Abstract
In this paper, a solution to the problem of theoretical terms is developed that is based on Carnap’s doctrine of indirect interpretation of theoretical terms. This doctrine will be given a semantic, model-theoretic explanation that is not given by Carnap himself as he remains content with a syntactic explanation. From that semantic explanation, rules for the truth-value assignment to postulates, i.e. sentences that determine the meaning of theoretical terms, are derived. The logical status of postulates will be clarified thereby in such a way that the problem of theoretical terms disappears.
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Notes
For the original exposition of the problem see Sneed (1979, pp. 31–40).
The problem of theoretical terms has received surprisingly little attention in the context of the debate on realism and antirealism. I may contemplate shortly on the question of whether it has some bearing on this debate. Antirealism may be defined as the view that the meaning of a linguistic expression is identified with our means of determining the extension of that expression. (This definition goes back to Dummett (1978, p. 146). That the focus of Dummett’s explanation is on sentence meaning but not on the meaning of linguistic expressions in general seems inessential to our discussion.) Let us assume that scientific terms such as “force" in classical mechanics do have sense, where the sense, or intensional meaning, of a linguistic expression is what determines its extension. In the context of the present discussion the question arises of whether we shall account for the sense of a theoretical term in an antirealist or realist fashion. The antirealist will clearly refer to the axioms of T as these are essential to our means of determining the extension of theoretical terms. By contrast, it is far from clear how a realist account of the sense of theoretical terms may look like. I am not aware of a satisfying answer to this question to be found in the literature. In particular, possible worlds semantics seem of limited help as these semantics take the extension of scientific terms as something that is already given. Rejecting the assumption that scientific terms do have sense seems not promising either because a purely extensional account of the meaning of scientific terms has its own shortcomings. For example, such an account would require a student of physics first to know the extension of the expression “force” in order to understand the meaning of that expression. By contrast, the teaching of physics and other natural sciences focuses rather on axioms and general explanations of how the extension of scientific terms can be determined when such terms are introduced.
Of course, these remarks are insufficient by far to rule out a realist approach to the meaning of theoretical terms. But they seem to indicate that the problem of theoretical terms is potentially apt to be used in a strong case for antirealism. For this reason we should not be surprised if the solutions proposed to this problem remain in an antirealist framework. Needless to say, there is a large number of philosophers of science who disagree totally with an antirealist approach to scientific theories. I am, however, not going to discuss metaphysical issues in the remainder of the paper.
For proof see Tuomela (1973, pp. 57–58).
Carnap’s dictum that the interpretation of theoretical terms necessarily remains open to further strengthening seems to imply that the interpretation of a theoretical term by postulates does not amount to a unique determination of the extension of that term. For a closer examination of this point see Andreas (2007, p. 157).
It has to be admitted that there are versions of the dual-level conception in which a direct interpretation of theoretical terms is assumed; this is the case with the ones of Hempel (1965) and Tuomela (1973). Carnap’s version is nevertheless, according to his own explanations, bound to an indirect interpretation of theoretical terms (See Carnap 1939, pp. 65–69; 1956, p. 46n).
If there is more than one interpretation that satisfies an axiomatic system, then there might be one single interpretation such that the axioms of that system were set up to account for the truths of that interpretation. This interpretation is referred to by the term “intended interpretation". For example, there are several interpretations satisfying the Peano axioms, but only the natural numbers are considered as the intended domain of interpretation for the language in which that axiomatic system is formulated. Likewise, it is sensible to speak of an intended intended interpretation of an observational language. Such an interpretation is such that a certain truth-value assignment to the sentences of the formalized language results from it; it can be given by expressions of a non-formalized meta-language.
The theorem just proved is also stated by Church (1956, p. 328). The proof is nevertheless left to the reader.
The notion of a canonical structures has been adopted from Barwise (1977, p. 31). Carnap himself requires that, for the observational language, every value of an individual variable is designated by an expression of L(V o ) (Carnap 1956, p. 41n). The restriction to canonical structures for the interpretation of L(V o ) is therefore justified. Irrespective of Carnap’s particular concept of an observational language it is reasonable to require for such a language that our linguistic means be sufficient to refer to every individual of the observational domain. For reasons of simplicity it is assumed in the present paper that every individual of the observational domain is designated by an individual constant. If there were individuals of the observational domain which are designated by a closed function expressions but not by an individual constant, the argumentation would remain valid, though it would have to be extended.
This version is to be preferred if one aspires to have a kind of truth-value semantics being equivalent to the model-theoretic one in which the domain of interpretations is restricted to canonical structures. It is adopted by Stegmüller in his (1984, p. 84) but rejected by Leblanc in his (1976, p. 17n). Leblanc, unlike Stegmüller, intends to have a truth-value semantics that is equivalent to the model-theoretic one without restriction to canonical structures. (Leblanc uses the term “Henkin structure" to refer to interpretations that are, in the present paper, called canonical structures. See Leblanc (1976, p. 20).) The reason for my adherence to Stegmüller’s version is that the interpretation of L(V o ) is restricted to canonical structures. This restriction has been explicitly mentioned as a premise of the theorem currently to be proved.
It may well happen that there is a function that takes empirical objects as arguments and has individuals of the theoretical domain as values. To give an example, we need such functions to express formally that a particular empirical object has a certain mass. The necessity to distinguish between theoretical terms with mixed and pure theoretical argument structure is pointed out by Ketland (2004, p. 290).
That the Ramsey sentence is equivalent to a proposition of the meta-language is not a completely new insight. Ketland has proved that TC R to a the claim that \({\mathfrak{A}}_o,\) the intended interpretation of L(V o ), can be expanded to an L(V o , V t ) structure satisfying the postulates. See Ketland (2004, p. 293). However, Ketland assumes a direct interpretation of the theoretical terms and thus makes no contribution to the problem under consideration in the present paper.
References
Andreas, H. (2007). Carnaps Wissenschaftslogik. Eine Untersuchung zur Zweistufenkonzeption. Paderborn: mentis.
Barwise, J. (Ed.) (1977). Handbook of mathematical logic. Amsterdam: North-Holland Publishing Company.
Carnap, R. (1934). Logische Syntax der Sprache. Wien: Springer-Verlag.
Carnap, R. (1937). The logical syntax of language. London: Routledge.
Carnap, R. (1939). Foundations of logic and mathematics. Chicago: University of Chicago Press.
Carnap, R. (1942). Introduction to semantics. Cambridge, Mass: Harvard University Press.
Carnap, R. (1956). The methodological character of theoretical concept. In H. Feigel & M. Scriven (Eds.), Minnesota studies in the Philosophy of Science I (pp. 38–76). Minneapolis: University of Minnesota Press.
Carnap, R. (1958). Beobachtungssprache und theoretische Sprache. Dialectica, 12, 236–248.
Carnap, R. (1975). Observational language and theoretical language. In J. Hintikka (Ed.), Rudolf Carnap. Logical Empiricist (pp. 75–85). Dordrecht: D. Reidel Publishing Company.
Church, A. (1956). Introduction to mathematical logic. Princeton: Princeton University Press.
Dummett, M. (1978). Realism. In Truth and other enigmas (Chapt. 10, pp. 145–165). Cambridge, Mass.: Harvard University Press.
Hempel, C. G. (1965). The theoretician’s dilemma. In: Aspects of scientific explanation. New York-London: Collier Macmillan Publishers.
Ketland, J. (2004). Empirical adequacy and ramsification. British Journal for Philosophy of Science, 55, 287–300.
Leblanc, H. (1976). Truth-value semantics. Amsterdam: North Holland Publishing Company.
Priest, G. (2001). An introduction to non-classical logic. Cambridge: Cambridge University Press.
Ramsey, F. P. (1950). Theories. In R. B. Braithwaite (Ed.), The foundations of mathematics and other logical essays (pp. 212–236). New York: Humanities Press.
Sneed, J. (1979). The logical structure of Mathematical Physics. Dordrecht: D. Reidel Publishing Company.
Stegmüller, W. (1970). Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie. Bd. 2 – Theorie und Erfahrung, Teilbd. 1. Berlin, Heidelberg, New York: Springer-Verlag.
Stegmüller, W. (1976). The structure and dynamics of theories. Berlin, Heidelberg, New York: Springer-Verlag.
Stegmüller, W. (1984). Strukturtypen der Logik. Berlin, Heidelberg, New York: Springer-Verlag.
Stegmüller, W. (1985). Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie. Bd. 2 – Theorie und Erfahrung, Teil D. Berlin, Heidelberg, New York: Springer-Verlag.
Tuomela, R. (1973). Theoretical concepts. Wien: Springer-Verlag.
van Fraassen, B. (1969). Presuppositions, supervaluations and free logic. In K. Lambert (Ed.), The logical way of doing things (pp. 67–92). New Haven: Yale University Press.
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Andreas, H. Another Solution to the Problem of Theoretical Terms. Erkenn 69, 315–333 (2008). https://doi.org/10.1007/s10670-008-9119-9
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DOI: https://doi.org/10.1007/s10670-008-9119-9