Figure 64: Rudolf Carnap: "Interpretations" of calculi (1939/42)

 

 

Cambridge, Mass.: Harvard University Press

with : Formalization of Logic (1943) in one volume 1961.

 

pp. 202-204

 

E. RELATIONS BETWEEN SEMANTICS AND SYNTAX

The sentences of a calculus K may be interpreted by the truth-conditions stated in the semantical rules of a system S, provided S contains all sentences of K. Therefore, if this condition is fulfilled, S is called an interpretation for K.

Different kinds of interpretations are distinguished. If the direct C-concepts of K (and, hence, also the other C-concepts of K) are in agreement with the corresponding radical concepts in an Interpretation S for K, then S is called a true interpretation for K; otherwise, a false interpretation. If there is a similar agreement with the L-concepts in S, S is called an L-true interpretation for K. In this case, the rules of K and S suffice to show that S is a true interpretation for K. Other kinds of interpretations: L-false, L-determinate, factual, F-true, F-false, logical, and descriptive interpretations.

These concepts of different kinds of interpretations are useful for the logical analysis of science; for what is usually called the construction of a model for a set of postulates is the same as the semantical interpretation of a calculus.

 

 

§ 33. True and False Interpretations

S is called an interpretation for K if S contains all sentences of K. An interpretation S for K is called a true interpretation for K if in any case where a direct C-concept (e. g. direct C-implication) holds in K the corresponding radical concept (e. g. implication) holds in S; otherwise, S is called a false Interpretation for K. If S is a true interpretation for K, then in any case where a C-concept (e. g. 'C-true') holds in K the corresponding radical concept (e. g. 'true') holds in S (T8).

 

In this chapter, certain relations between semantical systems and calculi will be investigated, especially relations of interpretation. Since these relations belong neither to semantics nor to syntax, we are here in a wider field, which comprehends both pure semantics and pure syntax but goes beyond them (for terminological remarks, see § 37, Theory of Systems).

 

A calculus K is constructed and analyzed within syntax in a formal way. As long as we stay in syntax there do not arise questions as to the meaning of the expressions and sentences occurring in K, i. e. as to the designata of the expressions and the truth-conditions of the sentences. But, if a calculus K is given, we may go over to semantics and assign designata to signs of K and truth-conditions to sentences of K by semantical rules. Hereby sentences of K become interpreted.

And if we lay down a sufficient set of such rules or, in other words, a semantical system S containing all the sentences of K, then all these sentences become interpreted. In this case we call S an interpretation for K (D1). S may contain many more sentences than those of K, but it must not contain fewer. We will call S, moreover, a true interpretation for K if the semantical rules of S are in accordance with the syntactical rules of K in such a way that if, according to a rule of K, a direct C-concept holds in a certain case, then the corresponding radical semantical concept holds in this case in S (D2). This definition involves that every primitive sentence of K is true in S (T8a).

It will be shown that if this condition is fulfilled an instance of any C-concept, whether direct or not, becomes an instance of the corresponding radical concept. That is to say, if S is a true interpretation for K, then every _i which is C-true in K becomes true in S; and likewise every C-false _i becomes false; when C-implication holds, implication holds, etc. (T8).

 

Interpretations of calculi play an important role in the method of science. In mathematics, geometry, and physics, systems or theories are frequently constructed in the form of postulate sets., And these are calculi of a special kind (see [Foundations] § 16).

For the application of such systems in science it is necessary to leave the purely formal field and construct a bridge between the postulate set and the realm of objects. This is usually called constructing models for the postulate set or laying down correlative definitions for it ('Zuordnungsdefinitionen', Reichenbach). It is easily seen that this procedure, described in our terminology, leads from syntax to semantics and is what we call here constructing an interpretation for a calculus.

 

 

p. 243-244

 

Postulate systems (or axiom systems), whose construction has been found so useful in mathematics, geometry, and physics, are, regarded from the point of view of semantics and syntax, combinations of two parts, basic system and a specific addition.

The basic system usually contains logical words like `not', 'and', 'every', etc.; these are not dealt with in a formal way but taken in their ordinary meaning. Therefore this part of the system is semantical, though not explicitly. The specific part, however, is taken in a formal way; deductions are carried out without presupposing a particular interpretation. Therefore this part is syntactical.

What is usually called the construction of a model for a postulate set is the construction of an interpretation for this syntactical part (compare [Foundations] § 16).

Hence, the general theory of postulate systems, dealing with the various kinds of such systems and with problems of consistency, independence, completeness, monomorphism, existence of models, etc., is a branch of general Syntax and general semantics. Hence, when general syntax and general semantics has been developed sufficiently, the theory of postulate systems will be in a position to make use of the concepts and theorems of these fields.

 

 

p. 14

This treatise is devoted to pure semantics and pure syntax, or rather to the field in which semantical systems and syntactical systems, and in addition their relations, are analyzed. (There is so far no suitable name for this field; see terminological remarks, § 37, 'Theory of Systems'.)

 

 

p. 240 (Appendix, § 37: Terminological remarks)

Theory of Systems. It would be convenient to have a term for the field in which semantical and syntactical systems are investigated. This field contains pure semantics and pure Syntax and, in addition, the study of the relations between syntactical and semantical systems (e. g. interpretation); the latter belongs neither to syntax nor to semantics.

Terms which might be considered:

1. 'systematics' (but the adjective 'systematical' could hardly be used in this sense);

2. 'systemics' (suggested by K. R. Symon);

3. ' (logical) grammar' (Wittgenstein seems to use this term and likewise ' (logical) syntax' for an analysis which, in our terminology, combines syntactical and semantical questions but also covers what we call descriptive syntax and descriptive semantics, and perhaps even something of pragmatics).

 

(If I had known a suitable term, I should have taken it instead of 'semantics' in the title of these studies.)

 

 

International Encyclopedia of Unified Science, Vol. 1, No. 3. Chicago: University of Chicago Press.

separate print

 

p. 37-38

 

16. General Remarks about Nonlogical Calculi (Axiom Systems)

 

In later sections we shall discuss certain other calculi which are applied in science. The logical calculus explained previously is distinguished from them by the fact that it serves as their basis. Each of the nonlogical calculi to be explained later consists, strictly speaking, of two parts: a logical basic calculus and a specific calculus added to it.

The basic calculus could be approximately the same for all those calculi; it could consist of the sentential calculus and a smaller or greater part of the functional calculus as previously outlined.

The specific partial calculus does not usually contain additional rules of inference but only additional primitive sentences, called axioms. As the basic calculus is essentially the same for all the different specific calculi, it is customary not to mention it at all but to describe only the specific part of the calculus.

 

What usually is called an axiom system is thus the second part of a calculus whose character as a part is usually not noticed.

For any of the mathematical and physical axiom systems in their ordinary form it is necessary to add a logical basic calculus. Without its help it would not be possible to prove any theorem of the system or to carry out any deduction by use of the system. Not only is a basic logical calculus tacitly presupposed in the customary formulation of an axiom system but so also is a special interpretation of the logical calculus, namely, that which we called the normal interpretation.

An axiom system contains, besides the logical constants, other constants which we may call its specific or axiomatic constants. Some of them are taken as primitive; others may be defined. The definitions lead back to the primitive specific signs and logical signs. An interpretation of an axiom system is given by semantical rules for some of the specific signs, since for the logical signs the normal interpretation is presupposed. If semantical rules for the primitive specific signs are given, the interpretation of the defined specific signs is indirectly determined by these rules together with the definitions.

But it is also possible - and sometimes convenient, as we shall see - to give the interpretation by laying down semantical rules for another suitable selection of specific signs, not including the primitive signs. If all specific signs are interpreted as logical signs, the interpretation is a logical and L-determinate one; otherwise it is a descriptive one. (Every logical interpretation is L-determinate; the converse does not always hold.)

 

 

Aus dem Amerikanischen übersetzt, mit einem Nachwort und einer kritischen Bibliographie versehen von Walter Hoering. München: Nymphenburger Verlagshandlung.

 

pp. 53-55

 

16. Allgemeine Bemerkungen über nichtlogische Kalküle (Axiomensysteme)

 

In späteren Abschnitten werden wir gewisse andere Kalküle diskutieren, die in der Wissenschaft Anwendung finden. Der bisher eingeführte logische Kalkül unterscheidet sich von diesen dadurch, dass er deren Basis darstellt. Jeder der nichtlogischen Kalküle, die später eingeführt werden, besteht genau gesprochen aus zwei Teilen: einem logischen Grundkalkül und einem hinzugefügten spezifischen Kalkül.

Der Grundkalkül könnte für all diese Kalküle ungefähr der gleiche sein; er könnte aus dem Aussagenkalkül und einem grösseren oder kleineren Teil des bereits skizzierten Prädikatenkalküls bestehen.

Der spezifische Teilkalkül enthält normalerweise keine zusätzlichen Schlussregeln, sondern nur zusätzliche Grund-Sätze, die Axiome genannt werden. Da der Grundkalkül für all die verschiedenen spezifischen Kalküle im wesentlichen der gleiche ist, ist es üblich, ihn gar nicht erst zu erwähnen, sondern nur den spezifischen Teil des Kalküls zu beschreiben.

 

Was gewöhnlich ein Axiomsystem genannt wird, ist daher der zweite Teil eines Kalküls, bei dem man normalerweise gar nicht daran denkt, dass er nur ein Teil ist.

Bei allen mathematischen und physikalischen Axiomensystemen in ihrer gewöhnlichen Form ist es nötig, einen logischen Grundkalkül hinzuzufügen. Ohne seine Hilfe wäre es nicht möglich, ein Theorem des Systems zu beweisen oder eine Ableitung mit Hilfe des Systems vorzunehmen. Nicht nur wird ein logischer Grundkalkül stillschweigend bei der üblichen Formulierung eines Axiomensystems vorausgesetzt, sondern auch eine spezielle Interpretation des logischen Kalküls, nämlich jene, die wir die Normalinterpretation genannt haben.

Ein Axiomensystem enthält neben den logischen Konstanten andere Konstanten, welche wir seine spezifischen oder axiomatischen Konstanten nennen können. Einige werden als Grundkonstanten angenommen, andere können definiert sein. Die Definitionen führen zurück auf die spezifischen Grundzeichen und die logischen Zeichen. Eine Interpretation eines Axiomensystems wird durch semantische Regeln für die spezifischen Zeichen gegeben, da für die logischen Zeichen bereits die Normalinterpretation vorausgesetzt wird. Wenn semantische Regeln für die spezifischen Grundzeichen angegeben werden, dann wird die Interpretation der definierten spezifischen Zeichen indirekt durch diese Regeln zusammen mit den Definitionen festgelegt.

Es ist jedoch auch möglich - und manchmal, wie wir sehen werden, angebracht - eine Interpretation durch Angabe von semantischen Regeln für eine andere Auswahl aus den spezifischen Zeichen festzulegen, die nicht die Grundzeichen enthält. Wenn alle spezifischen Zeichen als logische Zeichen interpretiert sind, dann ist die Interpretation eine logische und L-determiniert. Sonst ist sie deskriptiv. (Jede logische Interpretation ist L-determiniert; die Umkehrung gilt nicht immer.)