1. The Liar 说谎者悖论 英文解析 逻辑学教材(2)

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 : (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.

全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking

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 that 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

7 But, see the handout “Deviant Logic, No. 3: Bo var s 3-Valued Logic.” A sentence like these that describes its own content or construction is called an autogram.

全面解析the paradox of the liarfrom the different components of modern logic to the ancient greek philosophical thinking

V. A Proposed Solution

One way to escape would be to somehow guarantee that liar sentences could never 8arise. 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 …… 此处隐藏：5812字，全部文档内容请下载后查看。喜欢就下载吧 ……