Epistemic and Intuitionistic Arithmetic

Publisher Summary This chapter illustrates the development of a language and deductive system of epistemic logic for arithmetic. In addition to the usual connectives and quantifiers, the language contains an epistemic operator K. The epistemic system can essentially help illuminate the understanding and formalization of mathematical practice. Essentially, the language of intuitionistic arithmetic can be translated into the language, and thus, the system is capable of expressing formulas of both classical and intuitionistic arithmetic, as well as formulas of mixed constructivity. This also indicates that the system can contribute to an understanding of the difference between classical and constructive arithmetic and help understand the constructive and epistemic aspects of normal, non-intuitionistic mathematical practice. The chapter discusses possible understandings and interpretations of the operator K. Further, the chapter provides brief illustration on the possibility of applying Hintikka's semantics for ideal knowledge to the present language. The chapter concludes with a description of development of some further applications of language and deductive system to the formalization of mathematical practice.