III Choice of Metascience and Metalanguage

I11 CHOICE OF METASCIENCE AND METALANGUAGE $ 15. INTRODUCTION OF THE CONCEPTS

“METASCIENCE”

AND “META-

LANGUAGE”

I n according with existing usage we now introduce the expressions “metalanguage” and “metascience” (Cf. Ta., p. 22). We regard the term “syntax language” as synonymous with “metalanguage”. The two expressions designate relations ; the first a relation between languages and the second a relation between sciences. The converse of these relations is designated by the expressions “object language” and “object science”; i.e. if x is the metalanguage with respect to y, y is the object language with respect to x ; the concept object science is determined in the same way. The essence of the concept metalanguage can be expressed as follows : If x is the metalanguage with respect to y, x contains expressions which designate the expressions of y. Furthermore, if x is the metalanguage with respect to y, x can be either a different language than y or the same language as y. The concept metascience is closely connected to the concepts truth and falsity. We can express this connection by saying: Science x is called the metascience with respect to y if the proof that sentences of y are true or false as the case may be can be carried out in 5. There is obviously a connection between the concepts metascience and metalanguage. It may be said of any science that it is represented in a certain language. If x is the metascience with respect to y, then the language in which x is represented is the metalanguage with respect to the language in which y is represented. We will now refer to a law which is of great significance to the history of the sciences. Science as a whole and the individual sciences in particular are conceived of as being in progressive

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CHOICE OF METASCIENCE A N D M E T A L A N G U A G E

development. If the history of science x at time t is written, the form that science x has achieved at time t serves as metascience. To illustrate this law we notice that in a modern history of mathematics, modern mathematics serves as the metascience. I n each of the two books of Boethius, i.e. “de syllogism0 hypothetico” and the commentary on Cicero’s Topics, a science is represented. These sciences are here designated object sciences. Modern logistic will serve as the metascience with respect to these sciences. The language in which these two object sciences is represented is a form of Latin created by Boethius. As metalanguage we will use a language which contains the language of logistic.

3

16.

THE

SYSTEMS

OF

MATERIAL

IMPLICSTION

AND

STRICT

1MPLICATION

We have indicated that we will here regard modern logistic as o w metascience; we must also tell which of the modern representations of logistic we have in mind. There are two systems to be considered; we will call them briefly “the system of material implication” and “the system of strict implication”. The system of material implication is represented in the inathematical logic of the Principia Mathematica; in this connection we are concerned especially with the section entitled “The Theory of Deduction”. The system of strict implication is represented in the work of Lewis and Langford entitled “Symbolic Logic”. The sixth chapter is essential to our purpose. We note that this chapter was composed by Lewis. It may seem strange that we regard it necessary to consult two differing systems of modern logic, but we have considerable reason for doing this. It seems to us that we must consider two possibilities in the interpretation of the inference schemes of Boethius ; firstly, there is the possibility of interpreting certain of the implications that appear in these inference schemes as material implications,on the other hand, there is the possibility of considering them as strict implications. Therefore, in testing, we will use both the system of material implication and the system of strict implication as a basis.

DESCRIPTION O F THE METALANGUAGE

0

17.

DESCRIPTION O F THE

21

METALANGUAQE TO BE USED I N THE

INVESTIGATIONS TO FOLLOW

We will now consider the language to be used as metalanguage more closely. I n constructing the principles of the metalanguage we will use the teachings of the Polish logisticians Jan Lukasiewicz and Alfred Tarski. We will point out in advance of the presentation what ideas we have borrowed and from which of the two representatives of logistic they are taken. From Lukasiewicz we borrowed the idea of parenthesis-free notation, or, in other words, the principle that the functors are always to precede their argument (Cf. Lu., p. 125-126). From Tarski we borrowed the idea that every expression of the object language is correlated to two expressions of the metalanguage, namely (1) An expression which may be called the translation of the correlated expression of the object language into the metalanguage. (2) An expression which is the name of the correlated expression of the object language (Cf. Ta., p. 28).

In the two books of Boethius we find logical formulae; we have already indicated that these formulae are our principal concern (Cf. 5 2 supra). Our metalanguage must therefore be of such a nature as to make it possible to translate into this language every logical formula which occurs in the two books of Boethius. In the formulae presented by Boethius we find expressions that are to be designated propositional variables. “Propositional variable” is an expression of our metalanguage which is a general name for certain expressions of the object language. We can distinguish two different systems of propositional variables, one simple and one extended. The simple system has only two propositional variables, viz. the two expressions “hoc est” (this is) and “illud est” (that is). We call attention to the fact that not the simple signs “hoc” and “illud” but the complex signs “hoc est” and “illud est” are to be regarded as propositional variables.

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CHOICE OF METASCIENCE A N D METALANGUAGE

Sometimes Boethius uses the simple expressions “hoc” and

“illud” in place of the complex expressions “hoc est” and “illud est”; in these cases we will regard the simple signs “hoc” and

“illud” as propositional variables. The expression “hoc est” (or “hoc”) may be termed the first propositional variable and the expression “iltud est” (or “illud”) the second propositional variable. This simple system of propositional variables agrees in essentials with the one used by Cicero in the Topics. It is to be noticed that Cicero uses the simple signs “hoc” (this) and “illud” (that) as propositional variables (Cf. Ci. 14, 56-57). Boethius uses the simple system of propositional variables in two places : (1) A t the beginning of “de syllogismo hypothetico” (Cf. Boe., p. 608). (2) In the fifth book of the commentary on Cicero’s Topics immediately following Cicero’s presentation (Cf. Boe., p. 831-833). The extended system consists of four distinct propositional variables viz. the following four expressions : “a est” or “b est” or << c est” or “ d est” or

“est a” “est b” “est c” “est d”.

In the formulae which Boethius constructs the two simple signs of which the expressions are composed ordinarily appear in the second order. We stress again that the complex signs and not their elements are to be regarded as propositional variables. Boethius uses the extended system of propositional variables only in “de syllogismo hypothetico” ; this system completely replaced the simple one. I n order to explain this fact we note that a large number of the formulae that Boethius presents in “de syllogismo hypothetico” cannot be expressed if only two distinct propositional variables are allowed, since in many of these formulae three or even four propositional variables occur.

DESCRIPTION O F THE METALANGUAGE

23

In addition to the propositional variables, there occur only signs that may be designated functors ; we will enumerate them and, at the same time, introduce their names. These names belong to our metalanguage : ( 1 ) The sign of negation: “mn” (not). (2) The signs of implication: “si” (if) and “cum” (when). ( 3 ) The sign of disjunction: “aut . . . aut” (either . . . or). (4) The signs of conjunction: “ a t p i ” (but), “at” (but), “autem” (but) and “et” (and). ( 5 ) The sign of inference : “igitur” (therefore). (6) The modal signs: “necesse est” (it is necessary) and “contingit” (it is possible). Boethius says that the two conjunctions “si” and ‘Lcum”,i.e. the two signs of implication, are synonymous (Cf. Pr. I, p. 702, note 143). We believe we may assume that Boethius used two different signs for implication out of stylistic considerations only. It could, for example, be considered poor style if the same sign was immediately repeated and this can be avoided if one has two synonymous signs. This can be shown by an example. I n “de syllogismo hypothetico” we find the following formula: “si cum sit a, est 6 , est

G”

(Cf. Boe., p. 621). I n this formula the two signs of implication occur in immediate succession. If there were only one sign of implication, the same sign would have to occur twice in immediate succession. We remark that we mll not copy this characteristic of Boethius’ language in our metalanguage; it will be our principle to keep the number of signs as small as possible. We believe we may also regard the four signs of conjunction as synonymous. I n “de syllogismo hypothetico” we find some places where expressions are interpretable as conjunctions, although no sign of conjunction occurs (Cf. Boe., p. 626, p. 628, p. 629 and p. 676). It seems permissable to add a sign of conjunction, e.g. “et”, a t these places.

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CHOICE OF METASCIENCE A N D IKETALANGUAGE

The two modal signs are not synonymous; however, this will not oblige us to use two modal signs in our metalanguage. Boethius puts the modal signs before “mse u” and “non esse u” (Cf. Boe., p. 612). Strictly speaking, these expressions are not identical with the expressions “est u” and “non est a”. We believe we may disregard this difference in our metalanguage and proceed as if the expressions “est a” and “non est u” were at these places instead of “esse a” and “non esse a”. We will now proceed to indicate how Boethius’ logical formulae are to be translated into our metalanguage. We distinguish arguments and functors and adopt, as a syntactical rule, that functors are always to precede their arguments. The following may be arguments : ( 1 ) propositional variables. (2) well formed expressions constructed from functors and propositional variables. Our language contains four distinct propositional variables, viz. the four letters “p”, “q”, “r” and “s”. The use of these letters as propositional variables is recommended by the fact that in modern logistic, especially in the works named in § 16, they are used in this sense. As the sign of negation we will use the upper-case letter “ N ” ; this agrees with the practise of the Polish logisticians. (Cf. Lu., p. 126; Ta., p. 23). We state, as a syntactical rule, that the sign of negation is a functor which always takes one argument. On the basis of the two syntactical rules laid down, we can establish that the following expressions are well-formed :

“Np” and “ N N p ” . As the sign of implication, we use the lower-case letter “c”. We note that Lukasiewicz used the corresponding upper-case letter “G” as the sign of implication (Cf. Lu., p. 176). We have already indicated in Q 16 that there are two possibilities for the interpretation of Boethius’ inference schemes : certain implications that occur in them are to be understood either as

DESCRIPTION O F THE METALANGUAGE

25

material or as strict implications. This holds analogously for the logical formulae that are not inference schemes. We can refer to Boethius’ logical formula stated above. In that formula, the synonymous signs “gi” and “cum” can both be interpreted either as signs of material or strict implication. The sign “c” is ambiguous and can therefore be identified neither with a sign of material, nor with a sign of strict, implication. The use of the sign “c” in the sense referred to recommends itself because, on one hand, it is reminiscent of “G” and yet is distinct from it. That the sign of implication of the object language is ambiguous can be regarded as a fault of this language; it is however obviously not a fault of our metalanguage that the expression used to translate that sign is ambiguous. With respect to the sign of implication we also lay down a syntactical rule, namely, that the sign of implication takes two arguments, preceding the first of the two arguments while the second argument immediately follows the first. We can now identify the following as well-formed :

“cpq” and “cNpq”.

As sign of disjunction we use a lower-case letter “a”. The sign (‘a” is ambiguous, like the sign “c” is. The sign “a” is, on the one hand, a sign of material exclusive disjunction and, on the other, a sign of strict exclusive disjunction. It is possible to construct a system where both of the expressions “material exclusive disjunction” and “strict exclusive disjunction” receive legitimite definitions. We will however restrict ourselves t o indicating them by referring to an example. If the sentence ‘(Socratesis either well or ill” is interpreted as a material exclusive disjunction it says “it is not the case that Socrates is both well and ill and i t is also not the case that Socrates is not well and not ill”; when the same sentence is interpreted as a strict exclusive disjunction, it says “it is impossible that Socrates is both well and ill and it is also impossible that Socrates is not well and not ill”. I n addition, it should be remarked that the sentence “Socrates

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CHOICE OF METASCIENCE AND METALANGUAGE

is either well or ill” was used by Peter Abelard as an example of disjunction (Cf. Ab. C., p. 441-442) and that Boethius himself in “de syllogismo hypothetico” presents the incomplete sentence “is either ill or well” as an example of disjunction (Cf. Boe., p. 636). The sign of disjunction, like the sign of implication, takes two arguments and immediately precedes the first of the two arguments. It is easy to see that the two expressions “apq” and “aNpq” are well-formed. As sign of conjunction, we will use the Latin word “et”. We note that this word is not ambiguous as are the signs of implication and disjunction but has a single meaning. The sign of conjunction also takes two arguments and immediately precedes the first one. In logistic literature, the expression “logical product” is in general use (Cf. PM, p. 6 and L.a.L., p. 123). It seems therefore appropriate to state that if ‘9’’ and “u” are any arguments whatever, the expression “et t ZL” represents the logical product of “t” and “u”. We have borrowed the conjunction “et” from the object language, but we have changed what might be called its syntax. It is not the case in Latin that the functor “et” is placed before the arguments; on the contrary, it is placed between its arguments. We may however say that the syntactical rule that we have adopted is not too different from Latin syntax and that this syntax has a place for it, since Latin also uses as sign of conjunction the two membered conjunction “et . . , et” and in this case at least a part of the functor precedes the two arguments. We have also borrowed the sign of inference, the Latin word “igitur” from the object language; here too the syntax of the word is changed. The word “igitur” is in our language a functor which takes two arguments and immediately precedes the first. As opposed to this, the word “igitur” of the object language is placed immediately after the first or second sign of that sentence which may be called the conclusion.

DESCRZPTION OF TEJT METALANGUAUE

27

I n logical formulae the functor “igitur” can occur only at the beginning; in this it differs from all the other functors. This rule represents a serious limitation; we should like to point out why this limitation seems desirable. We will construct no logical formulae in which the functor “igitur” occurs any place but at the beginning of the formula. If we lay down rules for our language which did not prohibite the functor “igitur” at any place but a t the beginning, possibilities would occur of which we do not make use; and that contradicts the principle of parsimony which we would like to follow. If we will be allowed to state the maxim we will follow in a short and impressive form, me may say: what we do not use, we will not possess. A logical formula is called an inference scheme, if and only if, it begins with the functor “igitur”. It seems convenient to illustrate here the same inference scheme in the object language and the metalanguage; thus it will be possible to compare the original with the translation. We will choose as an example the first inference scheme of “de syllogismo hypothetico”. The inference scheme is stated in the object language as follows:

“si est a, est b, atqui est a , est igitur b” (Cf. Boe., p. 615). This expression corresponds to the following formula of the metalanguage :

“igitur et cpqpq” ; the first argument of the functor “igitur” is

“et cpqp” ; the second argument of this functor is

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CHOICE O F METASCIENCE A N D M E T A L A N G U A G E

the first argument of the functor “et” is “Cpp;

the second argument of this functor is

,,P”; the first argument of the functor ((c” is

((P” and the second argument of this functor is “G!”.

From this analysis we can see that the formula is well-formed. In order to tra,nslate the two modal signs of the object language into our metalanguage, we introduce a sign borrowed from Lewis’ system of strict implication. This sign is a rhombus so placed that one of the two diagonals is horizontal, and, as a result, the other is vertical. Following the usage of C. West Churchman in his article “On finite and infinite modal systems”, we call this sign “the modal operator” (Cf. Ch., p. 77). The modal operator is a functor which, like the sign of negation, always takes a single argument and precedes this argument. The modal operator will serve as the translation of the expression “contingit” (it is possible). The expression “contingit e8se u” of the object language corresponds t o the expression

“OP” of the metalanguage and the expression (‘necesse est a s e u” of the object language corresponds to the expression

“NONp” of the metalanguage. As an analysis of this expression, we note: the argument of the first sign of negation is the expression

“ONp’’;

DESCRIPTION OF TEE METALANGUAGE

29

the argument of the modal operator is the expression

‘“P” and the argument of the second sign of negation is the expression

“P”. We can easily see that the whole expression is well formed. This finishes the statement of the principles of our metalanguage.