Abstract

The main objective of this study is to seek for type-free systems based on proper analyses of the paradoxes. For this purpose, two methods are adopted: a mainly model-theoretic analysis has been applied in order to obtain a type-free semantical theory which extends classical semantic theories and, a mainly proof-theoretic analysis has been applied in order to obtain constructive type-free systems. ;In Chapter I, some problematic consequences of orthodox resolutions of the paradoxes are pointed out and, the need for type-free systems is underscored. ;In Chapter II, an axiomatic theory KF is developed, comparing the axiomatic approach with Kripke's model theoretic account. Some objections against the adequacy of this theory have been examined. Some of them are met by distinguishing the internal and the external truths. This distinction also motivates a "hierarchy" of theories, KF$\sb {\rm n}$. Based on this theory, we provide a solution to the problem of expressive incompleteness, which has been regarded as the most serious defect of truth-value gap theories. Finally, a para-consistent system KP is obtained as a dual of KF. ;In Chapter III, we seek for resolutions of paradoxes acceptable to constructivists. The phenomenon of non-normalizability of paradoxical arguments--mainly the non-normalizability of Curry's paradox --has been analyzed. Our analysis reveals some problematic features in standard accounts of constructive implications. It also leads to a resolution of Curry's paradox based on clarification of constructive meanings of implication and proofs. Our analysis is implemented by Myhill's levels of implication in order to obtain an intuitionistic system incorporating the logical conception of set. Finally, Gilmore's NaDSetI, which combines the method of this chapter with an idea behind KF, is examined as an example of a classical system incorporating the logical conception of set