reduction the replacement of one expression by a second expression that differs from the first in prima facie reference. So-called reductions have been meant in the sense of uniformly applicable explicit definitions, contextual definitions, or replacements suitable only in a limited range of contexts. Thus, authors have spoken of reductive conceptual analyses, especially in the early days of analytic philosophy. In particular, in the sensedatum theory (talk of) physical objects was supposed to be reduced to (talk of) sense-data by explicit definitions or other forms of conceptual analysis. Logical positivists talked of the reduction of theoretical vocabulary to an observational vocabulary, first by explicit definitions, and later by other devices, such as Carnap’s reduction sentences. These appealed to a test condition predicate, T (e.g., ‘is placed in water’), and a display predicate, D (e.g., ‘dissolves’), to introduce a dispositional or other ‘non-observational’ term, S (e.g., ‘is water-soluble’): (Ex) [Tx / (Dx / Sx]), with ‘/’ representing the material conditional. Negative reduction sentences for non-occurrence of S took the form (Ex) [NTx / (NDx / — Sx)]. For coinciding predicate pairs T and TD and -D and ND Carnap referred to bilateral reduction sentences: (Ex) [Tx / (Dx S Sx)]. Like so many other attempted reductions, reduction sentences did not achieve replacement of the ‘reduced’ term, S, since they do not fix application of S when the test condition, T, fails to apply. In the philosophy of mathematics, logicism claimed that all of mathematics could be reduced to logic, i.e., all mathematical terms could be defined with the vocabulary of logic and all theorems of mathematics could be derived from the laws of logic supplemented by these definitions. Russell’s Principia Mathematica carried out much of such a program with a reductive base of something much more like what we now call set theory rather than logic, strictly conceived. Many now accept the reducibility of mathematics to set theory, but only in a sense in which reductions are not unique. For example, the natural numbers can equally well be modeled as classes of equinumerous sets or as von Neumann ordinals. This non-uniqueness creates serious difficulties, with suggestions that set-theoretic reductions can throw light on what numbers and other mathematical objects ‘really are.’ In contrast, we take scientific theories to tell us, unequivocally, that water is H20 and that temperature is mean translational kinetic energy. Accounts of theory reduction in science attempt to analyze the circumstance in which a ‘reducing theory’ appears to tell us the composition of objects or properties described by a ‘reduced theory.’ The simplest accounts follow the general pattern of reduction: one provides ‘identity statements’ or ‘bridge laws,’ with at least the form of explicit definitions, for all terms in the reduced theory not already appearing in the reducing theory; and then one argues that the reduced theory can be deduced from the reducing theory augmented by the definitions. For example, the laws of thermodynamics are said to be deducible from those of statistical mechanics, together with statements such as ‘temperature is mean translational kinetic energy’ and ‘pressure is mean momentum transfer’. How should the identity statements or bridge laws be understood? It takes empirical investigation to confirm statements such as that temperature is mean translational kinetic energy. Consequently, some have argued, such statements at best constitute contingent correlations rather than strict identities. On the other hand, if the relevant terms and their extensions are not mediated by analytic definitions, the identity statements may be analogized to identities involving two names, such as ‘Cicero is Tully’, where it takes empirical investigation to establish that the two names happen to have the same referent. One can generalize the idea of theory reduction in a variety of ways. One may require the bridge laws to suffice for the deduction of the reduced from the reducing theory without requiring that the bridge laws take the form of explicit identity statements or biconditional correlations. Some authors have also focused on the fact that in practice a reducing theory T2 corrects or refines the reduced theory T1, so that it is really only a correction or refinement, T1*, that is deducible from T2 and the bridge laws. Some have consequently applied the term ‘reduction’ to any pair of theories where the second corrects and extends the first in ways that explain both why the first theory was as accurate as it was and why it made the errors that it did. In this extended sense, relativity is said to reduce Newtonian mechanics. Do the social sciences, especially psychology, in principle reduce to physics? This prospect would support the so-called identity theory (of mind and body), in particular resolving important problems in the philosophy of mind, such as the mind–body problem and the problem of other minds. Many (though by no means all) are now skeptical about the prospects for identifying mental properties, and the properties of other special sciences, with complex physical properties. To illustrate with an example from economics (adapted from Fodor), in the right circumstances just about any physical object could count as a piece of money. Thus prospects seem dim for finding a closed and finite statement of the form ‘being a piece of money i. . .’, with only predicates from physics appearing on the right (though some would want to admit infinite definitions in providing reductions). Similarly, one suspects that attributes, such as pain, are at best functional properties with indefinitely many possible physical realizations. Believing that reductions by finitely stable definitions are thus out of reach, many authors have tried to express the view that mental properties are still somehow physical by saying that they nonetheless supervene on the physical properties of the organisms that have them. In fact, these same difficulties that affect mental properties affect the paradigm case of temperature, and probably all putative examples of theoretical reduction. Temperature is mean translational temperature only in gases, and only idealized ones at that. In other substances, quite different physical mechanisms realize temperature. Temperature is more accurately described as a functional property, having to do with the mechanism of heat transfer between bodies, where, in principle, the required mechanism could be physically realized in indefinitely many ways. In most and quite possibly all cases of putative theory reduction by strict identities, we have instead a relation of physical realization, constitution, or instantiation, nicely illustrated by the property of being a calculator (example taken from Cummins). The property of being a calculator can be physically realized by an abacus, by devices with gears and levers, by ones with vacuum tubes or silicon chips, and, in the right circumstances, by indefinitely many other physical arrangements. Perhaps many who have used ‘reduction’, particularly in the sciences, have intended the term in this sense of physical realization rather than one of strict identity. Let us restrict attention to properties that reduce in the sense of having a physical realization, as in the cases of being a calculator, having a certain temperature, and being a piece of money. Whether or not an object counts as having properties such as these will depend, not only on the physical properties of that object, but on various circumstances of the context. Intensions of relevant language users constitute a plausible candidate for relevant circumstances. In at least many cases, dependence on context arises because the property constitutes a functional property, where the relevant functional system (calculational practices, heat transfer, monetary systems) are much larger than the propertybearing object in question. These examples raise the question of whether many and perhaps all mental properties depend ineliminably on relations to things outside the organisms that have the mental properties.
See also EXPLANATION, PHILOSOPHY OF SCIENCE , SUPERVENIENCE , UNITY OF SCIENC. P.Te.