# On the Extension of Beth's Semantics of Physical Theories

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<ul><li><p>On the Extension of Beth's Semantics of Physical Theories</p><p>Bas C. van Fraassen</p><p>Philosophy of Science, Vol. 37, No. 3. (Sep., 1970), pp. 325-339.</p><p>Stable URL:</p><p>http://links.jstor.org/sici?sici=0031-8248%28197009%2937%3A3%3C325%3AOTEOBS%3E2.0.CO%3B2-Y</p><p>Philosophy of Science is currently published by The University of Chicago Press.</p><p>Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.</p><p>Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/ucpress.html.</p><p>Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.</p><p>JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. Formore information regarding JSTOR, please contact support@jstor.org.</p><p>http://www.jstor.orgTue Jun 19 06:30:05 2007</p><p>http://links.jstor.org/sici?sici=0031-8248%28197009%2937%3A3%3C325%3AOTEOBS%3E2.0.CO%3B2-Yhttp://www.jstor.org/about/terms.htmlhttp://www.jstor.org/journals/ucpress.html</p></li><li><p>Philosophy of Science </p><p>September, 1970 </p><p>ON THE EXTENSION OF BETH'S SEMANTICS OF PHYSICAL THEORIES* </p><p>BAS C. VAN FRAASSEN1 University of Toronto </p><p>A basic aim of E. Beth's work in philosophy of science was to explore the use of formal semantic methods in the analysis of physical theories. We hope to show that a general framework for Beth's semantic analysis is provided by the theory of semi- interpreted languages, introduced in a previous paper. After developing Beth's analysis of nonrelativistic physical theories in a more general form, we turn to the notion of the 'logic' of a physical theory. Here we prove a result concerning the conditions under which semantic entailment in such a theory is finitary. We argue, finally, that Beth's approach provides a characterization of physical theory which is more faithful to current practice in foundational research in the sciences than the familiar picture of a partly interpreted axiomatic theory. </p><p>1. Beth's program. In his work in philosophy of science, E. W. Beth's aim was to apply the methods of formal semantics (as developed by Tarski et al.) to the analysis of theories in the natural sciences. When he wrote his book Natuurphilosophie 131, he was still of the common opinion that the semantics of a physical theory is constituted by a set of 'correspondence rules' linking theoretical with observation language. Just prior to publication, however, he added the following note to his chapter on the logical structure of quantum mechanics: </p><p>On rereading the material, which was written several years ago, I lean to the opinion that it is possible to give a semantic construction to the logic of quantum mechanics with the help of Hilbert space. .. .The interpretation [in terms of experimental results] is therefore not analogous to semantics. ([3], p. 133) </p><p>Beth then developed this point of view, which is related to those of von Neumann, Birkhoff, Destouches, and Weyl, in three articles (121 [4] [5]).2 As we shall suggest, it is a point of view which has close affinities to much contemporary foundational work in physics. </p><p>Beth's opinion, in which I concur, is that his semantic approach represents a much more deep-going analysis of the structure of physical theories than the </p><p>* Received May, 1969. This study was supported in part by NSF grant GS-1566. I also wish to express my debt to </p><p>Dr. F. Suppe, University of Illinois, for stimulating discussion. His doctoral thesis [27] develops a point of view closely related to Beth's. </p><p>The new statement by Suppes of his approach ([28] [29]) shows important agreement with this point of view notwithstanding earlier apparent differences. </p></li><li><p>326 BAS C . VAN FRAASSEN </p><p>axiomatic and syntactical analysis which depicts such a theory as a symbolic calculus interpreted (partially) by a set of correspondence rules. However, Beth did not present his new analysis in a general form, but only through specific examples. My hope is that a general framework for his approach is provided by the theory of what I have called "semi-interpreted languages" ([31] [32]). Sections 3-6 will provide a general exposition, except for the theorem in section 6, without much attention to what is and what is not Beth's. </p><p>Sections 2 and 7 are concerned with more general issues in philosophy of science, the former to introduce our point of view, and the latter to show its relation to current questions and problems. </p><p>2. The language of science. The language of science, as Carnap notes, is "mainly a natural language . . . with only a few explicitly made conventions for some special words or symbols" ([6], p. 241). Of course, there are many technical terms, and much use of mathematical language. Yet we do not have here a case of an artificially constructed symbolic language, but a naturally grown "variant of the pre-scientific language, caused by special professional needs" (loc. cit.). And the technical terms are most often old nontechnical terms given a new role. Thus, at some point the part of our language which concerns wave motion in liquids was taken over, almost bodily, to provide a new, technical way of talking about sound. There is, of course, a reason why this part of the language (rather than, say, the well-developed, sophisticated way in which horse lovers talk about the modes of motion of horses) was adapted for this new role. This is a subject which has been much discussed in recent years, for it concerns the use of analogies and models in theory constr~ction.~ </p><p>There are currently two general approaches to the formal study of language. One is syntactic and axiomatic, the other semantic in orientation. Within the first approach, a language or language game has as rational reconstruction a syntactic system plus an axiomatized deductive theory formulated within that syntax. In the second, the rational reconstruction consists in a syntactic system plus a family of interpretations of that syntax. In either case the construction may aptly be called an 'uninterpreted language': in neither case are the nonlogical terms provided with a specific interpretation, though in both cases the possible such interpretations are delimited to some extent (any interpretation must satisfy those axioms, any interpretation must fall within the described class). There are natural interrelations between the two approaches: an axiomatic theory may be characterized by the class of interpretations which satisfy it, and an interpretation may be characterized by the set of sentences which it satisfies; though in neither case is the characteriza- tion unique. These interrelations, and the interesting borderline techniques provided by Carnap's method of state-descriptions and Hintikka's method of model sets, would make implausible any claim of philosophical superiority for either approach. But the questions asked and methods used are different, and with respect to fruitful- ness and insight they may not be on a par in specific contexts or for special purposes. So we may reasonably hope to explore the semantic approach to the language of </p><p>Cf., e.g. [26], [27], 1291, Ch. 2. </p></li><li><p>science, and we shall begin with a brief account of our general position (developed in [31]). </p><p>Our view, to state it succinctly, is that in natural and scientific language, there are meaning relations among the terms which are not merely relations of extension. When a particular part of natural language is adapted for a technical role in the language of science, it is because its meaning structure is especially suitable for this role. And this meaning structure has a representation in terms of a model (always a mathematical structure, and most usually some mathematical pace).^ This language game then has a natural formal reconstruction as an artificial Ianguage the semantics of which is given with reference to this mathematical structure (called a "semi-interpreted language" in 1311; see section 3 below). </p><p>Before entering into the details of this kind of reconstruction, it may be helpful to discuss what is valid in such a language. First of all, of course, the admissible interpretations are such that those statements which are true in virtue of logic are true-if that familiar notion is applicable, i.e. if the language has among its expres- sions some which are meant to express the usual logical operations. Secondly, the meaning relations referred to above are such that certain logically contingent statements will always be true, in virtue of the meanings of the terms which occur in them. In other words, the mathematical structure with reference to which the language is partly interpreted plays a role in determining validity, and we may say in such a case that a statement is analytic or holds a priori in a broad sense. </p><p>There are many obvious and simple examples of this. In the case of simple discourse about color hues, the mathematical structure in question is the color spectrum, a segment of the real line. In the case of temperature it is the temperature scale being used, also a segment of the real number continuum. Thus </p><p>(1) Whatever is scarlet, is red. </p><p>holds a priori because the region of the colour spectrum assigned to the predicate "scarlet" is contained in the region of the spectrum assigned to "red," and </p><p>(2) Nothing is warmer than itself. </p><p>holds a priori because "is warmer than" is represented by the relation < (which is irreflexive) on the temperature scale. </p><p>That the language of science has such an inherent meaning structure-which may change in its historical development, however-has long been argued by Wilfrid Sellars ([24], [25], Chs. 4, 10, 1 I think that providing a formal representation such as we have attempted (intuitively here and more rigorously in [31]) helps to spell out this conception somewhat further. But on the other hand, I do not think this would be enough to make the view philosophically cogent; in addition, we must </p><p>The word "model" has many uses; the present sense is that found in discussions of the role of models in scientific theory, and differs from the sense in which it is used in formal semantics (and hence, below). </p><p>Hutten's point of view seems also in basic agreement with our own; compare "A model is a possible semantic interpretation; it is a picture of a situation which shows the semantic rules but does not state them explicitly. . . ." ([13], p. 120). </p></li><li><p>328 BAS C. VAN FRAASSEN </p><p>leave the abstract level of syntax and semantics, and provide a pragmatic counter- part to truth ex vi ternzinorum. In [31], [32] I have argued that the work of Sellars and Maxwell provides us with such a pragmatic retrenchment. </p><p>This general view concerning the structure of the language of science might perhaps be accompanied by quite different formal representations. Yet I think that the limitations of the axiomatic method are such that the semantic approach is the correct general approach here. For example, Carnap does not deny that principles like (I) and (2) hold; however, he feels that they can be made explicit in a set of "meaning postulates" to be laid down besides the axioms proper of the physical theory, from which they can be sharply distingui~hed.~ I must admit that I have not found myself equally capable of drawing such a sharp distinction between "meaning postulates" and "empirical postulates," beyond the rough and ready criterion that the meaning postulates are those not made explicit by the physicist. The divergence in approach may, however, be too fundamental to be accessible to simple direct arguments, or may not represent a disagreement but merely a difference in per- spective. We shall attempt to show the feasibility of our approach in a more detailed exploration before arguing its advantages. </p><p>3. Physical systems and physical theories. Like Beth we shall here address ourselves to the formal structure of notzrelativistic theories in physics, leaving the extension to the relativistic case for later. A physical theory then typically uses a mathematical model to represent the behavior of a certain kind of physical system. A physical system is conceived of as capable of a certain set of states, and these states are represented by elements of a certain mathematical space, the state-space. Specific examples are the use of Euclidean 2n-space as phase-space in classical mechanics and Hilbert space in quantum mecl~anics.~ To give the simplest example, a classical particle has, at each instant, a certain position q = (q,, q,, q,) and momentum p = (p,, pY, p,), SO its state-space can be taken to be Euclidean 6-space, whose points are the 6-tuples of real numbers (q,, q,, q,, p,, p,, p,). </p><p>Besides the state-space, the theory uses a certain set of measurable physical magizitudes to characterize the physical system. This yields the set of elementary statements about the system (of the theory) :each elementary statement U formulates a proposition to the effect that a certain such physical magnitude m has a certain value r at a certain time t. (Thus we write U = U(m, r, t), or U = U(m, r) if we abstract from variation with time, or U = U(t) if we wish to emphasize dependence on time.) </p><p>Whether or not U is true depends on the state of the system: in some states m has the value r and in some it does not. This relation betvieen states and the values of physical magnitudes may also be expressed as a relation between the state-space and the elementary statements. For each elementary statement U there is a region h(U) of the state-space H such that U is true if and only if the system's actual state is represented by an element of h(U). (We also say that these elements satisfy </p><p>See, for example, [6] ,Appendix B. </p><p>Other terms for "state-space" are "phase-space" and "system space" (Weyl). </p></li><li><p>ON THE EXTENSION OF BETH'S SEMANTICS OF PHYSICAL THEORIES 329 </p><p>U; thus in the case of the classical particle, (q,, q,, q,, p,, p,, p,) satisfies "The X-component of momentum is r" if p, = r). </p><p>The mapping h (the satisfaction function) is the third characteristic feature of the theory. It connects the state-space with the elementary statements, and hence, the mathematical model provided by the theory with empirical measurement results. This follows because, as we have said, the elementary statements concern measur- able physical magnitudes. We do not have in mind by this an operationalist identification of meaning. The exact relation between U(m, r, t) and the outcome of an actual experiment is the subject of an auxiliary theory of measurement, of which the notion of "correspondence rule" gives only t...</p></li></ul>

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