A well-known polymodal provability logic GLP is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic [9, 5, 8]. This system plays an important role in some recent applications of provabil-ity algebras in proof theory [2, 3]. However, an obstacle in the study of GLP is… (More)
Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omega-rule. We compare the information obtained by this kind of analysis with the results obtained by the more usual proof-theoretic techniques. In some cases the techniques of iterated reflection… (More)
We study an extension of Japaridze's polymodal logic GLP with trans-finitely many modalities and develop a provability-algebraic ordinal notation system up to the ordinal Γ0. In the papers [1, 2] a new algebraic approach to the traditional proof-theoretic ordinal analysis was presented based on the concept of graded provabil-ity algebra. The graded… (More)
Gurevich and Neeman introduced Distributed Knowledge Authorization Language (DKAL). The world of DKAL consists of communicating principals computing their own knowledge in their own states. DKAL is based on a new logic of information, the so-called infon logic, and its efficient subsystem called primal logic. In this paper we simplify Kripkean semantics of… (More)
We study reflection principles in fragments of Peano arithmetic and their applications to the questions of comparison and classification of arith-metical theories. Bibliography: 95 items.
We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some non-finitary principles such as transfinite induction up to ε0 or reflection principles. In fact, the closed fragment of GLP gives rise to a natural system of ordinal… (More)
A well-known result of D. Leivant states that, over basic Kalmar elementary arithmetic EA , the induction schema for n formulas is equivalent to the uniform reeection principle for n+1 formulas. We show that fragments of arithmetic axiomatized by v arious forms of induction rules admit a precise axiomatization in terms of reeection principles as well. Thus,… (More)