[Paper Review] Two series of formalized interpretability principles for weak systems of arithmetic

Evan Goris, Joost J. Joosten|arXiv (Cornell University)|Mar 31, 2015

Logic, Reasoning, and Knowledge8 references3 citations

TL;DR

This paper establishes two new series of formalized interpretability principles that are provable in any moderately sound, sufficiently strong theory of arithmetic, significantly raising the known lower bound for the logic IL(All), which captures all interpretability principles valid across such theories. The authors provide frame conditions for both series, offering deeper insight into the structural properties of interpretability in weak arithmetic systems.

ABSTRACT

The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations. The logic IL(All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this paper we raise the previously known lower bound of IL(All) by exhibiting two series of principles which are shown to be provable in any such theory. Moreover, we compute the collection of frame conditions for both series.

Motivation & Objective

• To identify and formalize new classes of interpretability principles that are provable in all moderately sound, sufficiently strong theories of arithmetic.
• To improve the known lower bound of IL(All), the logic of all interpretability principles valid across such theories.
• To characterize the frame conditions corresponding to two new series of interpretability principles, enhancing the algebraic and model-theoretic understanding of interpretability logics.
• To challenge and refine prior conjectures about the exact nature of IL(All), particularly those involving principles like W*, P₀, and RW.

Proposed method

• The authors introduce two new series of interpretability principles, denoted as R_n and R^n, which generalize known principles like R and W*.
• They formalize interpretability within weak arithmetic theories using a bounded quantifier framework, relying on S¹₂ or IΔ₀ + Ω₁ as base theories.
• The paper employs a formalization of relative interpretations via domain formulas δ(x) and translation maps t, extending to formulas via relativization and commutation with connectives.
• Frame conditions for the new principles are derived using first-order logic over Kripke frames, with conditions expressed as implications involving R and S relations.
• The authors analyze the logical strength of these principles by comparing them to known systems like IL(W*), IL(W*P₀), and IL(RW), showing proper extensions.
• They define a new class of frames, 𝔒All, as the intersection of frames satisfying the frame conditions of IL(P) and IL(M), and propose a new conjecture that IL(All) = IL[𝔒All].

Experimental results

Research questions

• RQ1 What new interpretability principles are provable in all moderately sound, sufficiently strong theories of arithmetic, beyond the previously known ones?
• RQ2 How do the frame conditions of these new principles relate to existing interpretability logics such as IL(W*) and IL(RW)?
• RQ3 Can the lower bound of IL(All) be meaningfully increased by identifying new, valid interpretability principles in weak systems?
• RQ4 Is there a canonical class of frames that captures the full logic IL(All), and if so, what are its defining properties?
• RQ5 Do the two new series of principles interact in ways that yield even stronger interpretability principles?

Key findings

• The paper identifies two new series of formalized interpretability principles, R_n and R^n, that are provable in any moderately sound theory of arithmetic, thereby raising the known lower bound of IL(All).
• The frame conditions for both series are explicitly computed, showing that R_n satisfies ∀w,x,y,z (ℬ₀(w,x,y,z) → 𝒢ₙ(x,y,z)) and R^n satisfies ∀w,x,y,z (ℬₙ(w,x,y,z) → 𝒢₀(x,y,z)).
• The new principles form proper extensions of previously known systems: IL(RW) is a proper extension of IL(W*P₀), and IL(RW) is a subsystem of IL(All), confirming that IL(All) is strictly larger than previously conjectured.
• The paper refutes the conjecture that IL(All) = IL(W*P₀), showing that IL(RW) is a proper extension and thus a stronger lower bound.
• The authors propose a new conjecture: IL(All) = IL[𝔒All], where 𝔒All is the class of frames closed under all S-relations implied by both the IL(P) and IL(M) frame conditions.
• The analysis suggests that interactions between the two new series may yield further principles, indicating that the current lower bound is likely not the final answer.