On the Extension of Beth's Semantics of Physical Theories Bas C. van Fraassen Philosophy of Science, Vol. 37, No. 3. (Sep., 1970), pp. 325-339. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28197009%2937%3A3%3C325%3AOTEOBS%3E2.0.CO%3B2-Y Philosophy of Science is currently published by The University of Chicago Press.
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Philosophy of Science
September, 1970 ON THE EXTENSION OF BETH'S SEMANTICS OF PHYSICAL THEORIES* BAS C . VAN FRAASSEN1 University of Toronto
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 semiinterpreted 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.
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: 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)
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. 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
* Received May,
1969. This study was supported in part by NSF grant GS-1566. I also wish to express my debt to Dr. F. Suppe, University of Illinois, for stimulating discussion. His doctoral thesis [27] develops a point of view closely related to Beth's. The new statement by Suppes of his approach ([28] [29]) shows important agreement with this point of view notwithstanding earlier apparent differences.
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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. 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.
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 prescientific 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.~ 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 characterization 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 fruitfulness 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 Cf., e.g. [26], [27], 1291, Ch. 2.
science, and we shall begin with a brief account of our general position (developed in [31]). 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). 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 expressions 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. 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 (1)
Whatever is scarlet, is red.
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 (2)
Nothing is warmer than itself.
holds a priori because "is warmer than" is represented by the relation < (which is irreflexive) on the temperature scale. 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 I think that providing a formal representation Sellars ([24], [25], Chs. 4, 10, 1 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 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). 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 ([13], p. 120). but does not state them explicitly.
.. ."
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leave the abstract level of syntax and semantics, and provide a pragmatic counterpart 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. 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 perspective. We shall attempt to show the feasibility of our approach in a more detailed exploration before arguing its advantages.
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 To give the simplest example, a classical and Hilbert space in quantum mecl~anics.~ 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,). 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.) 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 See, for example, [ 6 ] ,Appendix B.
Other terms for "state-space" are "phase-space" and "system space" (Weyl).
ON THE EXTENSION OF BETH'S SEMANTICS OF PHYSICAL THEORIES
329
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). 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 measurable 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 the shallowest characterization. But this is not a subject into which we intend to go deeper here (cf. Suppes t281). The elementary statements pertaining to a given kind of physical system constitute the set of well-formed formulas of a simple kind of formal language. This set of formulas, together with the state-space and the satisfaction function, constitutes what elsewhere we have called a semi-interpreted language. Before investigating the formal structure of this simple variety of languages, however, we wish to discuss certain fundamental questions about the notion of state, and the physical laws describing the behavior of such a system.
4. The role of the time variable. We have said that elements of the state-space represent possible states of the system. This is not unambiguous, however, for sometimes "state" is used in such a way that a system (though undisturbed) has different states at different times, and sometimes such that a system remains in the same state unless it is subject to interaction. In the second case, the magnitudes m may change even though the system remains in the same state. In our representation this amounts to: in the former case the satisfaction function is not itself a function of time, but the 'location' of the system in state-space changes with time; in the latter case, the satisfaction function must be time dependent. It must also be noted that if the first alternative is embraced, one finds that the term "state" is then also used to designate the function $ such that +(t) is the state (in the first sense) of the system at t. So in that case the usage of "state" is not unequivocal, though it is not confusing. These distinctions are easily illustrated with reference to quantum mechanics, since Schrodinger embraced the first alternative and Heisenberg the second. For this reason, the physicist speaks of the Schrodinger picture and the Heisenberg picture (cf. 1171, pp. 100-101; [9], pp. 35-36; [14], p. 44). In the case of quantum mechanics, the states of the system are represented by the elements (state-vectors) of a Hilbert space. For each measurable physical magnitude ("observable") m there is a linear operator M on this space, such that m has the value r if and only if Mx = rx, where x is the state-vector representing the state of the s y ~ t e r n .Thus ~ we have If U = U(m, r) then (3) h(U) = {x:Mx = rx)
8
If M x = rx, we call r an eigenvalue of M and x the corresponding eigenuector.
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BAS C. VAN FRAASSEN
The difference between the two pictures appears when we make explicit the time dependence of the magnitude m, writing U = U(m, r, t). In the Schrodinger picture, the state-vector x is a function of the time. Writing x, for the value of x at t, we have (4)
Schrodinger picture: If U = U(m, r, t) then h(U)
=
{x, :Mx, = rxt).
In the Heisenberg picture not the state-vector but the operator representing the magnitude m is time dependent. Writing M, for the value of M at t, we have (5)
Heisenberg picture: If U
=
U(m, r, t) then h(U) = {x:M,x
=
rx).
Finally we may note that the elements of the Schrodinger state-space are functions from the time (the whole real number continuum or a segment thereof) into an 'instantaneous state-space', which one may take to contain the set h(U) of equation (3). Each Schrodinger state describes an 'orbit' in this 'instantaneous state-space'. The opcrators in the Schradinger picture operate on the instantaneous states. Formally speaking, one might of course identify the Heisenberg state-vectors with the Schrodinger state-vectors, and regard the two pictures as but two frames of reference in a single Hilbert space. (Heisenberg's matrix mechanics resolves the state-vector on axes of definite energy, wave mechanics on axes of definite position.) This is what allowed Dirac's, and von Neumann's, unification of the theory ([12], section B). It also shows at once that we have here but several variants of a single concept of state, and that we can use one or the other as convenience or fancy dictates. 5. The representation of physical laws. In the case of a nonrelativistic theory, the function of a law is to describe the behavior of the kind of physical system with which the theory deals: to describe the possible states of which it is capable, its normal evolution through time when undisturbed, and its behavior in interaction. We shall therefore proceed in accordance with the traditional threefold distinction between laws of coexistence, laws of succession, and laws of interaction. Our discussion will first address itself to nonstatistical laws (section 5.1) and then to statistical laws (section 5.2). 5.1 Nonstatistical laws. What has traditionally been called a law of coexistence is a condition limiting the class of physically possible states. The most familiar example is the Boyle-Charles ideal gas law whicli functionally relates the volume, pressure, and temperature of a gas, in such a way that, given any two of these magnitudes at a time t, the value of the third at t is uniquely determined. It has the form where R is a constant, the "gas constant." What this means exactly is that at any given time the values of P, V, and T a r e related by equation (6). Hence if we use triples of real numbers (p, v, t) to represent the possible thermodynamic states of the gas, the law says that (p, v, t ) represents a physically possible state only if pc = Rt. Laws of coexistence select the physically possible subset of the state-space.
Of course, when the law of coexistence is general enough, it may be incorporated in the definition of the state-space. In our example, we may take the state-space of the ideal gas to be a region in Cartesian 3-space all of whose points satisfy (6)-and not the whole 3-space. There is nothing right or wrong about this procedure-and this is just to say that there is no objective distinction within physics between "meaning postulate" and "empirical postulates." Still using the instantaneous state picture, we can say that similarly, laws of succession select the physically possible trajectories in the state-space. As an example, we can take a simple harmonic oscillator: a classical particle moving along a straight line subject to a force which is proportional to the distance from, and directed to, a fixed point 0 on that line. Using q and p to range over the values of its position on the line and its momentum along that line, its motion is described by equations of the form
At each time t, the particle has a state described by the values of q and p; that is, a state represented by a point in the q-p plane (its state-space or phase-space). In time, the particle describes an 'orbit' or 'trajectory' in this plane, described by the equation ~ = - 1- q 2 +-p2 2n 2m
=
constant
(E being the mechanical energy). The law of succession is expressed by this equation (equivalently, by the previous equation) and the physically possible trajectories are the curves which satisfy this equation.1° We consider this subject briefly from a more abstract point of view. Let x, y, . . . range over the state-space. We shall call the system deterministic if and only if there is for each non-negative real number t a unique transformation U, such that (9)
If the state of the system at t' is represented by x, then its state at t' represented by U,(x).
+ t is
We clearly have here a one-parameter semi-group of transformations with Uo as identity element and group-multiplication defined by
This is called the dynamical semi-group of the system ([16], 1 ; [34], 80-81; [20], p. 418). If in addition the system is time-reversible, each operator Ut has a unique inverse U,-l = U-,. In that case the set of transformations forms a group, the dynamical group. Thus a system subject to a diffusion process (say, heat-flow) has a dynamical semi-group governing the evolution of its state, which is not a group Though this may be a histovicnl distinction; before a certain stage in the development of the theory, the law may not yet be an inherent principle of the language game. l o Thus if we use "state" in the sense of a function 4 such that $ ( t ) is the state (in our present sense) at t , then this law of succession appears with the formal character of a law of coexistence. Speaking generally, laws describe conditions on what is physically possible.
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BAS C. VAN FRAASSEN
(cf. [lo], pp. 233-234). But the simple harmonic system of our example has its motion governed by a group of transformations. We may state this condition equivalently in the form: there is only one physically possible trajectory through each point in the state-space. When the laws of the system are given by differential equations, we are essentially given the injinitesimal generators of the (semi-)group ([I 11, 6.1). The elements U, of the dynamical (semi-)group can then usually be found through integration. For the harmonic oscillator of our example, the infinitesimal transformation U,,(q, y) = (q + dq, p + dp) is therefore given by
(In the second form, this is easily generalized to more complex systems (cf. [3], p. 94-95).) Finally, we turn to the laws of interaction. Theoretically the most satisfying approach to interaction is no doubt the procedure of regarding the interacting systems as constituting a complex system. Thus an aggregate of n gravitationally interacting masses forms an n-body system subject to the same basic mechanical laws as individual bodies. Also, if two quantum mechanical systems have their (Schrodinger) states represented by 1 $,(T)) and 1 $,(T)) in Hilbert spaces H1 and H z , the complex system they constitute has a state-vector / $lz(~))in the product space H1@ H2. So in principle, the study of interaction of systems reduces to the study of syste~ns simpliciter. But this approach is often not feasible, especially if the interaction of the system studied is not with a like system but with a quite different kind of system. This leads to the typical problems of perturbation theory, scattering theory,ll and the theory of systems with input. Specifically, the abstract theory of systems with input (and possibly output) is one of the newest subjects of mathematical research ([19], 8, 1). A system with input has a state-space H a n d a nonempty set of inputs D. These inputs change the state of the system in a manner described by the 'next-state function'
We shall write I(x) = y for S(x, I ) = y. If n inputs are supplied successively, the resultant sequence of inputs is itself an input, whose effect is defined by (13)
I1...In(x)=xn+l
where x, = x, x,+, = I,(xj) for j = 1, 2,. . ., n. Thus if we accept also that the environment can leave the state unchanged, the inputs form a semi-group with identity : The work by Lax and Phillips [15] develops scattering theory via the notion of the dynamical group {Ut}.
(14)
= x for all x in H; if I,, I, belong to D, does an element I,, such that I12(x) = 12(11(x)).
D has a member I, such that I,(x) SO
In special cases, the inputs of an abstract machine may be a group, namely, if the action of the environment can reverse its previous actions. The theory can also be extended to the effect of a continuous or compact input applied throughout a time interval; see for example the study of 'compact acts' in [I]. 5.2 Staristical laws. The statistical analogue of a law of coexistence is a statement of a priori probabilities (also called, somewhat less misleading perhaps, "initial probabilities"). That is9 instead of dividing the (logically possible) states into the physically possible and the physically impossible, it assigns a probability to each state. An example of such a law is the Boltz~nannhypothesis that each microstate of a gas has equal probability (cf. the discussion of initial probability metrics in [7], Ch. XL or [22], section 10). Formally speaking, this amounts to the specification of a probability measure on the state-space (in [22], Reichenbach used the term "probability metric"). If U = U(m, r) is an elementary statement, andp the probability measure in question, we would expect p to be defined for the set h(U), and the law of coexistence would have the general form:
(15)
The probability that the physical magnitude m has the value r equals p(h(U)).
(One qualification is necessary here: "m has the value r" must here be understood as "the system is in an eigenstate of nz corresponding to value r," not as, e.g., "measurement of m would yield value r." This distinction is important only in the nonclassical case in which not all states are eigenstates of m.) Similarly, laws of succession have as statistical analogue statemei~tsof transition probabilities. As example, let us consider a finite Markov chain. We have here a finite state-space, for which the law of succession is given by describing a transition = 1 (cf. [19]). The probability matrix M = [Mtj], with 0 < Mi, < 1 and 2,Mi, law then has the general form : (16)
Given that the system is in state xi at time t, the probability that it is in state xi at t At equals Mi,.
+
Just as nonstatistical laws of coexistence can be used to redefine the state-space in such a way that then the law becomes analytic, its statistical analogue can be used to redefine the state-space; the problem of describing the behavior of the system is then essentially reduced to the nonstatistical case, as we shall see. Suppose that the original state-space is H; for convenience suppose H to be finite again. Let the elements of H be indexed as {x,, x,, . . .) and let p be the probability measure on H. We can now switch our attention to the new state-space H" whose elements are the vectors
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BAS C. VAN FRAASSEN
The new 'rule of correspondence' reads that a system is in state v if and only if the probability that it is in state xi equals pi (i = 1,2, .. .). If nothing special is known about the system, it is then in the state v such thatp, = p({xi}). But our information might be more definite, say enough to locate the system at a(0) = (1, 0, 0,. . .) at time t. Then the law of succession (16) uniquely determines its location at t At, as the vector v(1) = (MI,, MI,, . . .). In other words, the law of succession now assigns a unique orbit in HPto the system for all times (counted in intervals of At) from t on: with respect to Hp the system is deterministic. The law can now be written, using the notation v(m) = (pl(m), p,(m), . . .) to designate the state at t mat.
+
+
(17)
p,(m)
=
2 p,(m - I)Mij,
or equivalently,
a
Using the notation Mn to denote the nth power of the matrix M, this means that we have here a dynamical semi-group (U,), n ranging over the non-negative integers, defined by
This semi-group has U, = MO,the identity matrix, as identity element. We have only indicated informally what the satisfaction function comes to in the case of H". Denoting it as hP, we have (19)
hP(U) is the set of vectors v only if xi E h(U).
=
