Proof Theory, Modal Logic and Reflection Principles

Sergei Artemov Reflection vs. co-Reflection in Intuitionistic Epistemic Logic. This is joint work with Tudor Protopopescu. We outline an intuitionistic view of knowledge which maintains the Brouwer-Heyting-Kolmogorov semantics and is consistent with Williamson’s suggestion that intuitionistic knowledge is the result of verification. On this view, co-Reflection F → KF is valid since it encodes a fundamental property that intuitionistic truth is constructive and hence checkable. On the other hand, reflection KF → F demands that all verifications yield strict proofs which is not consistent with verification practice and hence is not among the basic principles of intuitionistic epistemic logic. Consequently we show that reflection of knowledge is a distinctly classical principle, too strong as the intuitionistic truth condition for knowledge, “false is not known,” which can be more adequately expressed by e.g., ∼(KF&∼F ), or, equivalently, ∼K⊥. We construct a system of intuitionistic epistemic logic IEL = IPC +K(F → G)→ (KF → KG) + F → KF +∼K⊥ and provide its provability/verification semantics by extending the well-known Gödel’s embedding. We also show that IEL enjoys a natural and explanatory Kripke-style semantics. Lev Beklemishev Positive modal logic and reflection principles