# 塔斯基与《形式化语言中的真理概念》- 第一章

《形式化语言中的真理概念》的目录

(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。

(2) x是一个真句当且仅当p

(3) “下雪了”是一个真句子，当且仅当天下雪了。

(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）。

C不是一个真句子。

'c不是一个真句子'c是相同的。

(b) 'c不是真句'是一个真句，当且仅当c不是真句。

(5) 对于所有的p’p'是一个真句子，当且仅当p

(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

(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

***

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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