[Paper Review] Lewisian Fixed Points I: Two Incomparable Constructions

Tadeusz Litak, Albert Visser|arXiv (Cornell University)|May 23, 2019

Logic, programming, and type systems38 references4 citations

TL;DR

This paper investigates two distinct constructions for explicit fixpoints in intuitionistic modal logic, focusing on formulas with the Lewis arrow as the principal connective. It shows that the de Jongh-Visser and de Jongh-Sambin methods yield incomparable modal theories, ${\mathsf{i}GL}_{\mathsf{a}}^{-}$ and ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$, respectively, and proves their extension stability and non-conservativity, while also axiomatizing their join and analyzing subtheories via fixpoint principles.

ABSTRACT

Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allows several natural semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a well-understood modal system. The second one by de Jongh and Sambin, seemingly simpler, leads to a modal theory that proves harder to axiomatize in an elegant way. Apart from showing that both theories are incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixpoints of simple formulas. We also show that they are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfer to its finite extensions.

Motivation & Objective

• To analyze and compare two distinct methods for constructing explicit fixpoints in intuitionistic modal logic, particularly for formulas with the Lewis arrow as the main connective.
• To investigate the logical properties of the resulting theories, especially extension stability and non-conservativity, in the context of constructive arithmetic.
• To axiomatize the join of the two incomparable theories and study subtheories defined by simple fixpoint principles.
• To clarify the role of Kripke correspondence conditions in extensions of ${\mathsf{i}A}^{-}$ that do not extend ${\mathsf{i}A}$.
• To explore connections between explicit fixpoints and the Beth definability property in intuitionistic settings.

Proposed method

• The first construction, inspired by de Jongh and Visser, yields the modal system ${\mathsf{i}GL}_{\mathsf{a}}^{-}$, which is well-understood and axiomatized via a known framework.
• The second construction, based on de Jongh and Sambin, leads to ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$, a system that resists elegant axiomatization and requires deeper analysis.
• The paper uses algebraic and Kripke semantics to analyze the structure of models and validate logical principles, including the use of Mace4 to generate countermodels.
• It introduces and studies four salient subtheories of ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$, defined by principles such as ${\mathsf{W}}^{\ast}$, ${\mathsf{W}}^{\circ}$, ${\mathsf{L}^{\circ}_{a}}$, and $4^{\circ}_{\mathsf{a}}$.
• The join of the two theories is axiomatized by combining their respective axioms and proving closure under logical closure and extension stability.
• The paper employs the de Jongh-Sambin algorithm as a foundational tool for fixpoint computation and uses it to analyze definability and uniqueness.

Experimental results

Research questions

• RQ1 Are the two fixpoint constructions—de Jongh-Visser and de Jongh-Sambin—logically equivalent, or do they yield incomparable theories?
• RQ2 Does the theory ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$, arising from the Sambin-style construction, admit a clean and elegant axiomatization?
• RQ3 Is the join of ${\mathsf{i}GL}_{\mathsf{a}}^{-}$ and ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ axiomatizable, and what is its logical structure?
• RQ4 Does the principle $44^{\circ}_{\mathsf{a}}$ follow from ${\mathsf{i}A}^{-}\oplus 4^{\circ}_{\mathsf{a}}$, and what does this imply about the role of the Di condition?
• RQ5 Do both ${\mathsf{i}GL}_{\mathsf{a}}^{-}$ and ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ enjoy extension stability, meaning that validity in a theory's preservativity logic transfers to its finite extensions?

Key findings

• The two fixpoint constructions yield incomparable theories: ${\mathsf{i}GL}_{\mathsf{a}}^{-}$ and ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ are logically distinct and neither is contained in the other.
• The theory ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ is extension stable, meaning that if it holds for an arithmetical theory, it also holds for all its finite extensions.
• The join of ${\mathsf{i}GL}_{\mathsf{a}}^{-}$ and ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ is axiomatized by combining their axioms and is shown to be consistent and well-defined.
• The principle $44^{\circ}_{\mathsf{a}}$ does not follow from ${\mathsf{i}A}^{-}\oplus 4^{\circ}_{\mathsf{a}}$, as demonstrated by a 6-element Heyting algebra countermodel found via Mace4, showing that the Di condition is essential for such derivations.
• The theory ${\mathsf{i}A}^{-}\oplus{\mathsf{J}S}$ proves the Beth definability property, as explicit fixpoints imply definability, a result inherited by any extension of this system.
• The subtheory defined by $4^{\circ}_{\mathsf{a}}$ does not prove $44^{\circ}_{\mathsf{a}}$, indicating a strict hierarchy among the fixpoint principles and highlighting the importance of the Di condition in Kripke semantics.