BOOK III - Page 3

Semantical Systems: Physics and the Reduction of Theories

Even before Carnap had published his Introduction to Semantics, he had formulated his concept of science as a semantical system, and this concept did not change fundamen­tally for the duration of his contributing career.  The early statements of this concept are set forth in his “Logi­cal Foundations of the Unity of Science” and “Foundations of Logic and Mathematics” in the International Encyclopedia of Uni­fied Science (1938).  In these works he asserts that philo­sophy of science is not the study of the activities of sci­entists, i.e., the pragmatics of science, but rather is the study of the results of the activity, namely the resulting linguistic expressions, which constitute semantical systems.  More specifically the philosopher treats the language of science as an object language, and develops a metatheory about the semantics and syntax of this object language.  The metatheory is expressed in a metalanguage.

A physical theory is an interpreted semantical system.  Procedurally a calculus is firstly constructed, and then semantical rules are laid down to give the calculus factual content.  The resulting physical calculus will usually presuppose a logical mathematical calculus as its basis, to which there are added the primitive signs which are descriptive terms, and the axioms which are the specific primitive sentences of the physical calculus in question.  For example a calculus of mechanics of mass points can be constructed with the fundamental laws of mechanics taken as axioms.  Semantical rules are laid down stating that the primitive signs designate the class of material particles, the three spatial coordinates of a par­ticle x at time t, the mass of a particle x, and the class of forces acting on a particle x or on a space s at time t.  Thus by semantical interpretation the theorems of the calcu­lus of mechanics become physical laws, that constitute phy­sical mechanics as a theory with factual content that can be tested by observations.  Carnap views the customary division of physics into theoretical and experimental physics as cor­responding to the distinction between calculus and interpre­ted system.  The work in theoretical physics consists mainly in the essentially mathematical work of constructing calculi and carrying out deductions with the calculi.  In experimen­tal physics interpretations are made and theories are tested by experiments.

Carnap maintains that any physical theory and even the whole of physics can be presented in the form of an inter­preted system consisting of a specific calculus, an axiom system, and a system of semantical rules for interpreta­tion.  The axiom system is based on a logicomathematical calculus with customary interpretation for the nondescrip­tive terms. The construction of a calculus supplemented by an interpretation is called “formalization”.  Formaliza­tion has made it possible to forgo a so-called intuitive understanding of the theory.  Carnap says that when abs­tract, nonintuitive formulas such as Maxwell’s equations of electromagnetism were first proposed as new axioms, some physicists endeavored to make them intuitive by constructing a “model”, which is an analogy to observable macroprocesses.  But he maintains that the creation of a model has no more than aesthetic, didactic, or heuristic value, because the model offers nothing to the application of the physical theory.  With the advent of relativity theory and quantum theory this demand for intuitive understanding has waned.

A more adequate and mature treatment of physics as a semantical system, and especially of the problem of abstract or theoretical terms in the semantical system, can be found in Carnap’s “The Methodological Character of Theoretical Concepts” (1956) and in his Philosophical Foundations of Physics: An Introduction to the Philosophy of Science (1966).  Firstly Carnap makes some preliminary comments about terms and laws: All the descriptive terms in the object languages used in science may be classified as either prescientific or scientific terms.  The prescientific terms are those that occur in what Carnap calls the physicalist or thing-langu­age.  This language is not the same as the phenomenalist language advocated by Mach.  Carnap had earlier in his career attempted to apply constructionalist procedures to the construction of a phenomenalist language in his Logical Struc­ture of the World (1928).  But later he decided to accept a language, in which the idea of a physical thing is not lin­guistically constructed out of elementary phenomena, because he came to believe that all science could be reduced to the thing-language.  This thing-language refers to things and to the properties of things. 

In Russell’s predicate calculus things and properties are symbolized as two distinct types of signs: instantiation signs and predicate signs.  But the thing language is also expressible in a natural language such as English.  The predicates or other descriptive signs referring to properties are of two types: observation terms and disposition terms.  Observation terms are simply names for observable properties such as “hot” and “red”.  These words are called “observable thing-predicates.”  Disposition terms express the disposition of a thing to a certain beha­vior under certain conditions.  They are called “disposition predicates” and are exemplified by such words as “elastic”, “soluble”, and “flexible”.  These terms are not observable thing-language properties, but by use of conditional reduc­tion sentences they are reducible to observation predicates.  Opposed to prescientific terms are scientific terms.  Carnap classified all scientific terms as “theoretical terms” in a broad sense, even though physicists, as he notes, customarily refer to such terms as “length” and “temperature” as observation terms, because their measure­ment procedures are relatively simple.  More abstract theo­retical terms are exemplified by “electron” or “electrical field.”

A discussion of theoretical terms requires some further discussion of semantical rules in physical theory.   There are two types of semantical rules: definitions and conditional reduction sentences.  A reduction sentence for a descriptive sign is a conditional statement that gives for the sign the conditions for its application by reference to other signs.  The reduction sentence does not give the com­plete meaning for the descriptive sign, but it gives part of its meaning.  It is a “method of determination” enabling the user to apply the term in concrete cases.  A definition is a special case of a reduction sentence that gives all of the meaning of a descriptive term, because it is an equivalence or biconditional sentence.  There is never more than one definition for a univocal term, but there may be many reduc­tion sentences for a univocal term, each of which contri­butes to the term a part of its meaning.  Unfortunately Carnap seems never to have elaborated on how the meanings of terms can have parts.  Both types of sem­antical rules – definitions and reduction sentences – introduce new terms into an object language.  If one language is such that every descriptive term in it is expressible by reduction sentences in terms of another language, then the second language is called a “sufficient reduction basis” for the first language.  For all scientific terms the scientist always knows at least one method of determination, and all such methods always either are reduc­tion sentences or are introduced into an axiomatic system of physics by explicit definition in the axiomatic system.

Carnap states that he disagrees with the philosophy of the physicist Paul W. Bridgman, who stated in his Logic of Modern Physics (1927) that, any concept is nothing more than a set of operations; it is synonymous with the corresponding set of operations. This principle is called “operationalism”, and it implies for example that there are as many different concepts of temperature or length as there are different ways of measuring temperature or length.  Carnap maintains that these different opera­tional rules for measurement should not be considered definitions giving the complete meaning of the quantitative concept.  He prefers his idea of reduction sentences in which statements of operational procedures are semantical rules giving only part of the meaning of the theoretical term.  In Carnap’s philosophy what distinguishes theoretical terms from observation terms is precisely the fact that the meanings of theoretical terms are always partial and incom­plete.  This view distinguishes Carnap from Heisenberg and from other positivists such as Nagel, who prefer equivoca­tion to partial meanings.  In Carnap’s view statements of operational rules understood as reduction sentences together with all the postulates of theoretical physics function to give partial interpretations to quantitative concepts.  These partial interpretations are never final, but rather are continually increased or “strengthened” by new laws and new operational or measurement rules that develop with the advance of physics.  Such in brief is Carnap’s taxonomy of terms.

Consider next Carnap’s taxonomy of scientific laws: Carnap classifies scientific laws as empirical laws and theoretical laws.  This division does not correlate exactly to the division between observation terms and theoretical terms in the broader and less abstract sense of his meaning of “theoretical term.”  The distinction is based on how the laws are developed.  Empirical laws are also called empi­rical generalizations, because they are developed by induc­tive generalization, which to Carnap means recognition of regularities by observation of repeated instances.  The empirical laws contain observation predicates or magnitudes that are measured by relatively simple procedures that can be expressed in reduction sentences or definitions.  Empirical laws therefore may contain theore­tical terms in the broad sense, such as “temperature”, “volume”, and “pressure”, as occur in the gas laws, as well as observation terms as may occur in such universal generalizations as “every raven is black.”  The scientist makes direct observations or repeated measurements, finds certain regularities, and then expresses the regularities in an empirical law.  Theoretical laws on the other hand cannot be made by inductive generalization, because they contain theoretical terms in the narrower or more abstract sense; these theoretical terms are too abstract for making laws by generalization.  Exam­ples of these terms are “electron”, “atom”, “molecule”, and “electromagnetic field.”  These are the descriptive terms that the physicists also call theoretical and unobservable, and measurements associated with these theoretical terms cannot be acquired in simple or direct ways.  The develop­ment of theoretical laws proceeds by the physicists’ imagi­native construction of theories in the object language of their science.

Having examined Carnap’s classification of the types of terms and of scientific laws, it is now possible to discuss the construction of physical theories.  Logically there is firstly a calculus.  Conceivably the calculus might be completely uninterpreted, but most often the calculus is supplied by what Carnap calls the logicomathematical cal­culus with its semantical rules for its logical terms with their “customary” interpretations.  In other words the physicist seldom develops his own logic or mathematics and may use some pre-existing mathematics that may never have previously been used in physics, e.g., a non-Euclidian geo­metry.  The physicist then postulates certain axioms, and the descriptive terms in the axiomatic system will either be primitive terms or will be completely defined by reference to primitive terms given in the axioms.  In the axiom system the primitive terms may be classified either as elementary terms or as theoretical terms in either the narrow or more the abstract sense.  Elementary terms are either observation terms, or are simple magnitudes which are theoretical terms in the less abstract sense.  The elementary terms are given their seman­tical interpretation by semantical rules that either define them or give methods of determination by conditional reduction sentences.

The aim of the early positivists was to make all the primitive terms elementary terms.  In this way the semantics of the primitive terms would be given by semantical rules that would either designate them as observation predicates, or designate them by reference to experimental measurement procedures.  And since none of the abstract theoretical terms are primitive in the axiomatic system, any such terms would have to be defined by reference to the primitive terms.  This method would completely satisfy the early positivist requirement that all the semantics in the physical theory be supplied by semantical rules that constitute an effective reduction of the theory to observations or to experimentally based measurements.  This would insure that there would be no contamination of science by metaphysical “nonsense”.

However, there is a problem with this approach, even though it would satisfy the requirements of the early posi­tivists.  The theories actually constructed by physicists contain abstract theoretical terms that cannot be defined by reference to elementary descriptive terms having seman­tical rules directly giving them their empirical meanings.  As Carnap states, what physicists actually do is not to make all the primitive terms elementary terms, but rather to make the abstract theoretical terms primitive in the axiomatic system and to make the axioms of the systems very general theoretical laws.  In this constructional procedure the se­mantical rules initially have no direct relation to the pri­mitive theoretical terms.  Carnap borrows Carl G. Hempel’s metaphor­ical language describing the axioms with their primitive terms as “floating in the air”, meaning that the theoretical hypotheses are firstly developed by the imagination of the physicist, while the elementary terms occurring in the empirical laws are “anchored to the ground.”  There then remains to connect the theoretical laws with the empirical laws. 

This connection is achieved by a kind of reduction sentence that relates the abstract theoretical terms in the theoretical laws with the elementary terms in the empirical laws.  This reduction sentence is called the “correspondence rule.”  It is a semantical rule that gives a partial and only a par­tial interpretation to the abstract theoretical terms.  Thus the axiomatic system is left open, to make it possible to add new correspondence rules when theories are modified and as physics develops, until one day the theory is completely replaced in a scientific revolution by a newer one with different axioms.  The new correspondence rules supply additional empirical meaning to the theoretical terms as the theory is developed, and they also enable the physicist to derive empirical laws from the theoretical laws.  The logical connection be­tween the two types of laws enables the theoretical laws to explain known empirical laws.  And Carnap maintains that the supreme value of a theory is its power to predict new empir­ical laws; explaining known laws is of minor importance in his view.  He states that every successful revolutionary theory has predicted new empirical laws that are confirmed by experi­ment.

But there still remains a problem for the logical positi­vist.  In this more complicated relationship between theory and experiment, there is a question of how abstract theore­tical terms can be distinguished from “metaphysical non­sense”.  Many philosophers of science, such as Popper, maintain that this is a pseudo problem that cannot be solved.  But it was resolved to Carnap’s satisfaction by the Ramsey sentence.  The Cambridge logician, Frank P. Ramsey, proposed that the combined system of theoretical postulates and correspondence rules constituting the theory be replaced by an equivalent sentence, which does not con­tain the theoretical terms.  In the Ramsey sentence the theo­retical terms are eliminated and are replaced by existen­tially quantified dummy variables.  The Ramsey sentence has the same explanatory and predictive power as the original statement of the theory, but without the metaphysical ques­tions that are occasioned by the original formulation with its theoretical terms.  Carnap reports that Ramsey did not intend that physicists should abandon their use of theore­tical terms; theory is a convenient “short hand” that is useful to the physicist.

Finally mention must be made of another application of the reductionist logic, the unity-of-science agenda.  Both Mach and Duhem expressed the belief that there is a basic unity to all science.  In the Vienna Circle the principal advocate of using constructional methods for advancing the unity of sci­ence was Otto Neurath, a sociologist who was interested in the sociology of science as well as its linguistic analysis.  In his autobiography Carnap stated that Neurath’s interest in this effort was motivated by the belief that the division between natural sciences and sociocultural sciences, a division that is characteristic of the romantic tradition, would be a serious obstacle to the extension of the empirical-logical method to the social sciences.  Neurath expressed a preference for the physicalist or thing language rather than the phenomenalist language, since the former is easier to apply in social sciences.  His own views are given in his “Foundations of the Social Sciences” in the second volume of the International Encyclopedia of Unified Science (1944). 

But before Neurath had published his views, Carnap had pub­lished his “Logical Foundations of the Unity of Science” in the first volume of the Encyclopedia (1938), where he set forth the constructionalist procedures for the logical re­duction of the descriptive vocabulary of the empirical sci­ences to the observational thing language.  The use of the thing language presumes in Carnap’s view a philosophical thesis called physicalism, the view that the whole of sci­ence can be reduced to the physical language, the language of physical things.  Carnap says that the physiological and behavioristic approaches in psychology and social science are reducible to the observational thing language, but that the introspective method may not be reducible.  The aim of Carnap’s constructionalist program is the logical reduction of only the descriptive terms in science to the observational thing language; this effort is not a reduction of the empirical laws of the sciences to one another.  The reduction of laws occurs as a part of the development of the sciences themselves, and is the task of the empirical scientist, not of the philosopher of science.   The constructionalist procedures for the reduction of descriptive terms for the unity of science are the same as those that Carnap had developed for the reduction of theoretical terms.

Semantical Systems: Probability and Induction 

In his article “Testability and Meaning” in Philosophy of Science (1936) Carnap abandoned the idea of verification, because he concluded that hypotheses about unobserved events in the physical world can never be completely verified by observational evidence.  Then he proposed instead the probabilistic idea of confirmation.  He became interested in the philosophy of probability in 1941, when he considered that the concept of logical probability might supply an exact quantitative explication of the concept of confirmation of a hypothesis with respect to a given body of evidence, such that it would become possible to speak of a degree of confirmation in a measurable sense.  Up to that time there were fundamentally two kinds of concepts of probability, which were proposed by their respective advocates as alternatives.  The earlier view is the frequency or statistical concept advanced by Richard von Mises and Hans Reichenbach.  The other view is the logical concept advanced by John Maynard Keynes and by Harold Jeffreys, and also considered by Ludwig Wittgenstein in his Tractatus, where he defined probability on the basis of the logical ranges of propositions.  Wittgenstein’s interpretation construes a probability statement to be analytic unlike the frequency concept, which construes it to be synthetic or empirical.  Carnap believed that the logical concept of probability is the basis for all inductive inference, and therefore he identifies the concept of logical probability with the concept of inductive proba­bility.

In 1950 Carnap published Logical Foundations of Proba­bility. This work on probability is not a development in either the calculus of probability or the techniques of statistical inference.  It is Carnap’s contribution to the interpretation of probability theory with his constructionalist approach and a further development of his metatheory of semantical systems.  Here his distinction between object language and metalanguage serves as the basis for his relating the con­cepts of logical and statistical probability.  Statements of statistical probability occur in an object language and are empirical statements about the world.  Statements of logical probability occur in the metalanguage and are about the degree of confirmation of statements in the object language.  Carnap also refers to the statements in the metalanguage for scientific theory as “metascientific” statements.  However, for Carnap metascientific statements are not empirical, but rather are analytic or L-true; he does not recognize an empirical metascience.  He accepts the frequency interpreta­tion for the statistical probability asserted by statements in the object language; statistical probability is the relative frequency of an occurrence of an event in the long run.  Logical probability is the estimate of statisti­cal probability, and it is the measure of the degree of con­firmation.  Symbolically he expresses this logical probability as:

c(h,e) = r

which means that hypothesis h is confirmed by evidence e to the degree r.  The variable r is the measure of the degree of confirmation, such that r can take values from 0.0 to 1.0; it is the estimate of the relative frequency and is expressed as:

r = m(e*h)/m(e)

where m(e*h) is the number of observation sentences descri­bing observed confirming instances e of hypothesis h, and m(e) is the number of observation sentences e describing the total number of observed instances, both confirming and disconfirming.  He calls m a measurement function.

In Carnap’s view the logical foundation of probability is logic in the sense of L-truth, and he therefore draws upon his metatheory of semantical systems, in which his ideas of state description and range have a central role.  A state description is a conjunction sentence containing for every atomic sentence that can be formed in a language, either its affirmation or its negation but not both.  Thus every L-true sen­tence is true in every state description and every L-false or self-contradictory sentence is false in every state description.  The F-true or factually true sentences are true in only some state descriptions but are not true in others.  When the idea of state description is related to the concept of logical probability, the L-true sentences have a degree of confirmation of 1.0, and the L-false sentences have a degree of confirmation of 0.0.  The F-true sentences on the other hand have a degree of confirmation between 1.0 and 0.0.  A closely related concept is that of the range of a statement.  The range is defined as the class of all state descriptions in which an empirical state­ment is true, and it may also be defined as those state descriptions that L-imply the statement.  Using the concept of range the equation r = m(e*h)/m(e) may be said to be the partial inclusion of the range of e in h as measured by r.   Therefore the equation c(h,e) = r is analogous to the statement that e L-implies h except that the range of e is not com­pletely contained in h.  Both types of statements are analy­tical or L-true statements in the metalanguage, because both are statements in logic, one in inductive logic and the other in deductive logic.  In Carnap’s philosophy the logical foundations of probability is logic in the sense of L-truth.

In 1952 Carnap published The Continuum of Inductive Methods, which was to be the volume on the theory of induc­tion that followed Logical Foundations of Probability, but he became dissatisfied with this treatment.  For many years he continued to work on induction.  At the time of his death in 1970 he had completed “Inductive Logic and Rational Decisions” and “A Basic System of Inductive Logic, Part I”, which were published in Studies in Inductive Logic and Prob­ability, Volume I (ed. Carnap and Jeffrey, 1971).  Carnap did not complete Part II of “A Basic System”, and it was edited for publication in 1980 by Jeffrey in Studies in Inductive Logic and Probability, Volume II.  In “Inductive Logic and Rational Decisions” Carnap is concerned with Bayesian decision theory. 

In this context the term “probability” does not mean relative frequency, but rather means degree of belief.  He distinguishes the psychological concept of actual degree of belief from the logical concept of rational degree of belief.  The former is empirical and descriptive, while the latter is normative for rational decision making.  Carnap considers the former to be subjective, since it differs from one individual person to another, while the latter is objective.  Carnap maintains that contrary to prevailing opinion relative frequency is not the only kind of objective probability.  He also calls the former “actual credence” and the latter “rational credence”.  Rational credence is the link between descriptive theory and inductive logic, and like inductive logic it is formal, deductive and axiomatic.  The concepts of inductive logic and of normative decision theory are similar but not identical.  The latter are quasi psychological, while the former have nothing to do with observers and agents, even as these are generalized so that the decision theory is not subjective.  Hence there are separate measure functions and confirmation functions for rational decision theory and for inductive logic.  In his “A Basic System of Inductive Logic” Carnap develops a set-theoretic axiomatic system, which uses set connectives instead of sentence connectives, and which is equivalent to the customary axiom systems for conditional probability.


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