1. The Liar 说谎者悖论 英文解析 逻辑学教材

The Paradoxes of Eubulides of Miletus (4th century B.C.)

1. The Liar

“When I lie and admit that I lie, do I lie or speak the truth?”1

I . The Paradox

A statement that directly or indirectly predicates falsehood of itself is logically

pathological in the worst, most mind-boggling way. Such sentences are called …liar sentences .? Version 1a : This sentence is false.

Proof of Paradoxicality

-That sentence, like every other, must be either true or false.

-If the sentence is true, then it follows that what it says is the case. It says that it is false, and, since what it says is the case, it therefore is false. So, if the sentence is true, then it follows that it is false. Contradiction .

-Therefore (by reductio ad absurdum /~I), the sentence …This sentence is false? cannot be true and, so, must be false.

-But if the proposition is false, then it follows that what it says is not the case. It says that it is false, and, as what it says is not the case, it follows that it is not the case that it is false. Therefore, the sentence is true. So, if the sentence is false, then it follows that it is true. Contradiction .

-Therefore (by reductio ad absurdum ), the sentence …This sentence is false? cannot be false and, so, must be true, which (as we proved) it cannot be…???!!!

As the argument shows, a proposition that directly or indirectly says of itself that it is false must be either true of false (like any other proposition), and, pathologically, cannot be either.

Version 1b : I am lying.

Version 2a : The following sentence is false.

The preceding sentence is true.

1 Aulus Gellius, Attic Nights , vol. 3, bk. XVIII, ch. II. Version 2b :

Hmwk . 1. Show in full and precise detail (on the model of my previous demonstration)

that these three situations are genuinely paradoxical, i.e., some statement must be false if it is true, must be true if it is false, and no third option is

possible.

In all four variations, the truth of a sentence directly or indirectly implies its own falsehood (which is a contradiction), and the sentence?s falsity directly or indirectly implies its own truth (which is a contradiction). By itself, one of these contradictory results merely would constitute a valid reductio ad absurdum/indirect proof for the non-contradictory denial of the assumption (of either truth or falsehood). But, both contradictory results together generate ineliminable contradictoriness: we must hold as true what we cannot hold as true…and the mind boggles. (Such paradoxes are also called …antinomies.?)

II. In the New Testament

In his Epistle to Titus, the Apostle Paul complains of the Cretans adding:2

One of themselves, even a prophet of their own, said, …The Cretans are always liars, evil beasts, slow bellies.? This witness is true.

According to St. Paul, then, there is a Cretan prophet who asserts that the Cretans always lie, which implies, of course, that the Cretan prophet himself is lying.3Now, that?s fine; but in adding that the Cretan prophet is telling the truth, Paul brutely contradicted himself.

Proof

1. Epimenides is a Cretan.

2. Epimenides said, “The Cretans are always liars.”

*3. This witness is true, i.e., what Epimenides said is true.

4. Therefore, “The Cretans are always liars” is true. [by (2) & (3)]

5. Therefore, the Cretans are always liars. [by (4)]

6. Therefore, Epimenides always lies. [by (1) & (5)]

7. Therefore, Epimenides lied when he said, “The Cretans are always liars.” [by (6)]

8. Therefore, what Epimenides said is false, i.e., this witness is false. [by (7)]

Contradiction: (3) & (8)! St. Paul?s assertion that Epi menides told the truth

implies that Epimenides lied.

Though the Apostle seems not to have realized it, we know that his self-contradiction constitutes a valid reductio ad absurdum (or is it a consequentia mirabilis?) that (3)—his assertion that the witness is true—is false.

Have we just disproven the thesis of Biblical literalism—every statement in the Bible is

literally true?

2 Titus 1: 12-13. The mentioned Cretan prophet was probably Epimenides, and the Liar Paradox is often called the Epimenidean Paradox.

3By the way, a Cretan is a person from Crete while a cretin is an idiot. You?re bound to insult somebody if you mix the two words up, so get it straight. (Originally, a cretin was someone who suffered the crippling mental and physical results of thyroid disease. The word …cretin? comes from the Swiss French …crestin’ meaning …Christian.? That is, these unfortunates, though retarded and deformed, are still human beings.)

It then follows that the witness is false, i.e., what Epimenides said is false. However, this conclusion does not entail a contradiction.To see that, let?s find out what that conclusion does entail.

Argument

a. The witness is false, i.e., what Epimenides said is false. [by Proof]

b. Epimenides said, “The Cretans are always liars.” [by (2)]

c. Therefore, …The Cretans are always liars? is false. [by (a) & (b)]

d. Therefore, …Some Cretans are not always liars? is tru

e.

Proof: The contradictory of an A proposition is the corresponding O proposition. Or, in

the language of predicate logic, ~(?x)(…x…) is equivalent to (?x)~(…x…) (by the

quantifier negation rule).

e. Therefore, there is at least one Cretan who does not always lie, i.e., who tells the truth

on at least one occasion.

[by (d)] f. There are three ways for (e) to be true:

i. Epimenides told the truth on this occasion,

ii. He told the truth on at least one other occasion, or

iii. There is at least one other Cretan who sometimes tells the truth.

Argument: I can?t think of any other.

g. (i) is false. [by (a)]

h. We have no grounds for denying either (ii) or (iii).

Argument: The fact that Epimenides lied on the relevant occasion alone implies nothing about the truthfulness of his other assertions or other Cretans? assertions.

i. Therefore, either (ii) or (iii) (or both) is true.

So, it follows from Epimenides? having spoken falsely when he said that the Cretans are always liars that at least one Cretan spoke, or will speak, truly on at least one occasion. And there is certainly nothing contradictory about that. In fact, I?m sure it?s true (and you are too).

If the only thing any Cretan ever had said, or ever would say, were, “The Cretans are

always liars,” then we?d be in trouble; but since that isn?t the case, we?re not.

Thus, Paul?s blunder isn?t genuinely paradoxical. So, it?s not really a version of the Liar, but I include it for its historical and religious interest.4

III. Reductio ad Absurdum?

4I can?t resist giving one more fascinating version of the liar paradox due to medieval logicians. So, for the sake of argument let?s accept the medieval claim that …God exists? i s necessarily true. Consider the following argument: God exists. Therefore, this argument is invalid. Is the argument valid? (Any necessarily true proposition may replace …God exists.?)

It?s worth being clear that there is no reductio ad absurdum escape from the Liar. For, what unrecognized, implicit assumption have we made such that we can take the Liar?s double-contradictory conclusion to constitute a reductio ad absurdum proof for its falsehood? Well, to begin, we?ve assumed that there is a liar sentence. So, perhaps the Liar proves there

is no such sentence. No, this is ridiculous! We didn?t assume that a liar sentence exists. The sentence is right there on page 1! (You don?t ne ed to turn back; the rumored-to-be-non-existent sentence will occur in my next sentence.)

We?ve also implicitly assumed that the sentence, …This sentence is false? actually says something, i.e., that the sentence is meaningful, it expresses a proposition. Maybe the Liar proves that that is a perfectly grammatical sentence that expresses no proposition, is meaningless.5 This suggestion fails for two reasons.

First Reason.If the sentence …This sentence is false? expresses no proposition, then it follows that it is meaningless. Therefore, that sentence is neither true nor false, and, so, we seem to have escaped. But, consider the sentence:

This sentence is not true.

This new sentence is also a paradoxical liar sentence.

Hmwk. 2. Prove it.

Now, according to the proposed solution, the new sentence is also meaningless, and, therefore, neither true nor false, and, thus, not true. But, the new sentence just says that it is not true. Therefore, it is true (for what it says is the case)! But if the sentence is true, then it cannot be meaningless. Consequently, the proposal does not resolve the paradox.

Second Reason. The proposal only ever had a chance of resolving versions 1a & 1b in which a sentence directly predicated falsehood of itself. In versions 2a & 2b, where a sentence indirectly predicates falsehood to itself by ascribing a truth-value to another sentence, it?s clear that the sentence is perfectly meaningful because you effortlessly understood it. After all, it was only as a result of reading it that you went on to consider precisely the sentence to which it referred.

Perhaps another example shows the point even more clearly:

The sentence printed in the box on p. 160 of Howard DeLong’s A Profile of

Mathematical Logic (Addison-Wesley, 1970) is true.

Go ahead; look it up. (I?ll wait.)

Finally, we?ve implicitly assumed the principle of bivalence: (i) there are only two truth-values, viz., the true and the false, and (ii) every sentence must have one, and only one,

of these. Therefore, we?ve implicitly assumed that our liar sentences have to be either true or false. Perhaps the Liar shows that some meaningful sentences have no truth-value. This is a

5 The Stoic logician Chrysippus suggested this resolution.

very complex and abstract reply, too complex and abstract for us to give it much consideration now.6 However, we can at least consider how difficult it is to make sense of the possibility that a sentence expresses something coherent about the way things are and, yet, it is neither true nor false, i.e., things neither are that way nor not that way. Which way, then are things supposed to be?

It seems, then, that the Liar, unlike the Barber or the Law Court, is a genuine paradox. IV. What the Liar Paradox Shows

Like questions about Russell?s barber shaving himself, the Liar reveals the surprising paradoxicality that self-reference sometimes produces. Now, self-reference clearly is not paradoxical in general, for most instances of it are perfectly coherent. Consider some examples.

1. Sax self-referentially asserts tha t he is bald. His statement is simply false, but it isn?t

paradoxical. (Indeed, a statement must be coherent even to be false.)

2. (i)-(iv) of the following self-referential sentences are just true; (v)-(vi) are just false:7

i. This sentence contains five words.

ii. This sentence contains thirty-six letters.

iii. There are fourteen vowels in this sentence.

iv. This sentence is written in English.

v. This sentence contains precisely fifty characters.

vi. Deis ist kein Deutscher Satz. [Translation: This sentence is not in German.]

3. Epimenides can self-referentially assert the self-referential sentence, “The sentence I am now

uttering contains nine words,” and his remark will be unproblematically true.

Still, the Liar shows that those instances of self-reference in which a statement directly or indirectly predicates falsehood of itself are logically pathological in the worst, most mind-boggling way: they generate ineliminable, unresolvable contradiction. And, the Liar also shows that a liar sentence can be constructed in any language with the resources to:

A. refer to its own sentences, and

B. predicate truth or falsity of those sentences.

Unhappily, all natural languages (like Gujarati, Magyar, Frisian, etc.) satisfy (A) and (B). That is very bad, for, as we?ve seen, liar sentences generate unpreventable contradiction, and, as we?ve learned, every statement validly follows and is derivable from a contradiction. In such a language, therefore, the entire practice of deductive reasoning as part of the search for truth is hopelessly corrupt. So, deductive rationality carried on in English (Basque, Korean, etc.) is a farce and a delusion…unless we can escape from the Liar.

6 But, see the handout “Deviant Logic, No. 3: Bo?var?s 3-Valued Logic.”

7A sentence like these that describes its own content or construction is called an …autogram.?

V.A Proposed Solution

One way to escape would be to somehow guarantee that liar sentences could never arise.8 In 1931, the logician Alfred Tarski argued that we could block the possibility of liar sentences if we:9

1. strictly distinguish object languages from metalanguages, and

2. restrict language so that it is only in the metalanguage that one can construct sentences about the sentences of an object language, and, so, predicate truth or falsity of those

object-language sentences.

Satisfying these conditions would prevent a language from doing (A) and (B).

For example, the English language would become an infinite sequence of distinct languages: English1, English2, English3, …where each succeeding language would be the meta-language for the previous one.

English1 would contain almost all of the vocabulary of the English we currently speak, but it would lack all semantic vocabulary: words like …sentence,? …proposition,? …assertion,? …meaning,? …lies,? …true,? etc. Thus, we could talk abo ut snow (and shoes, ships, and sealing wax, etc.) in English1 by asserting the English1sentence, …Snow is white.? However, English1 would lack the resources for us to be able to talk about any English1 sentences (or any other sentences) in English1.

So, if we would want to talk about the English1sentence …Snow is white? in order to say, for example, that it is true, then we would have to treat English1 as the object language and speak in its metalanguage English2. Among its vocabulary, English2 has the semantic terms needed to talk about sentences in English1. Only in English2, then, could we say that …Snow is white? is true, which we would do by asserting the English2sentence, “The English1sentence …Snow is white? is true.”

Now, English2 only has the semantic vocabulary needed to talk about sentences in English1, not its own sentences (or those of any other language in the sequence). Similarly, then, it would be only in English3—the metalanguage of English2—that we could construct a sentence to assert the truth of that English2 sentence, viz. “The English2sentence, …The English1sentence, “Snow is white” is true? is true.” And so on through the series of English metalanguages.

As no language in the infinite series could satisfy either (A) or (B), Tarski?s stratification banishes all same-level sentence reference and, thus, all sentence self-reference, 8 Of course, banishing all self-reference would guarantee that. But, that program would be ridiculously extreme. Almost all instances of self-reference are harmless and, indeed, very useful if not practically irreplaceable.

9In “The Concept of Truth in Formalized Languages,” but for a very readable version I recommend Tarski?s “Truth and Proof” [Scientific American, vol. 194, no. 6, pp. 63-77].

as ungrammatical and, therefore, meaningless. Consequently, no sentence in any of the languages could directly say anything at all about itself.

Hmwk. 3. Prove that version 1a of the Liar cannot be constructed in Tarski?s hierarchy.

Of course, one could utter or write the words …This sentence is false? or …I am lying,? but these putative self-referential constructions would be utterly ungrammatical—like the English construction …Shoes pointedly my sleep eat?—and, hence, not sentences; therefore, not meaningful; and, so, without truth-values.

Equally, indirectly self-referring sentences would be impossible to construct.

Hmwk. 4. Prove that version 2a of the Liar cannot be constructed in Tarski?s hierarchy.

Therefore, no sentence in any language in the series could (directly or indirectly) predicate falsity of itself. Consequently, no liar sentence could be constructed in any language in the series, and the paradox would be resolved.

Despite the fact that Tarski?s proposal would, if carried out, quash the liar paradox, the proposal has few adherents. (Yet, it?s impossible to overstate the historical importance o f Tarski?s work.) The problem is that Tarski?s proposal is too crude, for it banishes not only liar sentences but all (directly or indirectly) self-referring sentences. Consequently, it is too extreme, for, again, the vast majority of self-referring sentences are perfectly banal, and some are practically irreplaceable—consider the indirectly self-referring sentence, …All English sentences begin with a capital letter.? There is no good reason to proscribe all of these sentences and very good reasons not to proscribe some of them.

It is also interesting to note that successfully carrying out Tarski?s proposal makes that proposal impossible. If Tarski?s stratification were carried out, the description of the hierarchy of languages would have to be impossible, for the sentences of that description would have to belong to some finite level of the hierarchy in order to be meaningful, but in order to describe the whole hierarchy they couldn?t belong to any of its levels.