一，译文

1，皮尔士对问题的分析（1864-65）

在1864-65年发表的一系列演讲（在哈佛大学发表的科学哲学演讲）的第1讲和第3讲中，皮尔斯讨论了“说谎者悖论”。在第1讲中，他讨论了句子“此命题本身是假的（This very proposition is false）”；在第3讲中，他讨论了句子“这里所写的不是真的（What is here written is not true）”。正如我们所知，这样的句子会导致悖论性的结论。我将首先讨论皮尔士对这个问题的分析，然后讨论他对这个问题的解决。

1.1 问题陈述

S1：此命题本身是假的（This  very  proposition  is  false）

S2：这里所写的不是真的（What is here written  is  not  true）

皮尔士认为，这样的句子的问题在于在逻辑上是“无意义的(meaningless)”，或者说，在逻辑上是“无稽之谈（nonsense）”，其中nonsense被定义为“与符号有某种相似性而又不是符号” ，一个真正的符号受制于三个系统的形式法则：（1）语法，（2）逻辑，和（3）修辞学。一个符号要成为“有意义的（meaningful）”，必须满足语法、逻辑和符号可理解性的形式条件。这个符号在语法上是正确的，但却不能成为一个真正的符号，因为它不满足逻辑的形式条件。

在上述句子S1的情况下，一个逻辑原则，即排中律，并不适用。皮尔士说：

- 这是一个命题，排中律，即每个符号都必须是假的或真的，对它并不适用。因为如果它是假的，它就是真的；如果不是假的，它就不是真的。

一个有逻辑意义的句子将满足逻辑原则。皮尔士认为，这一逻辑原则不适用于S1，因为这个符号没有对象。皮尔士说，逻辑学关注的是断言式命题，在谈到断言式命题时他说：“断言式命题总是断言一个对象的某些东西，这个对象就是命题的主体”，然而，在S1的情况下，命题“本身并没有说明它没有对象，它谈论自己，而且只谈论自己，没有任何外部关系。”也就是说，命题的主体是命题本身，谓词不对命题所指对象进行断言。一个断言式的命题提到了一个外部对象，但这个命题“谈论它自己，而且只谈论它自己，没有任何外部关系。”然而， “逻辑原则”， “只对作为一个符号有一个对象的条件而成立。 ” 

同样，关于S2，皮尔士说，我们得到的命题的数量是无限的：

What is here written

The  statement that  that is false

The  statement that  that is false

The  statement that  that is false

这里写的是

这句话是假的

这句话是假的

这句话是假的

以此类推，无穷无尽。

这些陈述中的每一个都是关于什么的。每一个似乎如果关于任何东西，都是关于这个无限数列的最后一个声明。然而，作为一个无限序列，并没有最后一个陈述，因此，整个陈述集什么也没说，它们没有任何意义。

因此，这个符号同样没有对象，但如果命题要满足所有的逻辑原则，拥有对象是符号的必要条件，而符号要有逻辑意义，就必须满足逻辑原则的准则，那么这个没有对象的符号在逻辑上是没有意义的。皮尔士在1864-65年系列讲座的第6讲中认为，在一个肯定的命题中，所指的对象必须是一个存在的事物的类，而不是一个空类。他解释说，矛盾的谓词可以被断言为空类；因此，这样的命题不能满足逻辑原则，在逻辑上是没有意义的，“说谎者悖论”被分析为一个肯定命题，其主体是一个空类。在这一时期，他坚持认为这种无逻辑意义的命题是错误的。

在一篇未发表的论文（Ms 726）中，皮尔士在考虑“此命题是真的”这一命题时，基本上重复了这种分析，他说：

为什么这是荒谬的？因为它没有提到一个中间的或直接的对象，它的主体是命题本身，除了它自己，它没有主体。而且，由于它的谓语只是指它自己对一个对象的指称，而这个对象又是对一个对象的指称，如此反复，它没有对象。这表明，如果一个符号没有对象，它就是荒谬的，这与不合逻辑是一样的。

2，皮尔斯的问题解决方案（1864-65）

关于S1和S2，可以说：

（1）该陈述在逻辑上是无意义的，它没有对象，不能满足所有的逻辑原则，因此，它既不是真的也不是假的。

（2）该陈述在逻辑上是无意义的，它不是真的，但它断言它不是真的，因此它是真的。因此，这句话既是真的也是假的。

2.1 皮尔士拒绝了上述（1），认为S1和S2既非真也非假的观点是自相矛盾的。他说，“作为无意义的陈述不是真的；但它毕竟是真的，因为它说它不是真的。”按照皮尔士的说法，每个被视为断言的命题都有一个真值，所以说它既非真也非假，似乎是自相矛盾的。作为逻辑上的无意义，命题S1和S2至少从逻辑的角度来看不是真的。皮尔士认为，S2提出了一个独特的情况，因为在这个命题的内容是一个断言，它不是真的，它也是真的。这个解决方案只在提到S2的时候说过。在考虑S1的第1讲中，对这个问题进行了类似的分析，但没有提出解决方案。皮尔士并不致力于根据他对S2的陈述将“非真”等同于“假”，但在为支持上述(2)而写的下文中，他似乎还是这样做了。

事实上，在这个命题中，真理和非真理--肯定的和否定的--这个和另一个--是一致的。它站在真与假的边界上；因此，它在两者之中。

2.2 为了支持这一观点，皮尔士提出了一个类比的论证。他让我们考虑一张纸，一部分是红色，一部分是蓝色。这张纸上的每一个点不是红色就是蓝色。他说，如果是这样，那么这两种颜色之间的边界线是什么颜色？皮尔士回答说：

- 很明显，它既是一个，也是另一个。因此，我们必须说，它既是也不是。

他说，我们可以争辩说：

（1）它既不是，因为颜色驻留在一个表面而不是一条线。但这种观点是不够的，皮尔斯说，因为“正如运动不在任何瞬间，因为它是瞬间之间的关系，但它还是在任何特定的瞬间。同样，虽然颜色需要一个表面，但它是在这个表面的每一个点上，因此这条线可以被描述为有色的。

（2）边界既是红色又是绿色。皮尔斯认为： ”如果那条线不是红色的，它就不在纸的红色部分，因此，如果我只是把红色部分抽走，我就不能影响那条不在其中的线的颜色。因此，那条线和移动的边界所经过的东西既不是红色也不是绿色；但它可能经过整张纸，因此整张纸既不是红色也不是绿色。但它显然是绿色的。因此，在我看来，正确的答案是，边界既是红色也是绿色；它们之间的区别在这一点上消失了。 "

类似地，上述命题既是真的也是假的。我们被告知，说它既非真也非假是自相矛盾的；但说它既真又非真，作为肯定与否定之间的区别消失的一个限制性案例，却不是自相矛盾的。

既然这个命题在逻辑上是无意义的，它无对象可指，所以它类似于一个命题，其主体指的是一个空类。这样的主体可以接受矛盾的谓语，正是基于这些理由，我们说这个命题不代表实际的对象，在逻辑上没有意义，因此是错误的。但在“说谎者悖论”的情况下，这个命题断言这里写的是假的；因此，这个命题没有对象，因此在逻辑上是无意义的和假的，这个命题对它自己所说的，即这里写的是假的，被认为是真的。也就是说， “这个命题就它本身所讲的而言是假的，但就它所讲的本身而言是真的。 ” 皮尔斯补充说：

- 但这是一个没有区别的区分，问题是这个命题是否在所有方面都是真的。如果它不是在所有方面都是真的，那么它在所有方面都是真的；原因有二：第一，因为它所说的在这种情况下不可能是完全真的；第二，因为它被视为恰好符合在所有方面都是真的；即它在所有方面都不是真的。

从这个观点出发，皮尔士认为，我们必须说这个命题既是真的又是假的。

3，皮尔士对“说谎者悖论”的修订分析（1868年及以后） 

在1868年及以后，皮尔士认为（5.340），一个命题只有在其中所说的任何内容都是真的情况下才是真的，但如果其中所说的任何内容都是假的，则是假的。

每个命题都被说成是真的或假的。如果命题中的任何内容是假的，那么它就是假的，否则它就是真的。

关于“说谎者悖论”，他在1868年认为，这个命题所表示的内容或多或少断言明确。如果它意味着更少，那么它就没有任何意义，是没有意义的。但是，皮尔士认为，这个命题是有意义的，他拒绝了他在1864-65年的论点，认为这个命题的含义比明确陈述的多。

皮尔士认为，每个命题除了明确断言的内容外，还默示了自己的真实性。说谎者悖论明确地断言它自己是假的。也就是说，这个命题，我们称之为S2，“这里写的东西不是真的”，明确断言，S2不是真的。此外，皮尔斯认为，根据威尼斯的保罗的论点，每个命题也默示了它自己的真实性。一个命题的部分意义在于所断言的命题是真实的。也就是说，每个命题p都隐含着一个指代p的命题，这个命题说明p是真的。任何断言的命题的形式条件是：

p ⊢ ‘p’ is true

因此，该命题的含义比明确断言的要多。它既意味着“命题’p’不是真的”和“命题’p’是真的”，根据任何命题所明确断言和默示的内容来考虑该命题，一个命题明确地断言了命题中所陈述的内容，并暗示了该断言的真实性。在S2的例子中，这导致了一个矛盾的命题。明确断言的是’p’不是真的，而暗示的是'p'是真的。皮尔士的结论是：

- 因此，有关的命题在所有其他方面都是真实的，但在其对自身真理的暗示方面是真实的。由于该命题因此涉及到一个矛盾，所以它是假的。

1868年之后，皮尔士对“说谎者悖论”坚持这一立场，他这样说（3.447，1896年）：

- “此命题是假的”，这不仅没有意义，而且是自相矛盾的，也就是说，它意味着两个不可调和的东西。它涉及矛盾（也就是说，如果假定为真，就会导致矛盾），这一点很容易证明，因为如果它是真的，它就是真的；而如果它是真的，它就是假的。每个命题除了它明确断言的内容外，还暗示了它自己的真理。除非该命题明确断言的内容和暗示的内容都是真的，否则该命题就不是真的。这个命题是自相矛盾的，是假的；因此，它明确断言的东西是真的，但它暗示的东西（它自己的真理）是假的。

二，英语原文：

https://www.researchgate.net/publication/266930857_Peirce's_paradoxical_solution_to_the_Liar's_Paradox

PEIRCE’S PARADOXICAL SOLUTION TO THE LIAR'S PARADOX

EMILY MICHAEL

1 Peirce's analysis of the problem (1864-65) 

In Lecture 1 and Lecture 3 of a series of lectures presented in 1864-65 (Lectures on the Philosophy of Science delivered at Harvard), Peirce discusses the Liar's Paradox. In Lecture 1 he discusses the sentence, "This very proposition is false.";  in Lecture 3 he examines the sentence in the form "What is here written is not true." This sentence, as we know, leads to paradoxical conclusions. I will first consider Peirce's analysis of the problem and then his solution to it.

1.1  The  Problem  Stated

S1 This  very  proposition  is  false.

S2 What is here written  is  not  true.

Peirce argues that the problem with this sentence is that it is logically meaningless or logically nonsense, where nonsense is defined as « that which has a certain  resemblance to a symbol without being a symbol. » Each  genuine symbol is subject to three systems of formal laws;  these are the laws of (1) grammar, (2) logic, and (3)  rhetoric. Each symbol to be meaningful must satisfy  the  formal conditions of grammar, of logic, and of the  intelligibility of symbols. This symbol is grammatically correct but fails to be a genuine symbol because it does  not satisfy the formal conditions of logic.

In the case of the above sentence, S1,  a logical law, the law of the excluded middle, does not apply. Peirce  says,

This is a proposition to which the principle of the excluded  middle, namely that every symbol must be false or true,  does not apply. For if it is false, it is thereby true. And if  not false, it is thereby not true.

A logically meaningful sentence will satisfy the laws of  logic. Peirce argues that this logical law does not apply to S1 because this symbol has no object. Logic, Peirce says, is concerned with assertoric propositions. He says of assertoric propositions,  "Propositions which assert always  assert something of an object, which is the subject of the  proposition. » In the case of S1 however,  the proposition  "does itself state that it has no object. It talks of itself  and only of itself and has no external relation whatever. » That is, the subject of the proposition being the proposition itself, the predicate makes no assertion of an  object to which the proposition refers. An assertoric proposition, then, makes reference to an external object, but this proposition "talks of itself and only of itself and  has no external relation whatever."  "Logical laws,"  however,  "only hold good as conditions of a symbol having  an object. »

Similarly concerning S2 Peirce says that we get an infinite  number of propositions:

What is here written

The  statement that  that is false

The  statement that  that is false

The  statement that  that is false

and so on to infinity.

What are each of these statements about. Each one, it would seem, if about anything at all is about the last  statement of this infinite series. However, it being an infinite series, there is no last statement; as such the whole set of statements are about nothing—they have no  meaning whatever.

Thus, again the symbol has no object, but having an object  is a necessary condition of a symbol if the proposition is to fulfill all logical laws and the code of logical laws must be satisfied for a symbol to be logically meaningful. This  symbol having no object then is logically meaningless. Peirce  argues in Lecture 6 of the 1864-65 lecture series  that in an affirmative proposition the object referred to  must be an existent class of things, not a null class. He explains that contradictory predicates can be asserted of a null class; thus such of a proposition fails to satisfy the code of logical laws and is logically meaningless.The Liar's Paradox is analyzed, then, like an affirmative proposition  the subject of which is a null class. During this period he maintains that such logically meaningless propositions are false.

In an unpublished  paper  (Ms 726),  Peirce essentially  repeats this analysis when considering the proposition, "This very proposition is true. » He says,

Why is this absurd? Because it has no reference to an object mediate or immediate. Its subject being the  proposition itself, it has no subject except itself. And since  by its predicate it only refers to the reference of itself to an object, that object being in turn a reference to an  object and so on ad infinitum and it has no object. This shows that if a symbol can have no object it is absurd, which is the same as illogical. 

2  Peirce's solution to the problem  (1864-65)  

Concerning S1 and S2, it may be argued: 

1. The statement is logically meaningless, in having no object and thereby failing to satisfy all the laws of logic. As such it is neither true nor false.

2. The statement is logically meaningless and as such it is not true. But it asserts that it is not true, and thus it is true. The statement then is both true and not true.

2.1 Peirce rejects (1) above,  maintaining that the view that S1 and S2 are neither true nor false is self-contradictory. He says, "The statement as meaningless is not true; but then it is true after all for it says that it is not true. » It is self-contradictory, it would seem, to say that it is neither true nor false for every proposition considered as an assertion has a truth value according to  Peirce. As logically meaningless,  the propositions S1 and  S2 are at least not true from the point of view of logic. S2 presents a unique case, Peirce argues, for in that the content of the proposition is an assertion that it is not true, it is also true. This solution is stated only in reference to S2. In Lecture 1, where SI is considered, the  problem is similarly analyzed but no solution is presented. Peirce is not committed to equating 'not true with 'false'  on the basis of his statement of S2, but would seem nonetheless to do so in the following, written in support  of (2) above :

The fact is that in this proposition truth and not truth—affirmative and negative—this and other—coincide. It  stands upon the boundary of the true and the false;  and  is therefore in both.

2.2 In support of this view Peirce presents an  argument by  analogy. He instructs us  to consider a sheet of paper, part red, part blue. Every point on this sheet paper is either red  or blue. If this is the case, he says, then what color is the  boundary line between the two colors? Peirce answers,

It is plainly as much either one as it is the other. We must therefore say that it is both or neither.

He says we may argue that,

1. It is neither because color resides in a surface not a  line. But this view will not suffice, Peirce says, for "as motion is not in any instant in that it is a relation  between instants it nonetheless is at any given instant. Similarly, while color requires a surface, it is at every  point of that surface and thus this line can be characterized as colored.

2. The boundary is both red and green. Peirce argues, "If  that line is not red it lies without the red part of the sheet therefore, if I simply draw away the red portion, I cannot affect the color of that line which lies without  it. Accordingly that line and whatever the moving  boundary passes over is neither red nor green; but it  may pass over the whole sheet and therefore the  whole sheet is neither red nor green. But it clearly is  green. It  seems to me, therefore, that the proper  answer is that the boundary is both red and green;  —the distinction between them vanishing at this point. 

Analogously  the above  proposition  is both true  and false. To say that it is  neither  true  nor false,  we are told,  is  self-contradictory; but  saying  that it  is both true  and not-true,  as a limiting  case  where  the distinction between affirmation  and negation  vanishes,  is not  self-contradictory.

Since the proposition is logically meaningless, in referring to  no object,  it is  analogous  to a  proposition  the  subject  of which  refers  to a null class. Such a subject will admit of contradictory predicates; it  is  onthese  grounds  that  we say that the  proposition  represents  no actual object,is  logically meaningless,  and  thus  is  false. But  in the  case  of the  Liar'sParadox  this  proposition  asserts  that  what is  here  written  is false;  as  such,the  proposition,  having no  object  and  thus  being  logically meaningless andfalse,  what the  proposition  says  of itself,  i.e., what is  here  written  is  false,is  seen  to be  true. Tha  is,  "this  proposition  so  far  as it is  spoken  about by  itself  is  false  but so far  as it speaks  about  itself  is  true." Peirce  adds,

But this is a distinction  without  a  difference. The question  is whether this  proposition  is  in  all  respects  true. If  it  is  not  in  all  respects  true,  then it  is  in  all  respects  true;  for  two  reasons  1st  because what  it  says cannot  in that  case  be  altogether true and  2nd  because  it  is  seen  to  accord precisely with  what  is  in  all  respects  true;  namely, that  it  isn't in  all respects 

true.

From this viewpoint, Peirce argues that  we  must  say that this  proposition is  both  true  and  false.

3 Peirce s revised analysis of the Liar's Paradox  (1868ff.) 

In  1868  and thereafter, Peirce argues (5.340) that a  proposition  is  true  only  if  whatever is said  in  it is true, but is  false  if anything  said  in it  is  false.

Every proposition  is  said  to be either  true  or  false. If anything  said  in  the proposition  is  false,  then it  is  false. Otherwise it  is  true.

Concerning  the Liar's Paradox,  he  argues in  1868  that this proposition signifies either more or less than is explicitly asserted. If it signifies less then it signifies nothing  and  is meaningless. But, Peirce argues,  the

proposition  in  question  has  a  meaning. He rejects his  argument  of 1864-65, maintaining  that  this  proposition  means  more  than  is  explicitly stated.

Peirce argues that every proposition besides what it  explicitly asserts also tacitly implies its own truth. The Liar's Paradox expressly asserts about itself that it is false. That is, the proposition, which we shall call S2, "What is here written is not true. "explicitly asserts that S2 is not  true. Further, Peirce argues, following the argument of Paul of Venice, that every proposition also tacitly implies its own truth. Part of the meaning of a proposition is that the proposition asserted is true. That is, every proposition p implies a proposition referring to p which states that p is true. A formal condition of any asserted proposition is

p ⊢ ‘p’ is true

Thus the proposition means more than is explicitly asserted. It means both "The proposition ‘p’ is  not true . "  and "The  proposition ‘p’ is  true.", considering the proposition in the light of what is explicitly asserted and tacitly implied  by any proposition. A proposition explicitly asserts what is stated in the proposition and tacitly implies the truth of that assertion. In the case of S2 this results in a contradictory proposition. What is explicitly asserted is ‘p’ is not true and what is tacitly implied is ‘p’ is true. Peirce concludes,

The proposition in question, therefore, is true in all other  respects but in its implication of its own truth. In that the  proposition thus involves a contradiction, it is false.

Peirce maintains this position in regard to the Liar's Paradox after 1868. He thus says  (3.447, 1896)

“This proposition is false.”, far from being meaningless, is  self-contradictory. That is, it means two irreconcilable things. That it involves contradiction (that is, leads to contradiction if supposed true), is easily proved. For if it be true, it is true; while if it be true, it is false. Every proposition besides what it explicitly asserts, tacitly implies its own truth. The proposition is not true unless  both what it explicitly asserts and what it tacitly implies,  are true. This proposition, being self-contradictory, is false; and hence, what it explicitly asserts is true. But what it tacitly implies (its own truth) is false.