This work advances the use of nonclassical logics for developing qualitative models of real-world systems. Abstract mathematics is "qualitative" inasmuch as it relegates numerical considerations to the background and focuses explicitly on topological, algebraic, logical, or other types of conceptual forms. Mathematical logic, the present topic, serves to explicate alternative modes of reasoning for use in general research design and in model construction. The central thesis is that the theory of formal logical systems, and particularly, of logical systems based on nonclassical modes of reasoning, offers important new techniques for developing qualitative models of real-world systems. This thesis is supported in three major parts. Part I develops a semantically complete axiomatization of L. A. Zadeh's theory of approximate reasoning. This mode of reasoning is based on the conception of a "fuzzy set," by which means it yields a realistic representation of the "vagueness" ordinarily inherent in natural languages, such as English. All axiomatizations of this mode of reasoning to date have been deficient in that their linguistic structures are adequate for expressing only the simplest fuzzy linguistic ideas. The axiomatization developed herein goes beyond these limitations in a two-leveled formal system, which, at the inner level, is a multivalent logic that accommodates fuzzy assertions, and at the outer level, is a bivalent formalization of segments of the metalanguage. This system is adequate for expressing most of the basic fuzzy linguistic ideas, including: linguistic terms, hedges, and connectives; semantic equivalence and entailment; possibilistic reasoning; and linguistic truth. The final chapter of Part I applies the theory of approximate reasoning to a class of structural models for use in forecasting. The result is a direct mathematical link between the imprecision in a model and the uncertainty which that imprecision contributes to the model's forecasted events. Part II studies the systems of logical "form" which have beeen developed by G. Spencer-Brown and F. J. Varela. Spencer-Brown's "laws of form" is here shown to be essentially isomorphic with the axiomatized propositional calculus, and Varela's "calculus for self-reference" is shown to be isomorphically translatable into a system which axiomatizes a three-valued logic developed by S. C. Kleene. No semantically complete axiomatization of Kleene's logic has heretofore been known. Following on Kleene's original interpretation of his logic in the theory of partial recursion, this leads to a proof that Varela's concept of logical "autonomy" is exactly isomorphic with the notion of a "totally undecidable" partial recursive set. In turn, this suggests using Kleene-Varela type systems as formal tools for representing "mechanically unknowable" or empirically unverifiable system properties. Part III is an essay on the theoretical basis and methodological framework for implementing nonstandard logics in the modeling exercise. The evolution of mathematical logic is considered from the standpoint of its providing the opportunity to "select" alternative modes of reasoning. These general theoretical considerations serve to motivate the methodological ones, which begin by addressing the discussions of P. Suppes and M. Bunge regarding the role of formal systems in providing "the semantics of science." Bunge's work extends that of Suppes and is herein extended in turn to a study of the manner in which formal systems (both classical and nonclassical) can be implemented for mediation between the observer and the observed, i.e., for modeling. Whether real-world systems in fact obey the laws of one logic versus another must remain moot, but models based on alternative modes of reasoning to satisfy Bunge's criteria for empirical testability, and therefore do provide viable systems perspectives and methods of research.
| Identifer | oai:union.ndltd.org:pdx.edu/oai:pdxscholar.library.pdx.edu:open_access_etds-1590 |
| Date | 01 January 1981 |
| Creators | Schwartz, Daniel Guy |
| Publisher | PDXScholar |
| Source Sets | Portland State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Dissertations and Theses |
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