Einführung in die Metaethik: Reclams Universal-Bibliothek (German Edition)

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Braunschweig: Vieweg. Fifth Edition, Schlick, eds. Helmholtz: Schriften zur Erkenntnistheorie. Berlin: Springer. Translated as Hermann von Helmholtz: Epistemological Writings. Dordrecht: Reidel, Quine, W. Reprinted in From a Logical Point of View. Cambridge, Mass.

T Press. Davidson and J. Hintikka, eds. Words and Objections: Essays on the Work of W. Dordrecht: Reidel, pp. Buck and R. Cohen, eds. Reprinted in R. Creath, ed. La Salle: Open Court. The Philosophy of W. Buickerood, Eighteenth-Century Thought 1: Schuldenfrei, R. Wittgenstein, L. London: Routledge. The central notion to be investigated in this respect is that of a logical interpretation or a logical model. Carnap mentions the term several times in his writings from the time without specifying its exact meaning. His understanding of the logical interpretability of an axiom system will be examined in comparison with similar accounts in the work of Alfred Tarski and Friedrich Bachmann.

The subsequent investigation is guided by the following interpretive questions: 1. What is the specific role of the logical interpretability of axiom systems in his theory? Following this, the second question concerning his specific ties with logicism will be discussed in Section 3.

The details of this assumption will be discussed in Section 5 following the comparison of two similar accounts by Tarski Section 4. Finally, in Section 6, the third question will be addressed. Axioms and consequences of the theory make substantive claims about the logical relations of these interpreted terms. In the latter, the primitives are implicitly defined by the theory and do not have a fixed interpretation.

This difference from contentual axiomatics is specified in the following passage in Untersuchungen: One can also understand the primitive concepts as unspecified elements and as relations of an unspecified domain where it is only stipulated that they relate to each other as specified in the axioms. In case that elements and relations are found in different domains that satisfy these formal specifications, then the axiom system can be applied to each one of these domains; in each of these domains also the consequences of the axiom system hold under the respective interpretation.

Moreover, it also proposes a new understanding of the semantics of axiomatic theories. Without going into further historical detail, one can pin down at least three related semantic innovations in Grundlagen: i the schematic conception of the primitive mathematical terms i. Formally, this is done by the symbolization of the primitive symbols as free higher-order class and relation variables of STT. In consequence, axioms are expressed by propositional functions that range over the primitive terms. The axiom system is also presented as a propositional function expressing the conjunction of its axioms.

Systems with more than one primitive term of the form f P, Q, R, For this, it is essential It should be noted that this convention of symbolizing axiomatic theories was common practice at that time. A similar treatment of axiomatic primitives as variables is present in the works of Russell, the American postulate theorists, and Tarski. Carnap gives a detailed discussion of the notion in Section 2.

In Carnap , the following shortened definition is given: If f R is satisfied by the constant R1 , where R1 is an abbreviation of a system of relations P1 , Q1 ,. A model f R1 is a system of concepts of the basic system, generally a system of numbers number classes, relations and so forth. Moreover, Carnap is using in Untersuchungen a pure logical language with an empty signature, whereas, in the modern account, models are usually constructed for theories expressed in languages with a non-empty signature.

This has to be refined. More than his logicist predecessors, particularly Frege, Carnap was aware of key methodological innovations of modern axiomatics. Moreover, neither in Untersuchungen nor in Abriss was the relation to logicism addressed in any way. However, the situation is more intricate.

As is well known, the logicist reduction of mathematics to logic depends crucially on the formal notion of interpretability. A mathematical theory T1 expressed in formal language L1 is interpretable in a logical theory T2 expressed in the logical language L2 in the following way: 1 the mathematical terms of T1 are defined in L2 , i. Given conditions 1 and 2 , all axioms and theorems of T1 can be expressed as sentences of L2 that are deducible from the axioms of T2.

The first step of this was outlined above: primitive terms are expressed as variables, the axioms and sentences of the theory as propositional functions of STT. Thus, a full 8 9 For a more detailed definition see e. Burgess , Compare in particular Reck , on this point. Carnap does not discuss this additional condition in Untersuchungen or in Abriss. For every mathematical sign one or more interpretations are found, and in fact purely logical interpretations.

If the axiom system is consistent, then upon replacing each mathematical sign by its logical interpretation or one of its various interpretations , every mathematical formula becomes a tautology. If the axiom system is complete Hahn et al. Point 1 states the assumption that the primitive terms of an axiom system can be translated in logical, i.

Point 2 states that the resulting sentences are truths of the background logic. This logical interpretability of mathematical theories indicates a novel conception of logicism that is closely related to but not identical with the classical logical reductionism of Frege and Russell. There, the common method was to present logical and explicit definitions the mathematical primitives of a theory first and then, in a second step, to show that each axiom of the theory in question can be deduced from the logical axioms of the background logic.

Carnap starts with the formal specification of a particular axiom system and then, in a subsequent step, considers the logical construction of several possible models for this system. In commenting on a draft of the manuscript in a letter dated from January 1, , he remarks that: The existence of instances of application [is here] equivalent to the existence of models. This claim is hypothetical; I would like to name it the hypothesis of logicism: that no structure can be described axiomatically if it is not already describable in logic. In the first and weaker sense, the assumption simply requires that for every consistent mathematical axiom system, at least one logical model can be constructed.

In the second, stronger sense, it requires that all models can be translated into logical models. Im Endeffekt ist das ja das gleiche, aber dem Sinn nach ist Hilberts Abbildg. We will return to the limiting effects of the definability constraints on models in the last Section. In this and the next Section, the question will be addressed which of the two versions Carnap actually assumed in Untersuchungen. Here, in contrast to Untersuchungen, the additional assumption concerning the logical definability of models is explicitly mentioned.

Tarski and Lindenbaum paraphrase it in this way for their example of a theory of Euclidian geometry: in fact hold for theories with a logical interpretation. They are of particular importance for axiomatics. For this reason, an additional condition concerning the logical definability of the primitive terms of the axiomatic theory is mentioned. A second way to characterize the logical interpretability of a theory is also outlined in the article. It is based on an insightful reformulation of conditions i and ii of the above passage The translation involved here presupposes the construction of a particular model that consists solely of logical constants.

Unlike Carnap, however, Bachmann is explicit in his aim to provide a logicist presentation of axiomatic theories. Moreover, there exists an extensive discussion of the treatment of extremal axioms in Carnap , The close ties to classical logicism are evident in the specifics of these two tasks. Models are also conceived in essentially the same way as in Carnap Unlike Carnap, Bachmann is explicit about the type of constants admissible for this task. Nonetheless, it is obvious that they were considered to be semantic in nature.

It is thus understood in purely syntactic terms, i. Here again, the definability condition seems to be understood in the weak form. Nevertheless, there exists a crucial difference between Tarski and Bachmann that concerns the general motivation of their formal reconstructions of axiomatics. In this understanding, the stronger definability assumption concerning models turns out to be relevant. Given his logicist motivation, any model of elementary arithmetic that can be subject to this second task has to be logical in the sense specified above.

The stronger definability assumption is a natural consequence of this. A sentence with nonlogical constants is only deducible from logic if it is transformed into a universally valid sentential form in case the constants are substituted by variables. Consequently, Bachmann argues, any model constructed for the theory has to consist exclusively of logical constants in order to allow the proof of the validity the interpreted system.

Carnap stresses in the introduction of the manuscript that the principle objective is to provide formal explications, i. As a consequence of this, different kinds of model construction are possible. The logical interpretability of an axiom system outlined above is one possible case here. Consider the following passage in Untersuchungen where he further specifies the notion Here again, the underlying language is a pure and higher-order logic.

This substitutional conception of models is still present in the first English edition of Logical Syntax of Language from A definition of models and model domains is given here that is comparable to the ones in and in In the first method, the domain of the interpretations of a certain primitive symbol is the domain of the substitution-values of the variable. If, as is usual, it is a case of primitive variable within a system of types, then the same relations of types must hold between the symbols of the model as hold between the corresponding primitive variables. The expressibility condition for models effectively presupposes that there is a range of logical constants in STT comprehensive enough to express all classes, relations, etc.

Typically, constant expressions can be added in two ways to a formal language. They can be stipulated at the outset with the specification of the primitive symbols of the language. They can also be introduced via definitions based on the given vocabulary. For example, if a certain formula of the language specifies a certain class of or relation on the intended domain of individuals, a new constant can be added to the language to name this set or relation.

The main difference to a modern conception lies in the fact that the constant expressions are all logical here. Thus, the language does not contain non-logical expressions i. Two points should be noted here. First, the notion of logical concepts is obviously conceived here in a much more general sense than in the modern account. The logic includes not only the proper logical terminology of 1 but in addition the genuinely mathematical terms from arithmetic, set theory, and most importantly, relation theory. The generalized account of logical concepts e. Recall that, for Bachmann, the relevant logical constants used in a model of axiomatic arithmetic are particularly those mathematical terms that can be given a logicist definition in pure logic Bachmann , This point is explicitly stated in later work by Carnap, e.

A defined sign is descriptive if its definiens contains a descriptive sign, otherwise logical. These constants are i either conventionally fixed to be part of the basic vocabulary or ii introduced via explicative definitions in STT. Given this, to what extent does his theory capture the semantic innovations of modern axiomatics? Such results agree well with This is usually expressed in a separate metatheory in modern practice, e.

Given this, the generalization over models is then symbolized in terms of the higher-order quantification over relations of the type-level of M1. This decidedly metatheoretic use of higher-order quantification can be found at several places in his work on axiomatics from the time. According to him, an axiom system not only provides implicit definitions of its primitive terms. It also explicitly defines a higher-order concept, i.

Each of these tuples presents a model of the theory. Again, they are not expressed in a seperate metalanguage, however, but in a single, universal background logic. Note that Carnap understood STT as a fully interpreted system. It follows from this that each of the metatheoretic sentences above expresses a truth of STT, i.

More precisely, the generalization over models of a theory expressed in the notions above is strongly is directly related to the range of the higher-order quantifiers used in 1 to 4. Nevertheless, there is strong reason to believe that he conceived the higher-order domains of his background language in a nonstandard sense. Thus, quantification over models is effectively restricted to those higher-order entities of the type-theoretic universe whose elements are definable in the system.

This nonstandard treatment of quantifiers is also directly implied by his substitutional conception of models. Generally, it can safely be said that prior to Logical Syntax of Language Carnap , he upheld a substitutional conception of higher-order as well as of the first-order quantification. Here, consequence is characterized purely syntactically in terms of the provability, i. It was recently shown that several early contributions to modern axiomatics — e. Moreover, it was argued that the informal logic used for the discussion of these notions was understood to be fully interpreted.

One could add that the set-theoretical resources necessary for expressing this are best codified in a higher-order system or alternatively in ZFC with a standard interpretation. This holds in particular for his notion of formal models. The concept is not mentioned in Carnap More surprisingly, also his later work on semantics, most notably the three-volume book project Studies in Semantics, remains silent on the notion. For instance, it is well known that the proofs of categoricity for geometry, analysis, and Peano arithmetic are usually given in a set-theoretical metatheory that allows the generalization over all models of the respective theory.

See Shapiro , However, in light of the present paper, this shows to be more of a retrospective idealization than an adequate account of the theoretical evolution of this notion in his work. This is work for another day. Awodey and A. Awodey and E. Badesa, The Birth of Model Theory. Princeton: Princeton University Press Hintikka ed. Dordrecht and Boston: Reidel pp. Bonk and J. Carnap : pp. Burgess, Fixing Frege. Princeton: PUP Carnap, Abriss der Logistik.

Wien: Springer Benacerraf and H. Putnam eds. Cambridge: CUP pp. Carnap, Logische Syntax der Sprache. London: Open Court Carnap and F. Chicago: University of Chicago Press Carnap, Introduction of Semantics. Carnap, Meaning and Necessity. Carnap, Untersuchungen zur allgemeinen Axiomatik. Edited by T. Mosterin, Darmstadt: Wissenschaftliche Buchgesellschaft Cambridge: CUP Hahn, R.

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Carnap, K. Mancosu ed. Oxford: OUP , pp. Hilbert, Die Grundlagen der Geometrie. Teubner: Leipzig Hilbert and P. Bernays, Grundlagen der Mathematik, Vol. Berlin: Springer Gray eds. Oxford: OUP pp. Awodey and C. Klein eds. Chicago and La Salle: Open Court pp. Friedman and R. Creath eds. Schilpp ed. The Library of Living Philosophers. LaSalle: Open Court, Oxford: OUP Tarski, Logic, Semantics, Metamathematics. Second edition Indianapolis: Hackett Tarski and A. Menger ed. Tarski pp. In a certain sense, Carnap must be regarded as the one who rendered the whole issue irrelevant.

It will be shown that Carnap agreed with Reichenbach that realism has essentially to do with language, but that he disagreed with him over the significance of probability in defending the scientific realist stance. More exactly speaking, the realism issue as such is, for Carnap, incapable of being solved by rational means. They will agree on the relevant criteria concerning the empirical reality of the mountain.

Furthermore, given that the mountain exists, they will come to the same result about its position, shape, height, etc. While the realist will claim that the mountain, which they have both found, not only has the ascertained geographical properties, but is, in addition, also real, the idealist will contest this claim, arguing that the mountain itself is not real, but only our perceptions and conscious process are real. However, this ontological divergence between the two geographers does not occur in the empirical domain.

Neither of the disputants, says Carnap, would suggest that his thesis be tested experimentally or by other empirical means. It must be admitted that, in practice, linguistic realism [sprachliche Realismus], which is very useful in the empirical sciences, is frequently extended to a metaphysical realism; but this is a transgression of the boundaries of science […]. There can be no objection against such a transgression, as long as it influences only the mental representations which accompany the 1 2 3 Rudolf Carnap, Pseudoproblems in Philosophy, translated by R.

London: Routlegde , p. Realism as a Problem of Language 39 scientific statements; this transgression is objectionable only if it influences the content of the statements of science. But what is the criterion for making that distinction? It can be said without much exaggeration that this way of putting the issue leaves open more questions than it resolves. London: Routledge , pp.

Cambridge: Cambridge University Press , p. Dordrecht: Reidel , p. Carnap was the initiator of this restriction to the empirical facet of realism, because it was he who, in his contribution, declared that the metaphysical problem of an external world was nothing but a pseudo-problem. A theory of knowledge must consequently begin with a theory of language. Paris: Hermann , p. An Analysis of the Foundation and the Structure of Knowledge. Chicago: The University of Chicago Press , p.

His indebtedness to Carnap is more than obvious. This form of consideration, which has been applied particularly by Carnap, seems to be a means appropriate to the problem in question, and we shall make use of it for an illustration of our results.

Reichenbach writes: I should say, nevertheless, that such a definition of philosophy is not in opposition to the view that philosophy is concerned with the analysis of the more general relations holding for the physical world. This second interpretation is valid because language is not arbitrary but constructed in correspondence to facts. There are, however, other features of language which have their origin in certain features of the world. Thus an analysis of language is at the same time an analysis of the structure of the world. In opposition especially to the view defended by Schlick, Reichenbach explains that, and why the verifiability criterion, 16 17 18 19 20 Ibid.

This, it seems to me, is a decisive argument for preferring probability meaning. We may begin with a rather small domain of basic elements and construct upon it statements concerning elements of another domain without being obliged to borrow their meaning from statements about the basic domain. Thus probability meaning leads to the realistic language of actual science; we start from the rather small domain of our own observations and construct the whole world upon it. But the crucial point is that Reichenbach, in fact, intends to argue against the Aufbau project. Experience and Prediction, p.

Cambridge: Cambridge University Press See further Realism as a Problem of Language 43 relation with the relation between a wall and the bricks of which the wall is built. It can be said that the very existence of the complex is dependent on the existence of the elements in such a way that the complex may cease to exist without the elements ceasing to exist.

If the elements themselves cease to exist if, for example, the bricks are destroyed , then the complex in this case the wall can no longer exist either. He therefore imagines a setup in which the whole of mankind is confined to a huge cubical room with translucent walls. Outside the cube there are birds, the shadows of which are projected on the ceiling of the cube. The birds themselves can neither be seen nor heard. Furthermore, there is a mirror which causes a second set of shadows of the birds on one of the vertical walls of the cube.

But this, Reichenbach maintains, would be rejected by the physicist. Reichenbach writes: The physicist […] would not accept this […] theory. It is this probability inference which furnishes a different weight for the projective complex and the reducible complex. And indeed, there can be little doubt 35 Ibid. Realism as a Problem of Language 45 that Reichenbach is completely justified in criticizing the Carnap of the Aufbau for neglecting the role of probabilistic reasoning in science.

See in this respect especially the short discussion between Reichenbach and Carnap in Erkenntnis 1, , pp. But it seems to me to be at least doubtful whether this reduction to perceptual reports and pure logic exhausts everything we mean to include in our assertions about reality. It is a puzzle to me just how logical neo-positivism proposes to include assertions of probability in its system, and I am under the impression that this is not possible without an essential violation of its basic principles.

Cohen eds. Thus, reality and probability are tightly connected for Reichenbach but not for the Carnap of the Aufbau. It seems to be clear but cannot be further investigated here that quantum mechanics played a crucial role in this connection. Vol XI. La 46 Matthias Neuber And he immediately added: Later, Reichenbach gave to the thesis of realism an interpretation in scientific terms, as asserting the possibility of induction and prediction; a similar interpretation was proposed by Feigl.

On the basis of these interpretations, the thesis is, of course, meaningful; in this version, it is a synthetic, empirical statement about a certain structural property of the world. Realism as a Problem of Language 47 entities like properties, classes, numbers, and so on.

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Speaking in more specific terms, Carnap distinguished between two kinds of questions concerning the existence of entities: first, questions of the existence of certain entities within the framework — these he termed internal questions; and second, questions concerning the existence of the system of entities as a whole — these he termed external questions. External questions, on the other hand, were a matter of practical decision.

Those who raise the question of the reality of the thing world itself have perhaps in mind not a theoretical question as their formulation seems to suggest, but rather a practical question, a matter of a practical decision concerning the structure of our language. We have to make the choice whether or not to accept and use the forms of expression in the framework in question. The cubical world story in particular seems to suggest that it makes perfectly good sense to suppose that the assumption of the existence of an external world is a highly probable hypothesis.

London: Routledge , p. Yet, on the other hand, to make assertions about atoms and their existence in general is, for Carnap, not an internal but an external question, that is, a question which can only be answered by a practical decision. These criteria are definitely not dependent on facts about the world and therefore definitely not defendable on probabilistic grounds.

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To be sure, in a given specific situation, the concrete behavior of a given sample of atoms can be probabilistically inferred. But in order to be able to draw this sort of inference at all, the atomistic framework as such must first be introduced, and this is possible only on the basis of a practical decision. Speaking more generally, the foregoing introduction of the realistic framework with its assumption of the existence of an external world including unobservable entities first of all enables probabilistic inferences within it.

The two statements differ as to their predictive content. Experience and Prediction, pp. Oxford: Oxford University Press , p. Realism as a Problem of Language 49 If we now proceed to regard the differences of the positivistic and the realistic languages, we pass from the descriptive to the critical task of epistemology; with this turn we consider meaning as a matter of free decision, and ask for the consequences to which each form of decision leads, and thus for the advantages and disadvantages which may be used to determine our choice if we ourselves want to make a decision.

Rather, the adoption of the realistic language form is required in order to make sense of the kind of probabilistic argument Reichenbach has in mind. The introduction of new basic and irreducible concepts as, for example, in electromagnetics during the last century may be reconstructed as an expansion of the empirical language. Only after our language has thus been enriched, can we significantly assign probabilities 54 55 56 57 Experience and Prediction, p. We shall call the group of decisions involved by one decision its entailed decisions. The man who speaks the egocentric language cannot express certain ideas which the man with the realistic language may formulate; the decision for the egocentric language, therefore, entails the renunciation of certain ideas, and may, consequently, become highly relevant.

It is rather the method of entailed decisions which we have to apply here; we can show that the decision for the egocentric language leads to a scientific system of a restricted character which does not correspond to the system constructed by the realistic language in its full extension. The step of expansion of language cannot itself be justified on the grounds of probability, except perhaps in the sophisticated pragmatic sense of the question: Will this expansion be methodologically fruitful?


On the one hand the realist framework cannot be justified inductively because the very definition of such a framework implies its being introduced by a practical decision. Or, if this appears too crude, conceiving of realism as a problem of language means, in any case, that science does not conform to reality in an ontologically unconditioned manner. In the case of Kant, realism is dependent on the facul59 Ibid. Only after the introduction of the realistic frame can we legitimately argue inductively either from the theory to the outcome of as yet unperformed experiments; or vice versa from the results of experiments to specific postulates of the theory.

But the presupposed introduction of the realistic frame, i. New Haven: Yale University Press The answer seems to be clear: The view that realism is a problem of language must be abandoned. But would that not bring us back to metaphysical realism? But what exactly is the problem? Surely it would be absurd to aim at a reinstallation of speculative metaphysics in the sense of Hegel and Schelling. Michael Esfeld, one of the contemporary promoters of this project, writes in connection to this: The new metaphysics of nature distinguishes itself from the older essays in speculative metaphysics by being close to science: metaphysical claims are based on scientific theories.

Consequently, the metaphysical claims about nature are as hypothetical as our scientific theories: there is no more certainty to be gained in metaphysics than there is in science. In other words, scientific knowledge claims are fallible and metaphysics, insofar as it draws on those claims, is as fallible as science. Aldershot: Ashgate , esp. Chicago: Open Court Not in United States? Choose your country's store to see books available for purchase. Die Ethik behandelt die Frage, was wir tun sollen. Die Meta-Ethik geht einen Schritt weiter: Was tun wir, wenn wir sagen, wir sollten etwas tun?

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Einführung in die Metaethik

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Rudolf Carnap and the Legacy of Logical Empiricism

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