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.
[1]
Saul A. Kripke,et al.
Semantical Analysis of Intuitionistic Logic I
,
1965
.
[2]
George Weaver,et al.
Logical consequence in modal logic. II. Some semantic systems for S4
,
1974,
Notre Dame J. Formal Log..
[3]
S. Shapiro.
On the notion of effectiveness
,
1980
.
[4]
John T. Kearns.
Intuitionist logic, a logic of justification
,
1978
.
[5]
John Myhill.
Intensional Set Theory
,
1985
.
[6]
John Myhill,et al.
Some Remarks on the Notion of Proof
,
1960
.
[7]
Stephen Cole Kleene.
Disjunction and Existence Under Implication in Elementary Intuitionistic Formalisms
,
1962,
J. Symb. Log..
[8]
Saul A. Kripke,et al.
Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi
,
1963
.
[9]
Paul Benacerraf,et al.
God, The Devil, And Gödel
,
1967
.
[10]
Nicolas D. Goodman,et al.
Epistemic arithmetic is a conservative extension of intuitionistic arithmetic
,
1984,
Journal of Symbolic Logic.