Corcoran, Arist. Natural Deduction Syst

8/12/2019 Corcoran, Arist. Natural Deduction Syst

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8 JOHN CORCORAN

1.1. Mathematical LogicsLogicians are beginning to view mathematical logic as a branch of ap-plied mathematics which constructs and studies mathematical models inorder to gain understanding of logical phenomena. From this standpointmathematical logics are comparable to the mathematical models of solarsystems, vibrating strings, or atoms in mathematical physics and to themathematical models of computers in automata theory (cf. Kreisel,p. 204 . Thus one thinks of mathematical logics as mathematical modelsof real or idealized logical systems.In the most common case a mathematical logic can be thought of as amathematical model composed of three interrelated parts: a language , adeductive system and a semantics . The language is a syntactical systemoften designed to reflect what has been called the logical form of proposi-tions (cf. Church, pp. 2, 3). The elements of the language are called sen-tences. The deductive system, another syntactical system, contains ele-ments sometimes called formal proofs or formal deductions. Theseelements usually involve sequences of sentences constructed in accordwith syntactical rules themselves designed to reflect actual or idealizedprinciples of reasoning (cf. Church, pp. 49-54). Finally, the semantics isusually a set-theoretic structure intended to model certain aspects ofmeaning (cf. Church, pp. 54R), e.g., how denotations attach to nounphrases and how truth-values attach to sentences.2

Many theories of logic involve a theory of propositional forms, atheory of deductive reasoning and a theory of meaning (cf. Church, pp. 1,3,23). Such theories are intended to account for logical phenomena relat-ing to a natural language or to an ideal language perhaps alleged tounderlie natural language, or even to an artificial language proposed as asubstitute for natural language. In any case, it is often possible to constructa mathematical model which reflects many of the structural aspects ofthe system envisaged in the theory. Once a mathematical logic has beenconstructed, it is possible to ask definite, well-defined questions concerninghow well, or to what degree and in what respects, the model reflects thestructure of the system envisaged by the theory. Such activity usuallycontributes toward the clarification of the theory in question. Indeed anyattempt to construct such a model necessarily involves an organized and

AKISTOTLE S NATUR AL DEDUC TION SYSTEM 87detailed study of the theory and often raises questions not considered bythe author of the theory.1.1.1. Underlying logics Because some articulations of the above view-point admit of certain misunderstam~dings, few further comments may bein order. Consider a deductive science such as geometry. We may imaginethat geometry presupposes its own subject matter which gives rise to itsown laws, some of which are taken without deductive justification. Inaddition, geometry presupposes a geometrical language. The activity ofdeductively ustifying some laws on the basis of others further presupposesa system of demonstrative discourses (the deductions). The activity ofestablishing by means of reinterpretations of the language of geometrythat certain geometrical statements are independent of others furtherpresupposes a system of reinterpretations of the language. The last threepresupposed systems taken together from the underlying logic (cf. Church,p 58, 317; Tarski, p. 297) of geometry.

Although the underlying logic is not a science it can be the subjectmatter of a scientific investigation. Of course, there is much more to besaid about this approach to the study of deductive sciences, but what hasbeen said should be sufficient to enable the reader t o see that there is aclear distinction to be made between logic as a scientific study of underly-ing logics on one hand, and the underlying logic of a science on the other.It is roughly the difference between zoology and fishes. A science has anunderlying logic which is treated scientifically by the subject called logic.Logic, then, is a science (in our sense, not Aristotle s), but an underlyinglogic of a science (Aristotle s sense) is not a science; rather it is a complex,abstract system presupposed by a science. Some of the possibility forconfusion could be eliminated by using the term science in Aristotle ssense and the term metascience to indicate activities such as logic. Thenwe could say that a science presupposes an underlying logic which is thenstudied in a metascience, viz. logic.

It is unfortunate that in a previous article (Corcoran, Theories ) Ispoke of the science of logic for what I should have ternled the meta-science, logic or the science of logics . That unfortunate usage, amongother things, brought about Mary Mulhern s justified criticism (cf. herpaper below) to the effect that I am myself guilty of blurring a distinctionwhich I take to be crucial to understanding Aristotle s logic (mnetascience).


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Corcoran, Arist. Natural Deduction Syst

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