Abstract:
Strictly positive logics have recently attracted attention both in the description logic and in the provability logic communities for their combination of efficiency and sufficient expressivity. The language of Reflection Calculus, RCRC, consists of implications between formulas built up from propositional variables and the constant ‘true’ using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles.
The language of RCRC is extended by another series of modalities representing the operators associating with a given arithmetical theory TT its fragment axiomatized by all theorems of TT of arithmetical complexity Π0nΠ0n for all n>0n>0. It is noted that such operators, in a strong sense, cannot be represented in the full language of modal logic.
A formal system RC∇RC∇ is formulated that extends RCRC and is sound and (it is conjectured) complete under this interpretation. It is shown that in this system one is able to express the iterations of reflection principles up to any ordinal <ε0<ε0. Second, normal forms are provided for its variable-free fragment. This fragment is thereby shown to be algorithmically decidable and complete with respect to its natural arithmetical semantics.
In the last part of the paper the Lindenbaum–Tarski algebra of the variable-free fragment of RC∇RC∇ and its dual Kripke structure are characterized in several natural ways. Most importantly, elements of this algebra correspond to the sequences of proof-theoretic
Π0n+1Π0n+1-ordinals of bounded fragments of Peano arithmetic called conservativity spectra, as well as to points of Ignatiev's well-known Kripke model.
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This publication is cited in the following 7 articles:
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L. D. Beklemishev, F. N. Pakhomov, “Reflection algebras and conservation results for theories of iterated truth”, Annals of Pure and Applied Logic, 173:5 (2022), 103093
F. Pakhomov, J. Walsh, “Reflection ranks and ordinal analysis”, J. Symb. Log., 86:4 (2021), PII S0022481220000092, 1350–1384
Lecture Notes in Computer Science, 11541, Logic, Language, Information, and Computation, 2019, 195