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塔斯基与《形式化语言中的真理概念》- 第一章

已有 734 次阅读 2021-10-4 20:30 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

《形式化语言中的真理概念》的目录


引言

第一章 日常语言或口语中的“真句子”( true sentence )的概念 

第二章 形式化的语言,特别是语言或类语言

第三章 类语言中的真句子的概念

第四章 有限维度语言中的真句子的概念

第五章 无限维度语言中的真句子的概念

第6章 总结

第7章 后记



第一章 日常语言或口语中的真句子的概念 


为了向读者介绍我们的主题,似乎有必要考虑一下--即使只是一闪而过--用口语定义真理的问题。我特别希望强调解决这一问题的尝试所遇到的各种困难。


在为口语中的句子构建一个正确的真理定义所做的多种努力中,最自然的也许是对语义定义的探索。我指的是一个我们可以用下面的话来表达的定义:

(1)一个真句子是说,事情的状况是这样的,而事情的状况确实是这样的。


从形式正确性、清晰性和表达的无歧义性的角度来看,上述表述显然还有很多需要改进的地方。然而,它的直觉意义和一般意义似乎是相当清楚和明白的。使这个意图更加明确,并赋予其正确的形式,正是语义学定义的任务。


作为起点,某些特殊的句子可以作为句子真理性的部分定义,或者更正确地作为对“x是一个真句子这类言语的各种相关解释。这类句子的一般方案可以用以下方式来描述:


(2) x是一个真句当且仅当p


为了获得具体的定义,我们用句子代替这个方案中的符号’p’,用个体名称代替’x'


如果我们得到了一个句子的名字,我们就可以为它构建一个类型(2)的解释,只要我们能够写下这个名字所表示的句子。满足上述条件的最重要和最常见的名字是所谓的引号式名字。我们用这个词来表示一个句子(或任何其他甚至是无意义的表达)的每一个名字,它由引号、左和右以及位于它们之间的表达组成,而这个(表达)就是这个名字所表示的对象。作为这样一个句子名称的例子,下雪了”这个名称可以作为一个例子,在这种情况下,第(2)种类型的相应解释如下:


(3) “下雪了”是一个真句子,当且仅当天下雪了。


另一类我们可以构建类似解释的句子名称是由所谓的结构描述性名称提供的。我们将把这一术语用于描述构成该名称所表示的表达的词,以及每个词所包含的符号和这些符号和词之间的顺序。这样的名称可以在没有引号的帮助下制定。为此,我们必须在我们所使用的语言(在这种情况下是口语)中为所有字母和语言中的词语和表达方式所包含的所有其他符号提供某种单独的名称,但不是引号式的名称。例如,我们可以用 "A""E""Ef""Jay""Pe "作为字母 "a""e""f""j""p « 的名称。很明显,我们可以将一个结构描述性的名称与每一个引号名称相关联,一个不含引号的名称拥有相同的扩展(即表示相同的表达),反之亦然。例如,与“’”这个名字相对应的是一个由Es, En, O, Double-U这四个字母组成的词,它们彼此相邻。很明显,我们可以为句子的结构描述性名称构建(2)类型的部分定义。下面的例子说明了这一点:


(4)一个由三个词组成的表达式,其中第一个词由两个字母ITe(按顺序)组成,第二个词由两个字母IEs(按顺序)组成,第三个词由七个字母Es, En, O, Double-U, I, En, Ge(按顺序)组成,当且仅当天下雪了是一个真句子。


(3)(4)类似的句子似乎很清楚,而且完全符合(1)中表达的“true”一词的含义。就其内容的明确性和形式的正确性而言,它们一般不会引起任何怀疑(当然,假设我们在(2)中用符号’p'代替的句子中不涉及这种怀疑)。


但是,在这里还是有必要做一些保留。众所周知,在一些情况下,这种类型的断言与某些其他不那么直观明确的前提相结合,会导致明显的矛盾,例如J.Lukasiewicz提出的反义词的自反。


为了更清楚地说明问题,我们将使用符号’c’作为印在本页第5行的句子的印刷缩写。现在考虑下面这个句子:

C不是一个真句子。


考虑到符号’c'的含义,我们可以根据经验确定:

'c不是一个真句子'c是相同的。


对于句子c的引号名称(或其任何其他名称),我们设置了一个(2)类型的解释:

(b) 'c不是真句'是一个真句,当且仅当c不是真句。


前提(a)(b)结合起来就会产生一个矛盾:

当且仅当c不是一个真句时,c是一个真句。


这个矛盾的根源很容易发现:为了构建断言(b),我们在方案(2)中用符号’p’代替了一个本身就包含真句子的表达(因此,这样得到的断言(3)(4)相比不能再作为真理的部分定义。) 然而,没有任何合理的理由可以说明为什么要在原则上禁止这种替换。


在此,我将把自己限制在对上述自反的表述上,并将推迟到以后再得出这一事实的必要后果。把这个困难放在一边,接下来我将尝试通过概括(3)类型的解释来构建一个真句子的定义。乍一看,这项任务似乎很容易--特别是对于在某种程度上掌握了现代数理逻辑技术的人来说。也许我们需要做的就是在(3)中用一个句子变量(即一个任何句子都可以被替代的符号)来代替那里出现了两次的表达式 “天下雪了,然后确定所得到的公式对变量的每个值都成立,这样就不难得出一个将(3)类型的所有断言都作为特例的句子:

(5) 对于所有的p’p'是一个真句子,当且仅当p


但上述句子不能作为“x是一个真句的一般定义,因为符号’x’的全部可能替代物在这里被限制为引号名称。为了消除这一限制,我们必须求助于一个众所周知的事实,即每一个真句子(一般说来,每一个句子)都有一个引号名称,它只表示该句。考虑到这一事实,我们可以尝试对(5)的表述进行概括,例如,以下述方式进行:

(6) 对于所有的x来说,x是一个真句子,当且仅当对于某个p来说,x’p'相同,并且p


乍一看,我们也许会倾向于将(6)视为“真句子的正确语义定义,它以精确的方式再现了表述(1)的意图,因此接受它作为我们问题的满意解决方案。然而,事情并非如此简单。一旦我们开始分析(5)和(6)中出现的引号名称的含义,我们就会遇到一系列的困难和危险。


引号名称可以像语言中的单字一样对待,因此也像句法上的简单表达。这些名称的单一成分--引号和站在它们之间的表达方式--履行的功能与单字中的字母和连续字母的复合体相同。因此,它们不可能拥有独立的意义。因此,每一个引号名称都是一个确定的表达方式(引号所包围的表达方式)的恒定的个体名称,实际上是一个与人的专有名词相同性质的名称。例如, ‘p’这个名字表示字母表中的一个字母。有了这种解释,似乎是最自然的解释,而且完全符合使用引号的习惯方式,就不能使用类型(3)的部分定义了。 


因此,我们得到的结论是:(5)和(6)不是我们想表达的思想的表述,它们实际上显然是无意义的。此外,句子(5)立即导致了一个矛盾,因为除了上述给出的结果之外,我们还可以很容易地从句子中得到矛盾的结果:'p'是一个真句子,当且仅当天下雪。句子(6)本身并没有导致矛盾,但从它得出的明显无意义的结论是:字母'p'是唯一的真句子。


为了使上述考虑更加清晰,可以指出,根据我们对引号名称的概念,它们可以被消除,并在任何地方被例如相应的结构描述性名称取代。然而,如果我们考虑通过使用这种名称构建的第(2)种类型的解释(例如在上文第(4)种类型中所做的那样),那么我们就看不到概括这些解释的方法。如果在(5)(6)中,我们用结构描述性名称 "pe"(或 "由单个字母Pe组成的词")取代引号名称,我们马上就能看到由此产生的表述的荒谬性。


为了挽救第(5)和(6)句的意义,我们必须寻求对引号名称的不同解释。我们必须把这些名字当作语法上的复合表达,引号和其中的表达都是其中的一部分。在这种情况下,并非所有的引号表达式都是常量名称。例如,(5)(6)中出现的表达式“’p’”必须被看作是一个函数,其参数是一个句子变量,其值是句子的常数引号名称。我们将称这种函数为引号函数。这样,引号就成了属于语义学领域的独立词,在意义上接近于名字这个词,而从句法的角度来看,它们扮演着函数子(functor)的角色,但随后又出现了新的复杂情况。


引号函数和引号本身的意义还不够清楚。在任何情况下,这种函数子都不是扩展性的;毫无疑问,句子对于所有的pq,如果p是而且只有q是,那么’p’’q’是相同的与习惯上使用引号的方式是明显矛盾的。仅仅由于这个原因,定义(6)对于任何希望始终避免有意的函数子,甚至认为更深入的分析表明不可能给这种函数子以任何精确的意义的人来说是不可接受的。此外,使用引文函数子会暴露出卷入各种语义自反的危险,比如“说谎者”(liar)的自反。即使小心翼翼地我们只利用上面给出的那些似乎很高的引号函数的属性,也会如此,我们可以通过引入具有可变参数的引号函数,根本不使用真句子这个表达方式来表述。我们将给出这种表述的简图:

让符号’c’成为表达式“本页第6行从头开始打印的句子'”的印刷缩写。我们考虑以下语句:

对于所有的p,如果c与句子'p'相同,那么就不是p(如果我们接受(6)作为真理的定义,那么上述声明断言c不是一个真句子)。


我们根据经验建立:

句子’对于所有p,如果c与句子’p'相同,那么非p'与c相同。

此外,我们只做了一个关于引号函数的补充假设,似乎没有引起任何怀疑:

(b) 对于所有的pq,如果句子'p'与句子'q'相同,那么p当且仅当q


通过基本的逻辑规律,我们很容易从前提(a)和(b)中得出矛盾。


我想顺便提请注意一贯使用上述对引号的解释会使我们面临的其他危险,即某些表达式的模糊性(例如,在(5)和(6)中出现的引号表达式在某些情况下必须被重新归类为具有可变参数的函数,而在其他情况下它是一个表示字母的常数名称)。此外,我想指出,与句法基本规律的一致性至少是值得怀疑的,例如,有意义的表达式包含无意义的表达式作为句法部分(无意义表达式的每个引号名称都可以作为一个例子)。由于所有这些原因,定义(6)的正确性,即使对引号进行了新的解释,似乎也是非常值得怀疑的。


我们到目前为止的讨论使我们有权说,试图为真句子这一表达方式构建一个正确的语义定义,会遇到非常实际的困难。我们不知道有什么一般的方法可以让我们定义“x是一个真句子这种任意的具体表达方式的意义,在这里,我们用某个句子的名称来代替 ‘x’。例子(3)和(4)所说明的方法,在我们不能为一个给定的句子名称指出这个名称所表示的句子的情况下是失败的(作为这样一个名称的例子,“2000年印刷的第一个句子将被使用)。但是,如果在这种情况下,我们在定义(6)的表述中寻求庇护,那么我们就会让自己陷入上述的所有复杂情况。


面对这些事实,我们不得不寻求其他方法来解决我们的问题。我在此仅提请注意这样一种尝试,即试图构建一个结构性定义。这个定义的一般方案是这样的:一个真正的句子是一个拥有这样那样的结构属性(即关于表达的单个部分的形式和排列顺序的属性)的句子,或者可以通过这样的结构转换从这样的结构性描述的表达中得到的句子。作为一个起点,我们可以将形式逻辑中的许多法则用来推断句子的真假,这些法则使我们能够从句子的某些结构属性中推断出真假;或者从某些句子的真假推断出其他句子的类似属性,这些属性可以通过各种结构转换的方式从前者中获得。


以下是这类定律的一些微不足道的例子:每一个由四个部分组成的表达,其中第一个是字‘if’,第三个是‘then’,第二和第四个是同一个句子,是一个真句子;如果一个真句子由四个部分组成,其中第一个是‘if’,第二个是真句,第三个是‘then’,那么第四个是一个真句子。这种规律(尤其是第二种类型的规律)非常重要。在它们的帮助下,每一个零散的真理定义,其外延都包括可以从给定类别的句子中建立起来的句子,通过诸如 "如果......那么""如果且仅如果"""""" « 等表达方式,简而言之,通过属于句法计算(或演绎理论)的表达方式来组合它们。这导致了建立足够多的、强大的和一般的规律的想法,使每个句子都属于其中之一。通过这种方式,我们应该达到一个真句子的一般结构定义。然而,这种方式似乎也几乎是无望的,至少就自然语言而言是如此。因为这种语言不是完成的、封闭的或有明确界限的东西。它没有规定哪些词可以被添加到这种语言中,从而在某种意义上已经属于它的潜力。我们无法从结构上规定那些我们称之为句子的语言表达,更无法在其中区分出真句子。试图为真句子这个词制定一个结构性的定义适用于口语,但却遇到了无法克服的困难。


以前所有尝试的失败使我们认为,没有解决我们问题的满意方法。事实上,正如我现在要简要说明的那样,可以援引一般性质的重要论据来支持这一假设。


口语的一个特点(与各种科学语言相比)是其普遍性。如果在其他语言中出现了一个不能翻译成这种语言的词,那就不符合这种语言的精神了;可以说,如果我们能有意义地谈论任何东西,我们也能用口语谈论它。如果我们要在语义研究中保持日常语言的这种普遍性,为了保持一致,我们必须把这些语言中除了句子和其他表达方式之外,还包括这些句子和表达方式的名称和包含这些名称的句子,以及诸如真句子 “名称、“表示等语义表达方式。但是,大概正是这种日常语言的普遍性是所有语义自反的主要来源,就像说谎者的自反或异端词的自反。这些自反现象似乎提供了一个证明,即每一种在上述意义上具有普遍性的语言,如果正常的逻辑法则对其成立,就必须是不一致的。这尤其适用于我在第157页和第158页给出的说谎者的自反的表述,它不包含带有可变论据的引号函数。如果我们在上述表述中分析这个反义词,我们就会得出这样的信念:不可能存在一种一致性的语言,它既符合通常的逻辑规律,同时又满足以下条件。(I) 对于语言中出现的任何句子,这个句子的明确名称也属于该语言;(II) (2)中形成的每个表达式,通过用该语言的任何句子替换符号'p',用这个句子的名称替换符号'x',都将被视为该语言的真句;(III) 在有关语言中,具有与(a)相同含义的经验确立的前提可以被制定并接受为一个真句子。


如果这种观察是正确的,那么,对真句子这一表达方式的一致使用的可能性,与逻辑规律和日常语言的精神相一致,似乎是非常值得怀疑的,因此,对构建这一表达方式的正确定义的可能性也有同样的怀疑。






***

原文:


Chapter 1. The concept of the true sentence in everyday or colloquial language 

Chapter 2 Formalized language, especially the language or the calculus of classes

Chapter 3 The concept of true sentence in the language of the calculus of classes

Chapter 4 The concept of true sentence in languages of finite order

Chapter 5 The concept of true sentence in languages of infinite order

Chapter 6 Summary

Chapter 7 Postcript


Chapter 1. The concept of the true sentence in everyday or colloquial language 


For the purpose of introducing the reader to our subject, a consideration - if only a fleeting one - of the problem of defining truth in colloquial language seems desirable. I wish especially to emphasise the various difficulties which the attempts to solve this problem have encountered.


Amongst the manifold efforts which the construction of a correct definition of truth for the sentences of colloquial language has called forth, perhaps the most natural is the search for a semantical definition. By this I mean a definition which we can express in the following words :

  1. a true sentence is one which says that the state of affaire is so and so, and the state of affaire indeed is so and so.


From the point of view of formal correctness, clarity, and freedom from ambiguity of the expressions occurring in it, the above formulation obviously leaves much to be desired. Nevertheless its intuitive meaning and general intention seem to be quite clear and intelligible. To make this intention more definite, and to give it a correct form, is precisely the task of a semantical definition.


As a starting-point certain sentences of a special kind present themselves which could serve as partial definitions of the truth of a sentences or more correct as explanations of various concerte turns of speech of the type ‘x is a true sentence’. The general scheme of this kind of sentence can be depicted in the following way :

(2) x is a true sentence if and only if p.


In order to obtain concrete definitions we substitute in the place of the symbol ‘p’ in this scheme any sentence, and in the place of ‘x’ any individual name of this sentence.


If we are given a name for a sentence, we can construct an explanation of type (2) for it, provided only that we are able to write down the sentence denoted by this name. The most important and common names for which the above condition is satisfied are the so-called quotation-mark names. We denote by this term every name of a sentence (or of any other, even meaningless expression) which consists of quotation marks, left- and right-hand, and the expression which lies between them, and which (expression) is the object denoted by the name in question. As an example of such a name of a sentence the name « it is snowing » will serve. In this case the corresponding explanation of type (2) is as follows :

(3) ‘it is snowing’ is a true sentence if and only if it is snowing.


Another category of names of sentences for which we can construct analogous explanations is provided by the so-called structural-descriptive names. We shall apply this term to names which describe the words which compose the expression denoted by the name, as well as the signs of which each single word is composed and the order in which these signs and words follow one another. Such names can be formulated without the help of quotation marks. For this purpose we must haven in the language we are using (in this cas colloquial language), individual names of some sort, but not quotation-mark names, for all letters and all other signs of which the words and expressions of the language are composed. For example we could use ‘A’, ‘E », ‘Ef’, ‘Jay’, ‘Pe’ as names of the letters ‘a’, ‘e’, ‘f’, ‘j’, ‘p’. It is clear that we can correlate a structural-descriptive name with every quotation-mark name, one which is free from quotation marks and possesses the same extension (i.e. denotes the same expression) and vice versa. For example, corresponding to the name « ‘snow’ » we have the name « a word which consists of the four letters : Es, En, O, Double-U following one another ». It is the clear that we can construct partial definitions of the type (2) for structural-descriptive names of sentences. This is illustrated by the following example :


(4) an expression consisting of three words of which the first is composed of the two letters I and Te (in that order) the second of the two letters I and Es (in that order) and the third of the seven letters Es, En, O, Double-U, I, En, and Ge (in that order), is a true sentence if and only if it is  snowing.


Sentences which are analogous to (3) and (4) seem to be clear and completely in accordance with the meaning of the word ‘true’ which was expressed in the formulation (1). In regard to the clarity of their content and the correctness of their form they arouse, in general, no doubt (assuming of course that no such doubts are involved in the sentences which we substitue for the symbol ‘p’ in (2)).


But a certain reservation is nonetheless necessary here. Situations are known in which assertions of just this type, in combination with certain other not less intuitively clear premisses, lead to obvious contradictions, for example the antinomy of antinomy which is due to J. Lukasiewicz.


For the sake of greater perspicuity we shall use the symbol ‘c’ as a typographical abbreviation of the expression ‘the sentence printed on this page, line 5 from the top’. Consider now the following sentence  :

c is not a true sentence.


Having regard to the meaning of the symbol ‘c’, we can establish empirically :

‘c is not a true sentence’ is identical with c.


For the quotation-mark name of the sentence c (or for any other of its names) we set up an explanation of type (2) :

(b) ‘c is not a true sentence’ is a true sentence if and only if c is not a true sentence.


The premisses (a) and (b) together at once give a contradiction :

c is a true sentence if and only if c is not a true sentence.


The source of this contradiction is easily revealed : in order to construct the assertion (b) we have substituted for the symbol ‘p’ in the scheme (2) an expression which itself contains the term ‘true sentence’ (whence the assertion so obtained - in contrast to (3) or (4) - can no longer serve as a partial definition of truth). Nevertheless no rational ground can be given why such substitutions should be forbidden in principe.


I shall restrict myself here to the formulation of the above antinomy and will postpone drawing the necessary consequences of this fact till later. Leaving this difficulty on one side I shall next try to construct a definition of true sentence by generalizing explanations of type (3). At first sight this task may seem quite easy - especially for anyone who has to some extent mastered the technique of modern mathematical logic. It might be though that all we need do is to substitute in (3) and sentential variable (i.e. a symbol for which any sentence can be substituted) in place of the expression ‘it is snowing » which occurs there twice, and then to establish that the resulting formula holds for every value of the variable, and thus without further difficulty reach a sentence which inclues all assertions of type (3) as special cases :

(5) for all p, ‘p’ is a true sentence if and only if p.


But the above sentence could not serve as a general definition of the expression ‘x is a true sentence » because the totality of possible substitutions for the symbol ‘x’ is here restricted to quotation-mark names. In order to remove this restriction we must have recourse to the well-known fact that to every true sentence (and generally speaking to every sentence) there corresponds a quotation-mark name which denotes just that sentence. With this fact in mind we could try to generalize the formulation (5), for example, in the following way :

(6) for all x, x is a true sentence if and only if, for a certain p, x is identical with ‘p’ and p.


At first sight we should perhaps be inclined to regard (6) as a correct semantical definition of « true sentence », which relives in a precise way the intention of the formulation (1) and therefore to accept it as a satisfactory solution of our problem. Nevertheless the matter is not quite so simple. A soon as we begin to analyse the signification of the quotation-mark names which occur in (5) and (6) we encounter a sire of difficulties and dangers.


Quotation-mark names may be treated like single words of a language, and thus like syntactically simple expressions. The single constituents of these names - the quotation marks and the expressions standing between them - fulfil the same function as the letters and complexes of successive letters in single words. Hence they can possess no independent meaning. Every quotation-mark name is then a constant individual name of a definite expression (the expression enclosed by the quotation marks) and in fact a name of the same nature as the proper name of a man. For example, the name « p » denotes one of the letters of the alphabet. With this interpretation, which seems to be the most natural one and completely in accordance with the customary way of using quotation marks, partial definitions of the types (3) cannot be used for any significant generalizations. In no case can the sentences (5) or (6) be accepted as such a generalization. In applying the rule called the rule of substitution to (5) we are not justified in substituting anything at all for the letter ‘p’ which occurs as a component part of a quotation-mark name (just as we are not permitted to substitute anything for the letter ’t’ in the word ‘true’). 


Consequently we obtain as conclusion not (5) and (6) are not formulations of the thought we wish to express and that they are in fact obviously senseless. Moreover, the sentence (5) leads at once to a contradiction, for we can obtain from it just as easily in addition to the above given consequence, the contradictory consequence : ‘p’ is a true sentence if and only if it is not snowing. Sentence (6) alone leads to no contradiction, but the obviously senseless conclusion follows from it that the letter ‘p’ is the only true sentence.


To give greater clarity to the above considerations it may be pointed out that with our conception of quotation-mark names they can be eliminated and replaced everywhere by, for example, the corresponding structural-descriptive names. If, nevertheless, we consider explanations of types (2) constructed by the use of such names (as was done, for example, in (4) above), then we see no way of generalizing these explanations. And if in (5) or ((6) we replace the quotation-mark name by the structural-descriptive name ‘pe’ (or ‘the word which consists of the single letter Pe’) we see at once the absurdity of the resulting formulation.


In order to rescue the sense of sentences (5) and (6) we must seek quite a different interpretation of the quotation-mark names. We must treat these names as syntactically composite expressions, of which both the quotation marks and the expressions within them are parts. Not all quotation-mark expressions will be constant names in that case. The expression « ‘p’ » occurring in (5) and (6), for example, must be regarded as a function, the argument of which is a sentential variable and the values of which are constant quotation-mark names of sentences. We shall call such functions quotation-functions. The quotation marks then become independent words belonging to the domain of semantics, approximating in their meaning to the word ‘name’, and from the syntactical point of view, they play the part of functors. But then new complications arise. 


The sense of the quotation-function and of the quotation marks themselves is not sufficiently clear. In any case such functors are not extensional ; there is no doubt that the sentence « for all p and q, if p is and only if q, then ‘p’ is identical with ‘q’ » is in palpable contradiction to the customary way of using quotation marks. For this reason alone definition (6) would be unacceptable to anyone who wishes consistently to avoid intentional functors and is even of the opinion that a deeper analysis shows it to be impossible to give any precise meaning to such functors. Moreover, the use of the quotation functors exposes to the danger of becoming involved in various semantical antinomies, such as the antinomy of the liar. This will be so even if - taking every care - we make use only of those properties of quotation-functions which seem high has been given above, we can formulate it without using the expression ‘true sentence’ at all, by introducing the quotation-functions with variable arguments. We shall give a sketch of tis formulation.


Let the symbol ‘c’ be a typographical abbreviation of the expression «’the sentence printed on this page line 6 from the top ». We consider the following statement :

  • for all p , if c is identical with the sentence ‘p’, then not p (if we accept (6) as a definition of truth, then the above statement asserts that c is not a true sentence).

We establish empirically :

  • the sentence ‘for all p, if c is identical with the sentence ‘p’, then not p’ is identical with c.


In addition we make only a single supplementary assumption which concerns the quotation-function and seems to raise no doubts :

(b) for all p and q, if the sentence ‘p’ is identical with the sentence ‘q’, then p if and only if q.


By means of elementary logical laws we easily derive a contradiction from the premisses (a) and (b).


I should like to draw attention, in passing, to other dangers to which the consistent use of the above interpretation of quotation marks expose us, namely to the ambiguity of certain expressions (for example, the quotation-expression which occurs in (5) and (6) must be regraded in certain situations as function with variable argument, whereas in others it is a constant name which denotes a letter of the alphabet). Further, I would point out the agreement with the fundamental laws of syntax is at least doubtful, e.g. meaningful expressions which contain meaningless expressions as syntactical parts (every quotation-name of a meaningless expression will serve as a example). For all these reasons the correctness of definition (6), even with the new interpretation of quotation marks, seems to be extremely doubtful. 


Our discussion so far entitle us in any case to say that the attempt to construct a correct semantical definition of the expression ‘true sentence meets with very real difficulties. We know of no general method which would permit us to define the meaning of an arbitrary concrete expression of the type ‘x is a true sentence’, where in the place of ‘x’ we have a name of some sentence. The method illustrated by the example (3) and (4) fails us in those situations in which we cannot indicate for a given name of a sentence, the sentence denoted by this name (as an example of such a name ‘the first sentence which well be printed in the year 2000’ will serve). But if in such a case we seek refuge in the construction used in the formulation of definition (6), then we should lay ourselves open to all the complications which have been described above.


In the face of these facts we are driven to seek other methods of solving our problem. I will draw attention here to only one such attempt, namely the attempt to construct a structural definition. The general scheme of this definition would be somewhat as follows : a true sentence is a sentence which possesses such and such structural properties (i.e. properties concerning the form and arrangement in sequence of the single parts of the expression) or which can be obtained from such and such structurally described expressions by means of such structural transformations. As a starting-point we can press into service many laws from formal logic which enable us to infer the truth or falsehood of sentences from certain of their structural properties; or from the truth or falsehood of certain sentences to inter analogous properties of other sentences which can be obtained from the former by means of various structural transformations. 

Here are some trivial examples of such laws : every expression consisting of four parts of which the first is the word ‘if’, the third is the word ‘then’, and the second and fourth are the same sentence, is a true sentence; if a true sentence consists of four parts, of which the first is the word ‘if’, the second a true sentence, the third the word ‘then ‘, then the fourth part is a true sentence. Such laws (especially those of the second type) are very important. With their help every fragmentary definition of truth, the extension of which embraces sentences which can be built up from sentences of the given class by combining them by means of such expressions as ‘if …then’, ‘if and only if’, ‘or’, ‘and’, ‘not’, in short, by means of expressions belonging to the sentential calculus (or theory of deduction). This leads to the idea of setting up sufficiently numerous, powerful, and general laws for every sentence to fall under one of them. In this way, we should reach a general structural definition of a true sentence. Yet this way also seems to be almost hopeless, at least as far as natural language is concerned. For this language is not something finished, closed, or bounded by clear limits. It is not laid down what words can be added to this language and thus in a certain sense already belong to it potentially. We are not able to specify structurally those expressions of the language which we call sentences, still less can we distinguish among them the true ones. The attempt to set up a structural definition of the term ‘true sentence’ - applicable to colloquial language is confined with insuperable difficulties. 


The breakdown of all previous attempts leads us to suppose that there is no satisfactory way of solving our problem. Important arguments of a general nature can in fact be invoked in support of this supposition as I shall now briefly indicate.


A characteristic feature of colloquial language (in contrast to various scientific languages) is its universality. It would not be in harmony with the spirit of this language if in some other language a word occurred which could not be translated into it; it could be claimed that ‘if we can speak meaningfully about anything at all, we can also speak about it in colloquial language’. If we are to maintain this universality of everyday language in connexion with semantical investigations, we must, to be consistent, admit into the language, in addition to its sentences and other expressions, also the names of these sentences and expressions, and sentences containing these names, as well as such semantic expressions as ‘true sentence’, ‘name’, denote’, etc. But it is presumably just this universality of everyday language which is the primary source of all semantical antinomies, like the antinomies of the liar or of hererological words. These antinomies seem to provide a proof that every language which is universal in the above sense, and for which the normal laws of logic hold, must be inconsistent. This applies especially to the formulation of the antinomy of the liar which I have given on pages 157 and 158, and which contains no quotation-quotation-function with variable argument. If we analyse this antinomy in the above formulation we reach the conviction that no consistent language can exist for which the usual laws of logic hold and which at the same time satisfies the following conditions : (I) for any sentence which occurs in the language a definite name of this sentence also belongs to the language; (II) every expression formed from (2) by replacing the symbol ‘p’ by any sentence of the langage and the symbol ‘x’ by a name of this sentence is to be regarded as a true sentence of this language ; (III) in the language in question an empirically established premiss having the same meaning as (a) can be formulated and accepted as a true sentence.


If this observations are correct, then the very possibility of a consistent use of the expression ‘true sentence’ which is in harmony with the laws of logic and the spirit of everyday language seems to be very questionable, an consequently the same doubt attaches to the possibility of constructing a correct definition of this expression.








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