ISSN 1135-1349
Issue #16 -- December 2005. Pp. 46-53
Tarskian Metamathematics in Carnap's Metalogic
Copyright © by SORITES and Jesús Padilla Gálvez
Tarskian Metamathematics in Carnap's Metalogic
Jesús Padilla Gálvez

An important group of logicians working in the field of metatheory proposed a variety of different suggestions on the description and analysis of signs of formal structures. Thereby they influenced each other in different ways. This metatheory was then called «metamathematics», «metalogic», or «methodology of deductive sciences». Divergence certainly did not rely on the same mode for all logical significant aspects of a system, but with regard to the fundamental frame of references and composition of basic elements of the structure of formal systems, which should be described and analyzed. The aim of this paper is to view the contours of metalogic through a careful and exact analysis of the works of A. Tarski and R. Carnap and to differentiate between the various approaches, formulations and interests. In this way their differing positions on formalistic problems, divergence and convergence can be shown in full. This task demands a wide-ranging and exact revision, in the course of which it is easy to lose sight of the basic logical structure. The subject of this paper is to attempt a systematic explanation of metalogic.

1. Historical approach

Is it not surprising that one of the most influential members of the Vienna Circle, R. Carnap, supported metalogic despite his «metaphysical suspicions» and introduced the idea of looking at the problem of logical syntax of language from precisely this point of view? How could metalogic comply with the logistical basics of mathematics? How could different directions and philosophical points of view be connected to each other, without denying their positivistic aspect? This and other questions point to the problems which were discussed in the Vienna Circle in the Thirties.

In 1931 Carnap did not share Gödel's arithmetic view but held a descriptive view. The question which interests us is whether Tarski's descriptive view of metalogic was accepted. Carnap later changed his views in his «Syntax» and accepted and integrated Gödel's pioneering results into his program.

On 19th February 1930 Tarski gave a talk in Vienna with the title: `Über das Auswahlaxiom und die verallgemeinerte Kontinuumshypothese'. On the following two days Tarski spoke in the Schlick Circle on `Metamathematik und Metamathematik des Aussagenkalküls'.Foot note 1_1 The following question was up for debate: under what conditions can we speak of formal languages in a legitimate way free from objections, without hitting upon the common difficulties of self- references? The two great dangers of logical research were firstly those of autonymous ways of speakingFoot note 1_2 and secondly, drawing attention to the fact that expressions which we «indicate» have a relative character and therefore must always be relativised in a particular language. Metamathematics advocates a clear distinction between logic-mathematical formalism (the so-called object language) and the metamathematical considerations. This is particularly expressed by the difference between symbols and variables of formalism and signs of «communication». The formal theory is therefore opposed to a metalogic or metamathematics. Tarski attempted to show that although questions can be systematized by the expansion of expression in certain languages compared to the calculated language the systematic nature of which can be dealt with in an axiomatic form.

Both speeches divided the Vienna Circle. Carnap wrote in his diary that Schlick did not recognize the relevance of the metamathematical investigations.Foot note 1_3 It is possible that Neurath's resistance to Tarski's views developed during the lectures. At first it was suspected that Tarski's metalogical definitions (in particular the definition of truth) were not without requirements. The general mistrust within the Vienna Circle was not the same amongst all the members. In a frequently quoted passage from Carnap's letter to Neurath on 23rd of December 1933, he gave a summary showing which root his «syntax» had originated from. He wrote: «My syntax has historically two roots: 1. Wittgenstein, 2. Metamathematics (Tarski, Gödel).»Foot note 1_4 This reference is even more surprising as firstly, Wittgenstein had always been critical of metalogic and secondly, in the area of metamathematics, the Hilbert school was not mentioned. The meaning behind the Wittgensteinian reference can only be related to the fact that he had drawn attention to the importance of the problems which affect the language. In his research, however, Wittgenstein always rejected the view that it would be legitimate to talk about language and this argument took hold amongst some of the members. The second aspect, that the Hilbert school had not mentioned, requires an investigation of its own.

It is not our intention, however, to dwell on the negative results and discussions, rather it should be to emphasize the positive aspects. It is therefore appropriate to raise the following question: What positive effect did Tarski's lectures and discussions bring about? Was Gödel's work really so fundamental to Carnap's program in metalogic? In the secondary literature, opinions are divided. Alberto Coffa and others believe that Gödel's results, published in 1931, were the most important influence on Carnap.Foot note 1_5 Most academics do not agree about Carnap's critical position towards Gödel. In contrast to other interpretations, some authors believe that Tarski's Vienna speeches in February 1930 gave great impetus to Carnap's syntactical conception of language. This impetus is necessary for the so called «Erläuterungssprache» (i.e. explanatory language, language of elucidation or metalanguage) to be symbolized exactly.Foot note 1_6

It would be very simple in this view to indicate that the «Erläuterungssprache», is the language that we need in order to talk about the object language. Tarski spoke out very clearly against this simplification when he wrote: «the simple statement that sentences about sentences are legitimate, appears to be completely unfounded.»Foot note 1_7 In discussions a dichotomy between the «Erläuterungssprache» and metalanguage is indicated and therefore an important part of the discussion is blurred. To adopt a completely different approach, I would like to analyze the discussion and in so doing, explain that the differences in the `Erläuterungssprache' can be shown and that differences within the research areas between the Tarskian and the Carnapian programs can be highlighted. The question I would like to examine is this: what method did Tarski use to conceptualize his «Erläuterungssprache». Or to put it another way, how did the need for an «Erläuterungssprache» originate?

2. From the formal system to `Erläuterungsprache'

Yet can we ask ourselves what they understood then by the so-called «Erläuterungssprache» and how they reached that definition? In order to answer the questions in detail, we can see that it is necessary to look for sources and this is where we encounter difficulties. In the «Erläuterungssprache» no reference is made to meaning. If one looks at the meaning of a language, one can analyze the language mathematically. The resulting system contains both the description or definition as well as the study of the qualities of formal systems. In an abstract way, Tarski both describes and defines the deductive systemsFoot note 1_8 in a narrower and more general way than usual.Foot note 1_9 On the one hand the definition is narrower than normal as it touches on the definition of logical consequence which is defined in classical logic. On the other hand, it is more general as it refers to naive viewpoints of set theory. With regard to the second definition, a system is a set of propositions. For Tarski it was important to differentiate between the invariable deductive systems and the consequence relation. Tarski referred to the closure of all consequences.

The naive view of the finiteness of a set goes back to the well-known concept of the natural number. The starting point of a theory of finite sets forms a definition of the concept of the finite set. Depending on the choice of these starting definitions, the theories will develop differently and individually. Tarski's definition has the advantage that on the decision about the finiteness of a set, only this itself and its subsets, but also further sets of a general character do not need to be referred to.

In order to study the question of logical consequence in the correct structure, it is necessary for us to briefly look at the definition of formal systems. A formal system is a hierarchical grouping of sets of symbols or complete formulae, out of which other formulae can be generated, which can be accepted as valid. So that a complete system of interests can be created, a kind of lock must be made available, which can be viewed from two different and independent sides. On the one hand, not every complete formula of the system can be proved; on the other hand, though, it is impossible to constantly add new complete unprovable formulas to be produced, without every complete formula being provable. The precise definition of both qualities depends on the deduction rules which fixes the concept of proof.

By propositional statement or meaningful sentence we understand «... certain inscriptions of a well-defined form».Foot note 1_10 Somewhat later he published his thoughts which he had already written in 1931 and can be characterized in one sentence «... as a particular kind of expression, and thus as linguistic entities.»Foot note 1_11 From this a definition was drawn up to make clear the difference between «use» and «mention». This difference leads to the assumption that the statement does not denote concrete series of signs but the whole class of such series which are of like shape with the series given. After this, the quotation of names could be treated as individual names of expressions.

After his return from Poland, Carnap delivered three papers entitled `Metalogic'. In metalogic he saw the focus of his research as being the analysis of the quotation marks of a particular language.Foot note 1_12 The first priority of Metalogic is to answer (i) which signs appear and (ii) which row of symbols are formulae. A metalogical sentence serves as a description. He understood «description» as the presentation of «empirical data» which comes from «... Belegung eines Stellengebietes mit Qualitäten (oder Zustandsgrössen)» besteht.»Foot note 1_13 In Carnap's view it seemed that a formula is seen as an elementary disjunction from the metalogical definition of a concept of «elementary disjunction», which has a metalogical description.

Four years later in the `The Logical Syntax of Language' a sentence was defined as an expression which corresponds to a statement of natural language.Foot note 1_14 In this way a distinction was made between Language I and II. Language I only contains definite concepts, whereas Language II is much richer in terms of expression. Furthermore it also contains indefinite concepts, has classical mathematics and can also formulate sentences of Physics.Foot note 1_15 In the field of metalogic a language is a «...sort of calculus ..., a system of formation and transformation rules concerning what are called expressions, i.e. finite, ordered series of elements of any kind, namely, what are called symbols.»Foot note 1_16

Tarski and Carnap differ in their general views on what they understand as metamathematics or metalogic. For Tarski, metamathematics is a «General theory of mathematical sciences». He later showed very clearly that the strengths of the Warsaw School had been used in his metamathematical research. He himself said: «As an essential contribution of the Polish School to the development of metamathematics one can regard the fact that from the very beginning it admitted into metamathematical research all fruitful methods, whether finitary or not.»Foot note 1_17 It examined the «... mathematical theories in their entirety».Foot note 1_18 For Carnap, however, metalogic is a «... Theorie der Formen, die in einer Sprache auftreten, also die Darstellung der Syntax der Sprache.»Foot note 1_19 It is «... the theory of the forms of the expressions of a language».Foot note 1_20 In order to reveal the main similarities and differences, we see that we need to look much more closely at the concept of consequence. An exact definition of the concept can be given in metamathematics or metalogic where the research subject matter forms a solid formalized study. Tarski sees the concept of logical consequence as «a primitive concept» and characterized it by means of a number of axioms which we shall now look at in more detail.

3. The notion of «Logical Consequence»

Consequence always leads to new knowledge without the help of experience. If it is known for instance that Vienna lies to the east of Linz and Linz to the east of Salzburg then one can conclude from both these statements that Vienna is east of Salzburg. Every deductive conclusion is accordingly an example of this consequence. When it is known that judgements contain subject and predicate, and likewise that a proposition is also a judgement then it can be concluded from both premises that this proposition contains subject and predicate similarly. A logical consequence in the rules governing conclusions of relations and connections between the genus and the type is displayed. As that which makes a genus unique also makes its species unique, hence one must copy the model in the operative relations between genus and concepts of species.

Earlier Tarski had drawn attention to the evidence of a given set underpinned by certain operative functions, from which the consequences of the set of evidence can be constructed.Foot note 1_21 Nevertheless Tarski did not follow this up until later. This prompted Carnap to pick up the theme in the course of his investigations into the difference between the syntactical elements of derivation and the semantic elements of deduction.Foot note 1_22 Viewed systematically the concept of «consequence» should be dealt with at the outset of metalogic. If the concept of consequence specific to a language is determined, the logical connections within the language are laid down.

Carnap distinguishes two types of deductive processes, namely derivation (Ableitung) and consequence (Folgerung).Foot note 1_23 In the syntactic derivation d-terms are to be found: On the one hand, derivable, demonstrable, refutable, resoluble and irresoluble. On the other hand, there are the semantic consequences belonging to the family of the c-terms: consequence, analytic, contradictory, L-determinate and synthetic.Foot note 1_24 In Language I is used the concept of «consequence», to define notions of «analytical» and «contradictory». Every sentence and every class is either analytical or contradictory.Foot note 1_25 By analytical he understands a sentence involving a consequence of the null class of sentence and accordingly the consequence of every sentence. Every demonstrable sentence is analytical. The concept «analytical» refers to what is demonstrated to be logically valid or true for logical reasons. A sentence is regarded as contradictory when every sentence is the consequence of the null set. This also applies to a sentence class, when every sentence is the consequence of a negative sentence class. Every refutable sentence is contradictory. But the sentence class is only contradictory when at least one sentence belonging to it is contradictory.

In contrast the reverse process is suggested in a simplified form for the Language II. Furthermore the concepts analytical and contradictory are defined for sentences and sentence classes and by extension for the concept of consequence. Carnap's concept of an analytical sentence is dependent on every given class having an analytical quality or on the syntactic quality of expression, the analytical sentence which can be demonstrated in the following way from this sentence: every part of a sentence is constructed in the case of the sentence part belonging to the class through the 0=0Foot note 1_26 and otherwise through the negation of the 0=0.Foot note 1_27 Tarski referred to Carnap's first attempt also formulating a precise definition of the concept of consequence. He refuted Carnap's attempt, because his suggestions depended on special qualities of the formal language. According to Tarski Carnap's position runs as follows:

(FC) «The sentence X follows logically from the sentences of the class K, if and only if, the class consisting of all the sentences of K and of the negations of X is contradictory.»Foot note 1_28

Tarski attacked Carnap for shifting attention away from the concept of logical consequence towards the concept of the contradictory. The definition was complicated and highly specific. As an alternative, Tarski attempted to maintain an essential condition, namely statement X from a class of statements K following which he described thus: (FT) «If, in the sentences of the class K and in the sentence X, the constants - apart from purely logical constants - are replaced by any other constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by `K'', and the sentence obtained from X by `X'', then the sentence X' must be true provided only that all sentences of the class K' are true.»Foot note 1_29

Given that the condition (FT) is sufficient, then the appropriate definition of the concept of consequence would be positively decided. Certainly a proviso would have to be attached to the definition, namely the use of the semantic concept «true».Foot note 1_30 The use of the semantic concept of the predicate «true» is the essential difference to Carnap's definition of the concept of consequence which is founded on the concept of contradiction.

4. The peculiarities of analytic arguments

Why did Tarski refute Carnap's proposition about logical consequence? Tarski refuted Carnap because the latter stated in the supposition that certain objective problems can be reduced to linguistic problems. This reduction led to a false interpretation of the a priori point of view and of the role it plays in the analysis of reality. This opinion was taken of by Wittgenstein, who claimed that all a priori propositions, i.e. those that belong to logic and mathematics are of a tautological nature. Carnap however used other terminology and called all these propositions «analytic». Sentences which strictly adhere to the convention and which do not make material modes of speech in a statement are known as analytic.

All supposed a priori investigations are analytic, in the same way as conventionally accepted rules which govern the use of certain expressions in our language. Of course these interpretations of language are supported by Wittgenstein's interpretation which stated that all languages have the same logic. Therefore every analytical sentence in the context of Wittgenstein's interpretation of language, independent of any empirical proof, is correct. Thus the analytical concept can assume the role which in traditional philosophy is known as «a priori». In any case, the concept of analytic seemed to be a satisfactory representation of logic and mathematics. The problem arose when Wittgenstein's interpretation of language began to waver due to the development of an `Erläuterungssprache' on the metalevel.

Tarski moved away from the terminology of the earlier Vienna Circle because it had accepted an association which could have led to a misinterpretation. Both Wittgenstein and Carnap held that a priori propositions say nothing about reality, as they are simply instruments which make the recognition of reality possible. If necessary a scientific interpretation of the world can be given without having to refer to a priori elements. Tarski started with another assumption as he was very conscious that even since the «Grundlagenstreit», alternative logical systems were seen as evolving separately from each other. Furthermore they were thought to possess the quality that one can be translated into the other. Even today we cannot decide whether the relations between facts are best demonstrated through classical logic or through polyvalent logic's.

The great debate between the members of the Warsaw and the Vienna Circles was therefore the question of what effect Wittgenstein's conception of language would have. The Grundlagenstreit about logic had led to a new scenario in philosophy because the expressions of non-equivalent languages (such as those of logicism, of intuitionism and of axiomaticism) cannot be used interchangeably. As a consequence the supposition of a language- neutral concept of analytic was dismissed. Analeptics became (as Tarski had always implied) a language related concept.

Neither Carnap nor the Vienna Circle were prepared for this situation. It should be noted that in 1931 Carnap had recognized the relevance of the Erläuterungssprache for a formal system, but he did not perceive the philosophical consequences that this entailed. One consequence that we have looked at is the revelation of the unity of the language of sciences. Carnap also did not recognize all of the implications of this in 1935.Foot note 1_31 Metamathematical investigations, i.e., according to the Warsaw school, can then be carried out if the concept of statement and consequence is precise. Later in 1930 a program was devised which as its starting point had a set statement and was analyzed in relation to the lack of contradiction and axiomatic. The level of completion would then be tested.Foot note 1_32 In this program the work was carried out exactly as laid down in Hilbert's Basics of Arithmetic's: the so-called «geometric change»Foot note 1_33 was completed. Tarski began to interest himself in the supposition of the concepts «statement» and «consequence» and brought together these elements using certain axioms. This brought about the necessary consequence of showing the weakness of contradiction and axiomatic. Finally he tried to reach a more complete analysis of axioms. In this way Tarski came to accept the elementary-arithmetical character of metalogic.

Jesús Padilla Gálvez

Universidad de Castilla - La Mancha

Facultad de Ciencias Jurídicas y Sociales

E-45071 Toledo

<jesus.padilla at>

[Foot Note 1_1]

A. Tarski (1992): «Alfred Tarski: Drei Briefe an Otto Neurath» (ed. by Rudolf Haller. Translated into English by Jan Tarski), Grazer Philosophische Studien, 43, 1-31. S. 23.

[Foot Note 1_2]

According to the autonymous ways of speaking «... should constantly distinguish the sign from the designated object, especially in the cases, where the designated object is again a sign or, more generally, a figure of speech.» Tarski (1992), 26.

[Foot Note 1_3]

Tarski (1992), 4.

[Foot Note 1_4]

Carnap's letter to Neurath 23 December 1933. Hilman-Library, RC 29-03-06 A.

[Foot Note 1_5]

A. Coffa (1991): The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge.

[Foot Note 1_6]

R. Carnap's `Tagebuch'. See: Tarski (1992), 5.

[Foot Note 1_7]

Tarski (1992), 26.

[Foot Note 1_8]

A. Tarski (1935) «Der Wahrheitsbegriff in den formalisierten Sprachen», Studia Philosophica, 1, pp. 261-405/ 503-526; A.Tarski (1935-36) «Grundzüge des Systemenkalkül», Erster Teil, Fundamenta Mathematicae, 25 (1935), 503- 526; Zweiter Teil, Fundamenta Mathematicae, 26 (1936), 283-301; and A. Tarski (1936) «Über den Begriff der logischen Folgerung», Actes du Congrès International de Philosophie Scientifique, Paris 1935, Vol. VII, ASI, 394, Paris, 1-11.

[Foot Note 1_9]

Tarski (1992), 27.

[Foot Note 1_10]

A. Tarski (1930) «Über einige fundamentale Begriffe der Metamathematik», C. R. des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. III, 23, 22-29, see p. 23.

[Foot Note 1_11]

Tarski (1935), 269; note 5.

[Foot Note 1_12]

R. Carnap (1995) Metalógica-Metalogik [1931], Mathesis, 11, 137-192; see p. 139.

[Foot Note 1_13]

Carnap (1995) [1931], 140.

[Foot Note 1_14]

R. Carnap (1934) Logische Syntax der Sprache, Wien-New York, Springer Verlag; p. 13.; R. Carnap (1937) The Logical Syntax of Language, London, Routledge & Kegan Paul Ltd; p. 14.

[Foot Note 1_15]

Carnap (1934), 78/ (1937), 89.

[Foot Note 1_16]

Carnap (1934), 120. (1937), 167f.

[Foot Note 1_17]

A. Tarski (1986) Collected Papers (Eds. St. R. Givant, R. N. McKenzie). Vol. 4. Basel, Birkhäuser. p. 713.

[Foot Note 1_18]

Tarski (1995) [1939], 159.

[Foot Note 1_19]

Carnap (1995) [1931], 139.

[Foot Note 1_20]

R. Carnap (1963): «Intellectual Autobiography». In: P.A. Schilpp (Ed.), The Philosophy of Rudolf Carnap. La Salle, Ill. Open Court, 1-84; see: p. 54.

[Foot Note 1_21]

Tarski (1930), 97ff. and Tarski (1935-36).

[Foot Note 1_22]

Carnap (1934), 88ff.; 124ff.; 128. / (1937), 98ff.; 170ff.; 175.

[Foot Note 1_23]

Carnap (1934), 36 / (1937), 41f.

[Foot Note 1_24]

Carnap (1934), 88 / (1937), 101.

[Foot Note 1_25]

Carnap (1934), 34ff. / (1937), 37ff.

[Foot Note 1_26]

Carnap (1934), 75 / (1937), 84.

[Foot Note 1_27]

Carnap (1934), 88 / (1937), 102.

[Foot Note 1_28]

Tarski (1936), 6.

[Foot Note 1_29]

Tarski (1936), 7.

[Foot Note 1_30]

Tarski (1935), 261-405.

[Foot Note 1_31]

S. C. Kleene (1971): Introduction to Metamathematics. Amsterdam, North- Holland Publishing, p. 65.

[Foot Note 1_32]

Tarski (1930), 28 f. A. Heyting (1930) «Die formalen Regeln der intuitionistischen Logik», Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-Mathematische Klasse II, 42-56. K. Gödel (1930) «Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit», Anzeiger der Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Klasse, 67, 214-215.

[Foot Note 1_33]

P. Bernays (1923) «Erwiderung auf die Note von Herrn Aloys Müller: `Über Zahlen als Zeichen'». Mathematische Annalen, 90, 159-163.