logicians do not have any special knowledge about the truth of premises, they focus on validity. They happily endorse silly arguments such as “All tortoises belong to the genus Testudines. Plato is a tortoise. Therefore, Plato belongs to the genus Testudines.” If the premises were both true, then the conclusion would be true. Validity is merely a conditional guarantee of a true conclusion. If there is no way the premises could all be true, then the guarantee holds vacuously.
Euclides’ disciples in Megara (most famously Philo) pioneered the logical doctrine that the indicative conditional “If p, then q” is false only when the antecedent, p, is true and the consequent, q, is false. This implies two “paradoxes of material implication.” The negative version is that any indicative conditional with a false antecedent is true. For instance, “If Socrates visited the moon, then Plato visited the sun” comes out true. The positive version is that any indicative conditional with a true consequent is true. Thus, “If Socrates visited Atlantis, then Plato visited Megara” comes out true.
Some logicians softened the blow by emphasizing that strict conditionals have the form, “Necessarily, if p, then q.” These conditionals are not made automatically true by the antecedent being false or the consequent being true. However, subsequent logicians discovered “the paradoxes of strict implication”: a strict conditional is made automatically true by the necessary falsehood of its antecedent or the necessary truth of its consequent. Many minor paradoxes orbit the paradoxes of strict implication. My favorite is Jaakko Hintikka’s proof that it is immoral to do the impossible: One should never do anything that entails the destruction of the human race. One cannot do an impossible act without also destroying the human race. Therefore, one should not do the impossible.
An argument is valid if the conjunction of its premises strictly implies the conclusion. Thus, the paradoxes of strict implication affect some verdicts of validity. If the conclusion of an argument is a necessary truth, then the argument is automatically valid. If all the premises are true, then the argument will also be sound. This is the basis for some sophistical proofs of God’s existence. Medieval logicians believed that “God exists” is a necessary truth and so bemusedly regarded “Plato philosophized, therefore, God exists” as a sound argument. The devout mathematical genius Leonhard Euler (1707–1783) sprang this logical trick on the French atheist Denis Diderot. Addressing Diderot before the court of Catherine the Great, Euler solemnly said, “Sir, (a + bn) /n = x, hence God exists. Reply!.” Since Diderot was not mathematically sophisticated, he did not know what to say. He was laughed out of St. Petersburg and hurried back to France.
A believer in classical logic can simultaneously perceive an argument as sound and as a positive instance of the paradox of strict implication. For instance, medieval logicians regarded “Plato philosophized, therefore, God exists” as a paradoxical argument even though they agreed that the proof was sound. The paradox can be in how you prove something rather than in what you prove. This point causes indigestion for those who say that all paradoxes feature unacceptable conclusions. Their accounts are too narrow.
Jumble arguments exploit a consequence of the negative paradox of strict implication: Any argument with premises that combine to form a necessary falsehood must be a valid argument. When a jumble argument is tailored to the belief system of a modest classical logician, he will accept the reasoning because he thinks that the conjunction of the premises is necessarily false. (Whether it is actually false does not matter for the purposes of the counterexample.)
If jumble arguments qualify as paradoxes, then any negation of one of my beliefs is a paradox. For it would be an unacceptable conclusion backed by an argument whose premises I (individually) accept and whose reasoning I accept.
Of course, I regard each jumble argument as unsound. But that is common for paradoxes. The conclusions of Zeno’s arguments are plainly false and so Zeno’s arguments must be unsound. But this obvious unsoundness does not stop me from classifying Zeno’s conclusions as paradoxes.
We cannot exclude jumble arguments by requiring that paradoxes be short arguments. A sorites argument can have 10,000 premises. There are also paradoxical truths, such as Kurt Gödel’s incompleteness theorem, that take a whole semester to prove. Gödel’s proof is made lengthy by Gödel’s caution rather than by any sophistry.
For many of us, there are short jumble arguments. Paradoxes have convinced many philosophers that they have small sets of beliefs that are individually plausible but jointly consistent. If those beliefs are used as the premises of an argument, and the negation of some other belief is used as the conclusion, then that conclusion will satisfy Sainsbury’s definition of a paradox. Thus, from one genuine paradox we can grow as many short jumble arguments as we have beliefs.
I think the real difference between Zeno’s paradoxes and jumble arguments is that jumble arguments can be diagnosed as unsound without violating a Socratic commandment: Thou shalt not rely on the implausibility of the conclusion when explaining what is wrong with an argument. I abide by this requirement when I infer that a jumble argument is unsound directly from the inconsistency of its premises.
The rationale for the Socratic commandment is that Socrates wants to use argument as a method of discovery. He disagreed with Myson who “used to say we should not investigate facts by the light of arguments, but arguments by the light of facts; for the facts were not put together to fit the arguments, but the arguments to fit the facts.” (Diogenes 1925, I, 107–108) Socrates does not consider deduction as just an efficient way to unpack data that has been neatly stored in generalizations. Nor is deduction just a way to justify beliefs that you already hold. Socrates thinks of deduction in the same way that we think of an experiment: as a neutral method that can overturn our strongest convictions.
As we saw in his description of lawyers, Socrates despised “special pleading” in which one first has a conclusion and then comes up with reasons for holding that conclusion. The rationalizing of a propagandist is merely aimed at making a conclusion look reasonable. Socrates blinds himself to the outcome of his reasoning to ensure that he is being solely guided by the premises and the rules of inference. Admittedly, Socrates’ self-blinding throws away information. But this trade-off is familiar to us in the form of double-blind experiments, in which scientists prevent wishful thinking by keeping their subjects and themselves in the dark as to who receives the active treatment and who receives the placebo.
When we say Zeno’s bisection argument “looks sound,” we only mean it looks sound from a certain perspective—a perspective that does not include knowledge of the conclusion’s absurdity. The columns of the Parthenon look straight when viewed from the ground but do not look straight from the roof. From up high you can see that the columns slant inward to make them look straight to the people below. When Socrates follows an argument wherever it leads, he refuses to take a perspective that uses information about the conclusion’s plausibility. Socrates always reasons forward from the premises, never backward from the conclusion. (The only exception is when he reasons by reductio ad absurdum.) Even nowadays, many feel that a philosopher relying on outside evidence about the conclusion is like a student peeking at the answer book.
Many illusions require special viewing conditions. I have an “impossible crate” suspended from my office ceiling by an invisible thread. When an inquisitive visitor asks about this irregularly shaped mobile, I station him at a predetermined distance, have him close one eye, and then tell him to align his line of sight along the near and far corners of the “crate.” Under those special conditions, the crate looks like a three-dimensional counterpart of figure 8.1. Many paradoxical arguments require the same disciplined contemplation. Instead of evaluating them normally, you must start from premises that you only pretend to accept, proceed with inference rules that you regard as questionable transport, and then suppress the urge to backpedal from your destination.
No one believes that the crate is as it appears. Nor do they expect to. They make the effort partly for the sheer spectacle and partly in the hope of being edified. Cognitive paradoxes have the same mixture of aesthetic and theoretical allure. A scientist who goes through the observational contortions needed to witness a double rainbow will concede that rainbows are illusions. Yet, he will value the experience as much as an observation of a real phenomenon such as a solar eclipse.
The eminent Plato scholar Gregory Vlastos (1983) used to interpret Socrates as refuting his interlocutors by deriving a contradiction. This was the standard view as reflected in the Encyclopedia of Philosophy entry for “Dialectic”:
The Socratic elenchus was perhaps a refined form of the Zenoian paradoxes, a prolonged cross-examination which refutes the opponent’s original thesis by getting him to draw from it, by means of a series of questions and answers, a consequence that contradicts it. This is a logically valid procedure, for it corresponds to the logical law “if p implies not-p, then not-p is true.”
(Hall 1967, 386)
But Vlastos eventually noticed that Socrates usually achieves only the weaker result of showing that the interlocutor’s beliefs are jointly inconsistent. Vlastos finds this alarming because Socrates is under the impression that he has refuted a particular member of the inconsistent set. From a logical point of view, consistency can be regained by rejecting any member of the set. Therefore, Socrates has not refuted anything. All he has done is shown that his interlocutor needs to revise some of his beliefs.
Maybe not even that! News that a set of beliefs is inconsistent only justifies a hunt when there is some prospect of success. When a few false beliefs are concealed in a mass of true beliefs, then the cost of correction is high. Each belief enjoys a high probability. Under these poor hunting conditions, we resign ourselves to inconsistency.
Actually, Socrates does tend to ensure that good hunting conditions prevail. The inconsistencies exposed by Socrates consist of a small set of beliefs. They also tend to be minimally inconsistent: eliminating even just one member of the set would be enough to gain consistency. In other words, the beliefs in that small search space are nearly consistent. Just a little tinkering might solve the problem!
Socrates did not plan to create a minimally inconsistent set. His goal is to detect an inconsistency. He stops eliciting commitments when he thinks the job is done. A by-productof this dialectical efficiency is that you leave the inconsistent individual with the hope of becoming consistent by changing a single belief.
Socrates acts as if the elenchus is a universal method that allows him to take on all comers no matter what they believe. But the method does have some presuppositions. First, the elenchus presupposes that there are inconsistencies. Antisthenes of Athens argued that no statement can contradict another statement. The idea is inspired by Parmenides: A statement can be about Socrates only if it applies to Socrates. If it applies to Socrates, it is a true statement. Therefore, no false statement can be about Socrates. A statement about Socrates can be contradicted only if a false statement can be about Socrates. Consequently, no statement about Socrates can be contradicted. What goes for Socrates, goes for all things. Therefore, there are no contradictions.
Plato appears to target Antisthenes in the Euthydemus. Antisthenes is portrayed as a sophistical wrangler busy in the practice Protagoras advertised as “making the weaker argument appear the stronger.” Antisthenes in turn wrote a polemic against Plato under the naughty title Sathon or Little Willy. The Greek expression for “Little Willy” is a near pun of “Plato.” Antisthenes used Sathon as a name for Plato.
The elenchus assumes contradictions cannot be true. Since Antisthenes thinks there are no contradictions, he agrees that there are no true contradictions. However, there have been philosophers who have affirmed both the existence of contradictions and their truth. A contemporary example is Graham Priest (1987). He thinks a minority of contradictions are both true and false. According to Priest, paradoxes figure prominently in this provocative minority.
Priest realizes that his position is untenable in contemporary classical logic. As we saw with the negative paradox of strict implication, anything follows from an impossibility. When the impossibility is a strict logical contradiction, we can even formally derive the arbitrary consequence: From the premise p and not-p, we can infer p. From p we can infer p or q (where q is any proposition). Now we can go back to p and not-p and infer not-p. When not-p is combined with the earlier result of p or q, we can deduce q.
Classical logic is “explosive”; one contradiction implies every proposition. Is there any way for Priest to avoid an indiscriminate acceptance of everything?
Ludwig Wittgenstein (1889-1951) poked fun at the logicians who trumpeted contradictions as intellectual disasters. In real life, when people discover they have fallen into contradiction, they unceremoniously patch up the problem—if they do anything at all. Wittgenstein looked forward to the day when logicians would adapt to this anthropological reality: “Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from consistency.” (1964, 332)
And indeed there soon arose “dialethic logicians.” They take their name from Wittgenstein’s remark that the liar sentence is a Janus-headed figure facing both truth and falsity (1978, IV.59). A dialetheia is a two(-way) truth. As a dialetheist, Graham Priest values consistency but only in the comparative way he values simplicity, generality, and empiricalfruitfulness. Consistency is just one desirable feature among many. In the case of the liar paradox, Priest thinks we should trade a little consistency to get much simplicity. In particular, we should concede that “This sentence is not true” is both true and false and then use “paraconsistent logics” to stop the contradiction from spreading.
Paraconsistent logics are designed to safely confine the explosion. For instance, they reject the inference rule “p, therefore, p or q” on the grounds that a valid argument must have premises that are relevant to the conclusion. They extend this relevance requirement to conditionals in an effort to head off the paradoxes of implication.
Dialetheists portray themselves as friends of contradiction. They remind me of ranchers who present themselves as friends of the horses they castrate. A gelding is not just a tamer sort of stallion; it is not a stallion at all. The dialetheist’s “contradictions” may look like contradictions and sound like contradictions, but they cannot perform a role essential to being a contradiction; they cannot serve as the decisive endpoint of a reductio ad absurdum. At best they can be the q in a modus tollens argument: If p then q; not q, therefore, not p. So in the end, I think Priest falls into Antisthenes’ skepticism about contradictions.
Whether or not Priest agrees with me about his skeptical kinship with Antisthenes, he should agree that jumble arguments create a problem for Sainsbury’s definition of paradox. For I have been careful to subjectivize my examples. The reasoning in jumble arguments will be acceptable to those who believe in classical logic. Even if the reasoning in my jumble argument is unacceptable to Graham Priest, he will correctly calculate that the reasoning is acceptable to me—and to the majority of logicians. So according to Sainsbury’s definition, the jumble argument should count as a paradox to me. But since jumble arguments are not really paradoxes, they show that Sainsbury’s definition is too broad.