Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.

Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it is shown that (...) in all interesting cases "Archimedean axioms" are independent of other measurement axioms. (shrink)

We here present explicit relational theories of a class of geometrical systems (namely, inner product spaces) which includes Euclidean space and Minkowski spacetime. Using an embedding approach suggested by the theory of measurement, we prove formally that our theories express the entire empirical content of the corresponding geometric theory in terms of empirical relations among a finite set of elements (idealized point-particles or events) thought of as embedded in the space. This result is of interest within the general phenomenalist tradition (...) as well as the theory of space and time, since it seems to be the first example of an explicit phenomenalist reconstruction of a realist theory which is provably equivalent to it in observational consequences. The interesting paper "On the Space-Time Ontology of Physical Theories" by Ken Manders, Philosophy of Science, vol. 49, number 4, December 1982, p. 575-590, has significant affinities to this one. We both, in a sense, try to formally vindicate Leibniz's notion of a relational theory of space, by constructing theories of spatial relations among physical objects which are provably equivalent to the standard absolutist theories. The essential difference between our approaches is that Manders retains Leibniz's explicitly modal framework, whereas I do not. Manders constructs a spacetime theory which explicitly characterizes the totality of possible configurations of physical objects, using a modal language in which the notion of a possible configuration occurs as a primitive. There is no doubt that this is a more accurate realization of Leibniz's own conception of space than the embedding-based approach developed here. However, it also remains open to objections (such as those cited here from Sklar) on account of the special appeal to modal notions. Our approach here, by contrast, aims to avoid the special appeal to modal notions by giving directly a set of laws which are satisfied by a configuration individually, if and only if it is one of the allowable ones. One thus avoids the need for reference to possible but not actual configurations or objects, in the statement of the spacetime laws. We may then take this alternative set of laws as the actual geometric theory, and do away with the hypothetical entity called 'space'. Yet at the same time there is no invocation of modality, except in the ordinary sense in which every physical theory constrains what is possible. So that a relationalist is not forced to utilize a modal language (though Leibniz certainly does.). (shrink)

In the correspondence with Clarke, Leibniz proposes to construe physical theory in terms of physical (spatio-temporal) relations between physical objects, thus avoiding incorporation of infinite totalities of abstract entities (such as Newtonian space) in physical ontology. It has generally been felt that this proposal cannot be carried out. I demonstrate an equivalence between formulations postulating space-time as an infinite totality and formulations allowing only possible spatio-temporal relations of physical (point-) objects. The resulting rigorous formulations of physical theory may be seen (...) to follow Leibniz' suggestion quite closely. On the other hand, physical assumptions implicit in the postulation of space-time totalities are made explicit in the reconstruction of the space-time versions from the physical-relation versions. (shrink)

How many miles to Babylon? Three-score and ten. Can I get there by candle-light? Yes, and back again. If your heels are nimble dnd light, You may get there by candle-light. Any philosopher who takes more than a fleeting interest in the sciences and their development must at some stage confront the issue of incommensurability in one or other of its many manifes tations. For the philosopher of science concerned with problems of conceptual change and the growth of knowledge, matters (...) of incommensurability are of paramount concern. After many years of skating over, skimming through and skirting round this issue in my studies of intertheory relations in science, I decided to take the plunge and make the problem of incommensurability the central and unifying theme of a book. The present volume is the result of that decision. My interest in problems of comparability and commensurability in science was awakened in the formative years of my philosophi cal studies by my teacher, Jerzy Giedymin. From him I have learnt not only to enjoy philosophical problems but also to beware of simpleminded solutions to them. The vibrant seminars of Paul Feyerabend held at Sussex University in 1974 left me in no doubt that incommensurability was, and would remain, a major topic of debate and dispute in the philosophical study of human knowledge. (shrink)

The author's concept of incommensurability is explicated by elaborating the claim that some terms essential to the formulation of older theories defy translation into the language of more recent ones. Defense of this claim rests on the distinction between interpreting a theory in a later language and translating the theory into it. The former is both possible and essential, the latter neither. The interpretation/translation distinction is then applied to Kitcher's critique of incommensurability and Quine's conception of a translation manual, both (...) of which take reference-preservation as the sole semantic criterion of translational adequacy. The paper concludes by enquiring about the additional criteria a successful translation must satisfy. (shrink)