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VIII Formal Systems and Syntax
Vln FORMAL SYSTEMS AND SYNTAX A notion closely related to that of a formal system and now, thanks to Carnap, widely prevalent, is that of a syntax of a language. In this chapter we inquire into the relationships of that notion to the formal systems in the sense of Chapter IV. I shall argue that the two notions are essentially equivalent, and then shall turn to some criticisms of the syntactical point of view in general. These criticisms may be skipped without affecting anything in what follows. According to Carnap a language consists of a certain stock of symbols from which can be formed certain finite linear series called expressions. A rule, theory, or the like is called formal if it deals with these expressions as objects in such a way as to take account solely of the kind and arrangement of the symbols of which they are composed. Certain rules of this character are given to us in advance, and the syntax of the language is the systematic study of these rules and their consequences. The given rules are of two kinds. The first kind, called formation rules, determine a certain category of expressions called sentences, and also a classification of other expressions into categories with reference to their role in forming sentences; the second kind, called transformation rules, specify under what circumstances a sentence (we may call it the conclusion) is a direct consequence of a set of sentences called the premises. There may also need to be, in connection with these, definitions of auxiliary kinds of expressions. In all discussions of syntax we are concerned with two languages - or rather two different senses of 'language'. On the one hand there is the language under discussion, called the object language or O-language, - which mayor not be a language in the ordinary sense. On the other hand there is the language in which the discussion is conducted - which is a language in the ordinary sense
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- let us call it the S-language (i.e. syntax language). I shall use the prefixes '0' and'S' to distinguish notions belonging to these two languages. Is every formal system representable as the syntax of an 0language? Well, we can certainly find a representation in which the tokens are O-symbols and the operations are ways of combining O-expressions into linear series along with other O-symbols, i.e, we can assign to each operator q; an O-symbol, say '/' and then say that if aI' as, ... , an are any O-expressions, q; al as . .. a" is the O-expression got by writing in a row first '/' then the aI' az, ... ,an in order 1. The terms of the system will then be expressions of an O-language; but in general they will not constitute all the expressions of that language, so that at best we have only a part of the syntax. For the most general kind of formal system this is as far as we can go; because, although the predicates are formal in Carnap's sense, it is undetermined whether any of them will correspond to rO-sentencel, or whether there will be rules defining a term as a direct consequence of a class of other terms. Suppose, however, that we have a formal system with only one unary predicate. Then we can make the following definitions (for the terminology see Chapter VII): an O-sentence is the same as a formula; an O-sentence is a direct consequence of the null class when and only when it is an axiom formula; it is a direct consequence of the non-null class K when it is directly derivable from the members of K. Then the formal system will in fact become a part of the syntax of an O-language. It is evident that this will apply even for an indefinite system, since Carnap allows indefinite consequence rules. We note in passing that this language may be strangely and wonderfully made. In the case of a completely formalized system every term will be an O-sentence; and the O-symbols will consist of sentences and co-ordinating conjunctions only. This is not 1 This requires of course that the number of arguments to every operator be fixed. Then parentheses are unnecessary. Alternatively we may place the between commas, enclose the resulting expression in parentheses and prefix an '/' to the whole - this is the ordinary notation.
n,
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FORMAL SYSTEMS AND SYNTAX
surprising in the case of Example 3 because - taken intuitively as a logic - it had that character to start with; but in the case of Example 8 it is a little strange in view of the properties of that system mentioned at the end of Chapter VII. Incidentally, for systems just alluded to, the O-expressions which. are not terms are intrinsicly meaningless, like 'Bats live in caves and or'. Let us now consider the converse question, can the syntax of a language be set up as a formal system? The first step in such a procedure would be the formal definition of expressions as a category of terms. The fundamental operation in the given syntactical theory is combining two expressions by writing them consecutively; let us call this, after Quine, concatenation. This is an associative operation. Hence if we take it as a fundamental operation in our formalized syntax, we run into the difficulty that an expression is not formed from the O-symbols in a unique manner. This would make it necessary to take as terms not the analogues of the expressions themselves, but of methods of constructing them, and then to formulate, morphologically or theoretically, a relation of equality. However Hermes has shown that we can take instead a set of unary operations, one for each symbol, whose interpretation is writing that symbol after the operand 1. The idea of this is shown in Example 2; but it is not necessary to be so formal here, - we can simply take III A 2 and 3 as a recursive definition of a morphological operation. We then have the O-expressions formalized as expression terms. The formation rules can then be taken over bodily into the system, replacing each category, etc. of expressions by the corresponding category, etc. of expression-terms. Corresponding to the 0sentences we have the formulas among the expression-terms. Next take the rules of consequence. Let the axiom-formulas correspond to those O-sentences which are direct consequences of the null class, and let the other rules of direct consequence be set up as rules of procedure. We then have the syntax set up as 1 Of course one can take instead the symbols as tokens and a single operation of affixing a token to an expression. (This is an instance of the second reduction of § 7).
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formal system, definite or indefinite; if the rules of the syntax are definite the formal system will be definite also. The preceding argument has necessarily a somewhat heuristic character. It is therefore instructive to see how it goes through in a special case, say Carnap's Language II. In this case we have a type structure so complicated that it is advantagoous to take a special category of terms to represent the types. Then we take as syntactical predicates: r- is a variable', r- belongs to ---l, r_ is a type", r_ is a sentence'. We shall need in connection with argument series concatenation of types as well as of expressions. Using 'A' for the concatenation operation we get a series of rules of which the following are samples: There is one special type: Z (numbers). If t 1 and tz are types, so are t 1 A tz, t 1 : tz, and (t1 ) . If Q 1 belongs to t1 and ~ belongs to tz then Q1 A ',' A Q2 belongs to t1 A tz. If Q1 belongs to (t1 ) and b belongs to t 1 , then Q1 A '('A b A')' is a sentence. If Q1 is a sentence, Q2 belongs to Z and Q2 is a variable, then '('A K A ~ A')' A Q1 belongs to Z. It appears from the above discussion that "formal system' and rsyntax of a language' are essentially equivalent notions. Aside from the representation, the only difference is that a syntax is by definition tied to the fundamental operation of concatenation. From an abstract point of view, "the syntax of a language .... is concerned, in general, with the structures of possible serial orders (of a definite kind) of any elements whatsoever"; from the same point of view a formal system is concerned with the structures of arbitrary combinations of any elements whatever. The notion of formal system is thus more general; but this generality is rather trivial since an arbitrary formal systen can be embedded in a syntactical one. The syntax of a language is then, essentially, a formal system represented in a certain way. It is now necessary to deal with the view, which seems to be prevalent among those addicted to syntax, that a discourse is only formal insofar as it is syntactical, - or in other words that 8.
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a formal system must be represented syntactically if it is significant at all. If this were to mean simply that a reader, in considering a formal system, must have some such representation in mind, then it would be superfluous to add anything to what has been said in Chapter VI. But the syntacticians appear to have something more specific than this in mind. Thus Carnap writes as follows: 1 "In a treatise by a distinguished logician, the following sentence occurs: ,(~) a is a formula which results from the formula a when the variable x (if it occurs in a) is replaced throughout by the combination of symbols p' 2. Here we are from the beginning completely uncertain as to the interpretation. Which of the symbolic expressions in this statement are used as autonymous designations and are accordingly to be enclosed in inverted commas if the correct mode of expressing the author's meaning is to be achieved? '" We do not know to which object-language all the formulae, as syntactical formulae, are to refer." It appears that Carnap is demanding that a formal system be the syntax of a particular, explicitly stated, language 3. If so, there are several special criticisms to be added to those of Chapter VI. To begin with this would require a duplicity of symbolism. One must have the a-expressions themselves and also their Sdesignations, and if we are to be exact these must be kept distinct. Logical Syntax, p. 159. The sentence quoted does not occur in any logical treatise that I know of, and neither does the German translation of it in the original edition. But in Heyting's paper C 385.3, p. 58 the following sentence 1
2
occurs: "Zweitens fiihren wir die Bezeichnung
(~) a ein fur 'der Ausdruck,
der aus a entsteht indem man die Veranderliohe x uberall, wo sie auftritt, durch die Zeichenzusammenstellung p ersetzt' ". I am assuming this is what is referred to. 3 This language must be such as can be represented in the printed page, and not counters, noises, etc. of which Carnap indicates the possibility (Logical Syntax § 2). Such will be considered non-linguistic in what follows).
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Moreover the usual procedure is to take for a-language some more or less familiar set of marks; then for the S-designations of these we are confronted with wholly unfamiliar symbols of an outlandish kind. But - assuming the discussion is a formal one - the a-symbols do not appear in the main part of the argument at all; they only occur in the introductory discussions. This greatly increases the difficulties of comprehension 1. Why not, then, abolish the object language altogether! Certainly the theory would be more perspicuous if the ordinary symbols were used syntactically, while the object language is left to the reader's imagination or explained by means of other symbols 2. This would also be more in accord with our mathematical habits; for in mathematics we do not talk about our symbols, we use them. Again the syntactical point of view leads to an extreme nominalism which is foreign to mathematics. By this I mean that syntax tends to make purely linguistic accidents appear on a par with more substantial considerations. In mathematics we abstract not only from meanings of our symbols - from the nature of the objects which they denote - but also from the peculiarities of their external structure. Our symbols are not intrinsically meaningless, but their meaning is unspecified; and considerations which do not have a reference to such meaning are ignored. In doing such ample justice to the first of these abstractions the syntacticians perform only lip-service 3 to the second. Take for instance the matter of commas, parentheses, etc. To the ordinary mathematician these are just devices for symbolizing an operation not readily symbolized by concatenation alone; whereas to the syntactician they are symbols which are 1 A striking example of this is the system of Tarski, for example in C 285.13 or § 2 of C 285.16). In the case of systems using quotation marks we have a further confusion in that these symbols are used in a slightly different manner in ordinary discourse. S These symbols could be words of ordinary language. Thus we can say that if 'A' and 'B' are abbreviations of the sentences p and q (named by 'p' and 'q') respectively, then 'Opq' (or 'p:J q') is a name of 'if A then B'. a Cf. Logical Syntax, pp. 5 - 6.
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capable of being joined with other symbols to form expressional monstrosities 1. In a formal system we can leave these useless combinations out. Our terms then have a closer relation to the essential content of our ideas. A point much belabored by the syntax-addicts is the ambiguity of the "autonymous" mode of speech, - viz. that in which a symbolic expression is used to designate itself. If the expression already has a meaning in its own right, there then arises the possibility of doubt as to whether the expression refers to itself or to its original meaning. This is a particularly serious difficulty when the original meaning is itself symbolic in character. Frege pointed out this possibility of confusion, and introduced the systematic use of quotation marks to avoid it; but the idea can scarcely have originated with him, since the same difficulty arises whenever we talk about a language within that language, and English grammars were written in English long before Frege's parents ever dreamed of the blessed event. Yet, although the distinction is important, it is not necessary to make it constantly. In particular, when the subject under discussion is non-linguistic our occasional uses of the autonymous mode of speech are usually made clear in connection with the context by a careful use of the ordinary conventions of language. Again in the syntax of a wholly uninterpreted object-language - i.e. of a system of marks without meaning - there can be no confusion in using these marks as their own names, since they have no other meaning 2. Only in case we are dealing with the syntax of an actual, interpreted language is the danger of obscurity serious. These considerations suggest that if, in dealing with a formal system, we bear in mind the possibility of a non-linguistic represenr
1 In the syntax of a language these monstrous expressions are excluded by the formation rules. Hence in seting up a syntax as a formal system we have an alternative mode of procedure to that considered above defining only such categories etc. of terms as are significant in the formation in rules. In this way Example 6 arose from Church's system. The terms there are his "well formed expressions". I This appears to be the representation which Hilbert has in mind.
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tation, then we may be allowed a certain liberality, which we otherwise would not have, in connection with the autonymous mode of speech. This mode has long been customary in dealing with substitution; the sentence from Reyting, quoted above, is a definition in which he follows tradition. The trouble which Carnap has with it seems to be due to his trying to force a syntactical representation where it does not belong. If he were to admit a non-syntactical representation I think he would be able to understand what Reyting means. Reyting's statement, to be sure, is incorrect; but it is clear how it is to be made correct (Cf. Example 6, above). In such cases it is often a matter of considerable inconvenience to make the analysis necessary for absolute correctness, and to insist it be done at every stage is to hamper research. All we need is to know that the correct analysis can be made. The consideration of a non-syntactical representation - perhaps alongside a syntactical one - may therefore be in the interests of progress 1. Again, in the case where the object language is an interpreted language, then we do not need a syntax language at all. For we can regard the primitive frame as being the definition (by recursion, of course) of the symbols for the formal predicates 2; these are to be regarded, then, as new technical words of the object language. Of course these predicates may be liable to confusion with intuitive predicates already in the a-language; and in that case a syntax language may be a convenience; but it is not a necessity. I do not agree that the only things which can be treated formally are symbols. This process involves certain difficulties if the object language contains variables, but we now know, in principle at least, how variables can be eliminated. In most cases in which we are actually interested in the syntax of a language, we are interested, directly or indirectly, in its subject matter also; then there is always the possibility of representing a formal system 1 I have benefited by some discussion with Church in regard to these matters; but he is not to be held responsible for the views expressed. 2 Including morphological as well as theoretical predicates. The notion of "term" must also be so defined.
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directly in terms of the subject matter, rather than through the somewhat artificial medium of the syntax of the language. These considerations have a bearing on the discussions in the last part of Carnap's book. As to the philosophical aspects of that discussion I am not competent to speak. I can only say, however, as a mathematician, that the fundamental thesis viz. that the confusion mentioned is due to the material mode of speech - is not proved. For the sentences listed as examples are in many cases intrinsically vague - they contain words not defined in the context, - and it is conceivable that it may be this vagueness, rather than the material mode of speech per se, which is at fault. Moreover the translations into the syntactical mode of speech do not, in all cases, have the same meaning as the original 1. To illustrate these points consider the two sentences: (1) Seven is a number. (2) Seven is an odd number. If I understand Carnap correctly he would say that (1) is a "quasisyntactical sentence in the material mode of speech" while (2) is an object-sentence. Let us examine them more closely. In the first place it is necessary to inquire after the meanings of 'seven', 'number', 'odd number'; for without these meanings both (1) and (2) are nonsense. These meanings may naturally depend on the context in which the sentences are embedded. Let us first analyze these sentences on the hypothesis that they occur in the midst of an ordinary, non-technical discussion in the English language. Now in English 'seven' is primarily an adjective; when used as a noun it is essentially only a name for the adjective. Hence 'seven' in both (1) and (2) is autonymous. As for 'number' we find in the dictionary that a number is one of a series of words 1 For example the sentences 'Yesterday's lecture was about Babylon' and 'In yesterday's lecture "Babylon", or some synonymous word, occurred', are not equivalent. For suppose the lecture had been about the breeding habits of the polar bear, but the professor had told a joke in which there was an instance of 'Babylon'; then the first sentence would be false and the second true.
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(or symbols) used for counting, hence it is practically synonymous with 'number word'. (1) is therefore a syntactical sentence and expresses, practically, the same meaning as (3) 'seven' is a number word. Now what about (2)? Given any series of numbers in the above sense, we can define which ones are even and which odd by a recursive definition, and this can be made intelligible to a Chinaman having no knowledge of English provided that the explanations were made in Chinese. (2) is therefore just as much a syntactical sentence as (1) is. We get some light on the relations of (1) and (3) if we translate them into German 1. The translation of (1) is evidently (4) Sieben ist eine Zahl. In considering the translation of (3) consider the sentence' 'Seven' has five letters'. To translate this as "Sieben' hat fiinf Buehstaben' would evidently be incorrect, since the latter sentence does not have the same truth value; to preserve the meaning we should have to regard "seven" as its own translation. But if we translate (3) in this way we get a sentence which would be meaningless in a German context, and we have to add a phrase of explanation. Hence we get as translation of (3): (5) In der englischen Sprache ist 'seven' ein Zahlwort.
It is evident that (4) and (5) are not synonymous for the one refers to a fact of the English language, while the other does not. Yet it is evident that (4) is the translation of (1) in the sense of non-technical discourse. Therefore (3), - which, incidentally, would not occur in non-technical, but only in linguistic discourse - is not a true translation of (1). The difference is that (1) has an abstract connotation; 'seven' there refers, not to 'seven' as a linguistic phenomenon, but to the meaning of 'seven', i.e. to rsevenl. Whether one thinks of this as an abstract concept of seven-ness or as a class of equivalent words, or what not, is an 1
This suggestion is due to Langford.
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FORMAL SYSTEMS AND SYNTAX
irrelevant matter. But in any case to replace (1) by (3) is to distort its meaning 1. Suppose now, we take (1) in a different context. Church has shown how to define numbers in the system of Example 6 2 • In fact, let One S
=
Axl Ax2 ax l x 2 • I AX2 Axs ax2 aaxl x 2 xs'
= AX
This S is the successor-function, and in terms of it the other integers are defined successively 3. Now, suppose we take for object language the English language together with the technical words and symbols defined in the primitive frame of Example 6. Then 'number' can be defined recursively as follows: a) One is a number. b) If a is a number, aSa is a number. c) If a cnv b and a is a number, then b is a number. In that context 'Seven is a number' is an object-sentence"; Moreover in order to decide whether it is true you would have to know, not only the syntactical rules of the English language, but the definitions of 'seven' and 'number' also 5. In view of these considerations there is something which needs to be added to the indicated part of Carnap's book. The object of Carnap's whole discussion is, as I understand it, to clarify the vague sentences often propounded in philosophy. That the concept of syntax language can be a great help in that connection may 1
Another example is In antiquity seven was regarded as sacred.
'Seven' of course, did not exist in antiquity. What was regarded as sacred was rseven1. 2 This example is used, rather than Example 1, because 'number' is not a "universal word". 3 The definition of seven is then as as as as as as (one) whose normal form (in a sense defined by Church) is AX! Ax2 ax! ax! ax! ax! ax! ax! ax! x z. 4 It does not express an elementary proposition, but a metaproposition. (See below.) 5 These can, of course, be translated into the syntax language.
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be granted. But one can be formal without being syntactical, moreover the insistence on a syntactical point of view does not solve all the problems. To sum this all up: - the syntax of a language is essentially a formal system represented in a certain way. This representation is an importaht and fruitful one. One of its achievements is that it enables us to think of a formal system as something very concrete without losing sight of abstractness, and so incidentally to show that we do not need to presuppose mystical entities of a logical or other idealistic kind in order to be formal. But it has also certain disadvantages, and the time has come when we should sign a declaration of independence from it also.
FORMAL SYSTEMS AND SYNTAX
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- let us call it the S-language (i.e. syntax language). I shall use the prefixes '0' and'S' to distinguish notions belonging to these two languages. Is every formal system representable as the syntax of an 0language? Well, we can certainly find a representation in which the tokens are O-symbols and the operations are ways of combining O-expressions into linear series along with other O-symbols, i.e, we can assign to each operator q; an O-symbol, say '/' and then say that if aI' as, ... , an are any O-expressions, q; al as . .. a" is the O-expression got by writing in a row first '/' then the aI' az, ... ,an in order 1. The terms of the system will then be expressions of an O-language; but in general they will not constitute all the expressions of that language, so that at best we have only a part of the syntax. For the most general kind of formal system this is as far as we can go; because, although the predicates are formal in Carnap's sense, it is undetermined whether any of them will correspond to rO-sentencel, or whether there will be rules defining a term as a direct consequence of a class of other terms. Suppose, however, that we have a formal system with only one unary predicate. Then we can make the following definitions (for the terminology see Chapter VII): an O-sentence is the same as a formula; an O-sentence is a direct consequence of the null class when and only when it is an axiom formula; it is a direct consequence of the non-null class K when it is directly derivable from the members of K. Then the formal system will in fact become a part of the syntax of an O-language. It is evident that this will apply even for an indefinite system, since Carnap allows indefinite consequence rules. We note in passing that this language may be strangely and wonderfully made. In the case of a completely formalized system every term will be an O-sentence; and the O-symbols will consist of sentences and co-ordinating conjunctions only. This is not 1 This requires of course that the number of arguments to every operator be fixed. Then parentheses are unnecessary. Alternatively we may place the between commas, enclose the resulting expression in parentheses and prefix an '/' to the whole - this is the ordinary notation.
n,
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FORMAL SYSTEMS AND SYNTAX
surprising in the case of Example 3 because - taken intuitively as a logic - it had that character to start with; but in the case of Example 8 it is a little strange in view of the properties of that system mentioned at the end of Chapter VII. Incidentally, for systems just alluded to, the O-expressions which. are not terms are intrinsicly meaningless, like 'Bats live in caves and or'. Let us now consider the converse question, can the syntax of a language be set up as a formal system? The first step in such a procedure would be the formal definition of expressions as a category of terms. The fundamental operation in the given syntactical theory is combining two expressions by writing them consecutively; let us call this, after Quine, concatenation. This is an associative operation. Hence if we take it as a fundamental operation in our formalized syntax, we run into the difficulty that an expression is not formed from the O-symbols in a unique manner. This would make it necessary to take as terms not the analogues of the expressions themselves, but of methods of constructing them, and then to formulate, morphologically or theoretically, a relation of equality. However Hermes has shown that we can take instead a set of unary operations, one for each symbol, whose interpretation is writing that symbol after the operand 1. The idea of this is shown in Example 2; but it is not necessary to be so formal here, - we can simply take III A 2 and 3 as a recursive definition of a morphological operation. We then have the O-expressions formalized as expression terms. The formation rules can then be taken over bodily into the system, replacing each category, etc. of expressions by the corresponding category, etc. of expression-terms. Corresponding to the 0sentences we have the formulas among the expression-terms. Next take the rules of consequence. Let the axiom-formulas correspond to those O-sentences which are direct consequences of the null class, and let the other rules of direct consequence be set up as rules of procedure. We then have the syntax set up as 1 Of course one can take instead the symbols as tokens and a single operation of affixing a token to an expression. (This is an instance of the second reduction of § 7).
FORMAL SYSTEMS AND SYNTAX
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formal system, definite or indefinite; if the rules of the syntax are definite the formal system will be definite also. The preceding argument has necessarily a somewhat heuristic character. It is therefore instructive to see how it goes through in a special case, say Carnap's Language II. In this case we have a type structure so complicated that it is advantagoous to take a special category of terms to represent the types. Then we take as syntactical predicates: r- is a variable', r- belongs to ---l, r_ is a type", r_ is a sentence'. We shall need in connection with argument series concatenation of types as well as of expressions. Using 'A' for the concatenation operation we get a series of rules of which the following are samples: There is one special type: Z (numbers). If t 1 and tz are types, so are t 1 A tz, t 1 : tz, and (t1 ) . If Q 1 belongs to t1 and ~ belongs to tz then Q1 A ',' A Q2 belongs to t1 A tz. If Q1 belongs to (t1 ) and b belongs to t 1 , then Q1 A '('A b A')' is a sentence. If Q1 is a sentence, Q2 belongs to Z and Q2 is a variable, then '('A K A ~ A')' A Q1 belongs to Z. It appears from the above discussion that "formal system' and rsyntax of a language' are essentially equivalent notions. Aside from the representation, the only difference is that a syntax is by definition tied to the fundamental operation of concatenation. From an abstract point of view, "the syntax of a language .... is concerned, in general, with the structures of possible serial orders (of a definite kind) of any elements whatsoever"; from the same point of view a formal system is concerned with the structures of arbitrary combinations of any elements whatever. The notion of formal system is thus more general; but this generality is rather trivial since an arbitrary formal systen can be embedded in a syntactical one. The syntax of a language is then, essentially, a formal system represented in a certain way. It is now necessary to deal with the view, which seems to be prevalent among those addicted to syntax, that a discourse is only formal insofar as it is syntactical, - or in other words that 8.
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FORMAL SYSTEMS AND SYNTAX
a formal system must be represented syntactically if it is significant at all. If this were to mean simply that a reader, in considering a formal system, must have some such representation in mind, then it would be superfluous to add anything to what has been said in Chapter VI. But the syntacticians appear to have something more specific than this in mind. Thus Carnap writes as follows: 1 "In a treatise by a distinguished logician, the following sentence occurs: ,(~) a is a formula which results from the formula a when the variable x (if it occurs in a) is replaced throughout by the combination of symbols p' 2. Here we are from the beginning completely uncertain as to the interpretation. Which of the symbolic expressions in this statement are used as autonymous designations and are accordingly to be enclosed in inverted commas if the correct mode of expressing the author's meaning is to be achieved? '" We do not know to which object-language all the formulae, as syntactical formulae, are to refer." It appears that Carnap is demanding that a formal system be the syntax of a particular, explicitly stated, language 3. If so, there are several special criticisms to be added to those of Chapter VI. To begin with this would require a duplicity of symbolism. One must have the a-expressions themselves and also their Sdesignations, and if we are to be exact these must be kept distinct. Logical Syntax, p. 159. The sentence quoted does not occur in any logical treatise that I know of, and neither does the German translation of it in the original edition. But in Heyting's paper C 385.3, p. 58 the following sentence 1
2
occurs: "Zweitens fiihren wir die Bezeichnung
(~) a ein fur 'der Ausdruck,
der aus a entsteht indem man die Veranderliohe x uberall, wo sie auftritt, durch die Zeichenzusammenstellung p ersetzt' ". I am assuming this is what is referred to. 3 This language must be such as can be represented in the printed page, and not counters, noises, etc. of which Carnap indicates the possibility (Logical Syntax § 2). Such will be considered non-linguistic in what follows).
FORMAL SYSTEMS AND SYNTAX
43
Moreover the usual procedure is to take for a-language some more or less familiar set of marks; then for the S-designations of these we are confronted with wholly unfamiliar symbols of an outlandish kind. But - assuming the discussion is a formal one - the a-symbols do not appear in the main part of the argument at all; they only occur in the introductory discussions. This greatly increases the difficulties of comprehension 1. Why not, then, abolish the object language altogether! Certainly the theory would be more perspicuous if the ordinary symbols were used syntactically, while the object language is left to the reader's imagination or explained by means of other symbols 2. This would also be more in accord with our mathematical habits; for in mathematics we do not talk about our symbols, we use them. Again the syntactical point of view leads to an extreme nominalism which is foreign to mathematics. By this I mean that syntax tends to make purely linguistic accidents appear on a par with more substantial considerations. In mathematics we abstract not only from meanings of our symbols - from the nature of the objects which they denote - but also from the peculiarities of their external structure. Our symbols are not intrinsically meaningless, but their meaning is unspecified; and considerations which do not have a reference to such meaning are ignored. In doing such ample justice to the first of these abstractions the syntacticians perform only lip-service 3 to the second. Take for instance the matter of commas, parentheses, etc. To the ordinary mathematician these are just devices for symbolizing an operation not readily symbolized by concatenation alone; whereas to the syntactician they are symbols which are 1 A striking example of this is the system of Tarski, for example in C 285.13 or § 2 of C 285.16). In the case of systems using quotation marks we have a further confusion in that these symbols are used in a slightly different manner in ordinary discourse. S These symbols could be words of ordinary language. Thus we can say that if 'A' and 'B' are abbreviations of the sentences p and q (named by 'p' and 'q') respectively, then 'Opq' (or 'p:J q') is a name of 'if A then B'. a Cf. Logical Syntax, pp. 5 - 6.
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capable of being joined with other symbols to form expressional monstrosities 1. In a formal system we can leave these useless combinations out. Our terms then have a closer relation to the essential content of our ideas. A point much belabored by the syntax-addicts is the ambiguity of the "autonymous" mode of speech, - viz. that in which a symbolic expression is used to designate itself. If the expression already has a meaning in its own right, there then arises the possibility of doubt as to whether the expression refers to itself or to its original meaning. This is a particularly serious difficulty when the original meaning is itself symbolic in character. Frege pointed out this possibility of confusion, and introduced the systematic use of quotation marks to avoid it; but the idea can scarcely have originated with him, since the same difficulty arises whenever we talk about a language within that language, and English grammars were written in English long before Frege's parents ever dreamed of the blessed event. Yet, although the distinction is important, it is not necessary to make it constantly. In particular, when the subject under discussion is non-linguistic our occasional uses of the autonymous mode of speech are usually made clear in connection with the context by a careful use of the ordinary conventions of language. Again in the syntax of a wholly uninterpreted object-language - i.e. of a system of marks without meaning - there can be no confusion in using these marks as their own names, since they have no other meaning 2. Only in case we are dealing with the syntax of an actual, interpreted language is the danger of obscurity serious. These considerations suggest that if, in dealing with a formal system, we bear in mind the possibility of a non-linguistic represenr
1 In the syntax of a language these monstrous expressions are excluded by the formation rules. Hence in seting up a syntax as a formal system we have an alternative mode of procedure to that considered above defining only such categories etc. of terms as are significant in the formation in rules. In this way Example 6 arose from Church's system. The terms there are his "well formed expressions". I This appears to be the representation which Hilbert has in mind.
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tation, then we may be allowed a certain liberality, which we otherwise would not have, in connection with the autonymous mode of speech. This mode has long been customary in dealing with substitution; the sentence from Reyting, quoted above, is a definition in which he follows tradition. The trouble which Carnap has with it seems to be due to his trying to force a syntactical representation where it does not belong. If he were to admit a non-syntactical representation I think he would be able to understand what Reyting means. Reyting's statement, to be sure, is incorrect; but it is clear how it is to be made correct (Cf. Example 6, above). In such cases it is often a matter of considerable inconvenience to make the analysis necessary for absolute correctness, and to insist it be done at every stage is to hamper research. All we need is to know that the correct analysis can be made. The consideration of a non-syntactical representation - perhaps alongside a syntactical one - may therefore be in the interests of progress 1. Again, in the case where the object language is an interpreted language, then we do not need a syntax language at all. For we can regard the primitive frame as being the definition (by recursion, of course) of the symbols for the formal predicates 2; these are to be regarded, then, as new technical words of the object language. Of course these predicates may be liable to confusion with intuitive predicates already in the a-language; and in that case a syntax language may be a convenience; but it is not a necessity. I do not agree that the only things which can be treated formally are symbols. This process involves certain difficulties if the object language contains variables, but we now know, in principle at least, how variables can be eliminated. In most cases in which we are actually interested in the syntax of a language, we are interested, directly or indirectly, in its subject matter also; then there is always the possibility of representing a formal system 1 I have benefited by some discussion with Church in regard to these matters; but he is not to be held responsible for the views expressed. 2 Including morphological as well as theoretical predicates. The notion of "term" must also be so defined.
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directly in terms of the subject matter, rather than through the somewhat artificial medium of the syntax of the language. These considerations have a bearing on the discussions in the last part of Carnap's book. As to the philosophical aspects of that discussion I am not competent to speak. I can only say, however, as a mathematician, that the fundamental thesis viz. that the confusion mentioned is due to the material mode of speech - is not proved. For the sentences listed as examples are in many cases intrinsically vague - they contain words not defined in the context, - and it is conceivable that it may be this vagueness, rather than the material mode of speech per se, which is at fault. Moreover the translations into the syntactical mode of speech do not, in all cases, have the same meaning as the original 1. To illustrate these points consider the two sentences: (1) Seven is a number. (2) Seven is an odd number. If I understand Carnap correctly he would say that (1) is a "quasisyntactical sentence in the material mode of speech" while (2) is an object-sentence. Let us examine them more closely. In the first place it is necessary to inquire after the meanings of 'seven', 'number', 'odd number'; for without these meanings both (1) and (2) are nonsense. These meanings may naturally depend on the context in which the sentences are embedded. Let us first analyze these sentences on the hypothesis that they occur in the midst of an ordinary, non-technical discussion in the English language. Now in English 'seven' is primarily an adjective; when used as a noun it is essentially only a name for the adjective. Hence 'seven' in both (1) and (2) is autonymous. As for 'number' we find in the dictionary that a number is one of a series of words 1 For example the sentences 'Yesterday's lecture was about Babylon' and 'In yesterday's lecture "Babylon", or some synonymous word, occurred', are not equivalent. For suppose the lecture had been about the breeding habits of the polar bear, but the professor had told a joke in which there was an instance of 'Babylon'; then the first sentence would be false and the second true.
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(or symbols) used for counting, hence it is practically synonymous with 'number word'. (1) is therefore a syntactical sentence and expresses, practically, the same meaning as (3) 'seven' is a number word. Now what about (2)? Given any series of numbers in the above sense, we can define which ones are even and which odd by a recursive definition, and this can be made intelligible to a Chinaman having no knowledge of English provided that the explanations were made in Chinese. (2) is therefore just as much a syntactical sentence as (1) is. We get some light on the relations of (1) and (3) if we translate them into German 1. The translation of (1) is evidently (4) Sieben ist eine Zahl. In considering the translation of (3) consider the sentence' 'Seven' has five letters'. To translate this as "Sieben' hat fiinf Buehstaben' would evidently be incorrect, since the latter sentence does not have the same truth value; to preserve the meaning we should have to regard "seven" as its own translation. But if we translate (3) in this way we get a sentence which would be meaningless in a German context, and we have to add a phrase of explanation. Hence we get as translation of (3): (5) In der englischen Sprache ist 'seven' ein Zahlwort.
It is evident that (4) and (5) are not synonymous for the one refers to a fact of the English language, while the other does not. Yet it is evident that (4) is the translation of (1) in the sense of non-technical discourse. Therefore (3), - which, incidentally, would not occur in non-technical, but only in linguistic discourse - is not a true translation of (1). The difference is that (1) has an abstract connotation; 'seven' there refers, not to 'seven' as a linguistic phenomenon, but to the meaning of 'seven', i.e. to rsevenl. Whether one thinks of this as an abstract concept of seven-ness or as a class of equivalent words, or what not, is an 1
This suggestion is due to Langford.
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irrelevant matter. But in any case to replace (1) by (3) is to distort its meaning 1. Suppose now, we take (1) in a different context. Church has shown how to define numbers in the system of Example 6 2 • In fact, let One S
=
Axl Ax2 ax l x 2 • I AX2 Axs ax2 aaxl x 2 xs'
= AX
This S is the successor-function, and in terms of it the other integers are defined successively 3. Now, suppose we take for object language the English language together with the technical words and symbols defined in the primitive frame of Example 6. Then 'number' can be defined recursively as follows: a) One is a number. b) If a is a number, aSa is a number. c) If a cnv b and a is a number, then b is a number. In that context 'Seven is a number' is an object-sentence"; Moreover in order to decide whether it is true you would have to know, not only the syntactical rules of the English language, but the definitions of 'seven' and 'number' also 5. In view of these considerations there is something which needs to be added to the indicated part of Carnap's book. The object of Carnap's whole discussion is, as I understand it, to clarify the vague sentences often propounded in philosophy. That the concept of syntax language can be a great help in that connection may 1
Another example is In antiquity seven was regarded as sacred.
'Seven' of course, did not exist in antiquity. What was regarded as sacred was rseven1. 2 This example is used, rather than Example 1, because 'number' is not a "universal word". 3 The definition of seven is then as as as as as as (one) whose normal form (in a sense defined by Church) is AX! Ax2 ax! ax! ax! ax! ax! ax! ax! x z. 4 It does not express an elementary proposition, but a metaproposition. (See below.) 5 These can, of course, be translated into the syntax language.
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be granted. But one can be formal without being syntactical, moreover the insistence on a syntactical point of view does not solve all the problems. To sum this all up: - the syntax of a language is essentially a formal system represented in a certain way. This representation is an importaht and fruitful one. One of its achievements is that it enables us to think of a formal system as something very concrete without losing sight of abstractness, and so incidentally to show that we do not need to presuppose mystical entities of a logical or other idealistic kind in order to be formal. But it has also certain disadvantages, and the time has come when we should sign a declaration of independence from it also.