1. The Liar 说谎者悖论 英文解析 逻辑学教材

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

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 : This sentence is false.

-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.

: I am lying.

: The following sentence is false.

The preceding sentence is true.

Hmwk. 1. Show in full and precise detail (on the model of my previous demonstration) 1 Aulus Gellius, Attic Nights, vol. 3, bk. XVIII, ch. II.

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

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 . )

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.3 Now, that s fine; but in adding that the Cretan prophet is telling the truth, Paul brutely contradicted himself.

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)] 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 Epimenides 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

2literally true? Titus 1: 12-13. The mentioned Cretan prophet was probably Epimenides, and the Liar Paradox is often called the Epimenidean Paradox. 3 By 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.)

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

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.

a. The witness is false, i.e., what Epimenides said is false.

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

c. Therefore, The Cretans are always liars is false.

d. Therefore, Some Cretans are not always liars is true.

[by Proof] [by (2)] [by (a) & (b)] 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 leas …… 此处隐藏：5698字，全部文档内容请下载后查看。喜欢就下载吧 ……