Approximate reasoning, logics for self-reference, and the use of nonclassical logics in systems modeling
Daniel G. Schwartz · 2000
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