# Reflection calculus and conservativity spectra

## TL;DR

This paper extends the Reflection Calculus with new modalities representing fragments of arithmetical theories, explores their formal properties, and connects them to proof-theoretic ordinals, providing a decidable and complete logical framework.

## Contribution

It introduces RC$^
abla$, an extension of Reflection Calculus, with new modalities for arithmetical fragments, and establishes its properties and connections to proof-theoretic ordinals.

## Key findings

- RC$^
abla$ can express reflection principles up to any ordinal less than ε₀.
- The variable-free fragment of RC$^
abla$ is decidable and complete with respect to its semantics.
- Normal forms of RC$^
abla$ relate to collections of proof-theoretic ordinals for arithmetical theories.

## Abstract

Strictly positive logics 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 RC consists of implications between formulas built up from propositional variables and constant `true' using only conjunction and diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. We extend the language of RC by another series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity $\Pi^0_n$, for all n>0. We note that such operators, in a precise sense, cannot be represented in the full language of modal logic. We formulate a formal system RC$^\nabla$ extending RC that is sound and, as we conjecture, complete under this interpretation. We show that in this system one is able to express iterations of reflection principles up to any ordinal $<\epsilon_0$. On the other hand, we provide normal forms for its variable-free fragment. Thereby, the variable-free fragment is shown to be decidable and complete w.r.t. its natural arithmetical semantics. Whereas the normal forms for the variable-free formulas of RC correspond in a unique way to ordinals below $\epsilon_0$, the normal forms of RC$^\nabla$ are more general. It turns out that they are related in a canonical way to the collections of proof-theoretic ordinals of (bounded) arithmetical theories for each complexity level $\Pi^0_{n+1}$. Finally, we present an algebraic universal model for the variable-free fragment of RC$^\nabla$ based on Ignatiev's Kripke frame. Our main theorem states the isomorphism of several natural representations of this algebra.