Self-Reference and Modal Logic

A note on the interpretability logic of finitely axiomatized theories

“…Clearly, then, rE(El, ~) C vs ~)*. REMARK 2.3. s is hl fact the language of the bi-modal provability logic PRL1 discussed in [3] (with the modal operators interpreted as tableaux provability instead of ordinary provability). Using (.…”

Towards System: From Computation to the Phenomenon of Language

“…If T is consistent, then the sentence Con T , which expresses T:s consistency, is nonprovable in T: IT Con TConcerning minimal assumptions for the T-axiomatization that permit a proof of this theorem, we have the Lob derivability conditions, for which we refer to [30].…”

“…For an understanding of the possibilities of partial introspection of formal languages, the so called recursion theorems (cf [13J, [28]) are quite helpfuL One such is the following (cf [30]):…”

“…The theorem is provable from the Diagonalization Theorem and the following completeness and soundness assumptions for the T-axiomatization (see [30)): corn pleteness…”

All finitely axiomatizable subframe logics containing the provability logic CSM $_{0}$ are decidable

“…A subframe logic is a logic whose frames are closed under a certain type of substructures. All the provability logics investigated by A. Visser [8] and hence a number of systems from [7], [4], [1] and [2] are included in this class. We shall, however, not discuss the interpretation by provability predicates in this paper, but refer the reader to those papers for more information.…”