Abstract Artemov, building upon a tradition beginning with Kolmogorov and Gödel, developed a paradigm for understanding Constructive Reasoning in terms of classical proofs. Kolmogorov–Gödel–Artemov constructivism flies in the face of the usual understanding of Constructive Reasoning as being distinguished from Classical Reasoning in terms of its theory of truth. Is there something that stands to traditional Classical Reasoning as Kolmogorov–Gödel–Artemov constructivism stands to Constructive Reasoning? In this paper, we develop an affirmative answer to this question by presenting a justification account of Classical Reasoning in terms of explicit justification. The traditional truth paradigm account of Classical Reasoning leads to the well-known paradoxes of material implication. We show that the justification account of Classical Reasoning avoids this problem.1

Abstract In this paper, we investigate epistemic predicates in extensions of arithmetic. We use as our case study Kurt Gödel’s 1951 thesis that either the power of the human mind surpasses that of any finite machine or there are absolutely unsolvable problems. Because Gödel also claimed that his disjunction was a mathematically established fact, we must ask the following: what sort of syntactical object should formalize human reason? In this paper, we lay the foundations for a predicate treatment of this epistemic feature. We begin with a very general examination of the Gödel sentence in the arithmetical context. We then discuss two systems of modal predicates over arithmetic. The first, called coreflective arithmetic or ${\textsf{CoPA}}$, extends ${\textsf{PA}}$ with a coreflective modal predicate but does not contain a consistency statement. The second, called doxastic arithmetic or ${\textsf{DA}}$, has as its characteristic feature the consistency statement but does not contain coreflection or its instance, the ${\textsf{4}}$ axiom. We examine the logical properties of, motivations for and criticisms of both systems. We close with a brief comparison of the systems in the context of Gödel’s disjunction.