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arxiv: 2606.31872 · v1 · pith:GBHNHE3Nnew · submitted 2026-06-30 · 💻 cs.LO

Hyperformalism for Relevant Modal Logics

Thomas Macaulay Ferguson (Rensselaer Polytechnic Institute) , Shay Allen Logan (Kansas State University) This is my paper

Pith reviewed 2026-07-01 02:58 UTC · model grok-4.3

classification 💻 cs.LO
keywords relevant modal logichyperformalismMPos-hyperformalismvariable sharingB-BoxK-MPossoundnesscompleteness
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The pith

The weak relevant modal logic B-Box is MPos-hyperformal and K-MPos is the largest MPos-hyperformal sublogic of classical K.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends hyperformalism to relevant modal logics through MPos-hyperformalism, a property that closes logics under non-uniform substitutions respecting nesting inside modal operators. It establishes that B-Box satisfies this closure and examines the substitution classes for its extensions, along with corresponding refinements to the variable sharing property. The work defines K-MPos as the largest MPos-hyperformal sublogic of classical modal logic K and supplies soundness and completeness results for it.

Core claim

B-Box is closed under MPos substitutions, which treat occurrences of the same atom independently when their nesting within modal operators differs, and K-MPos is the largest sublogic of K that remains closed under these substitutions, with soundness and completeness established for the latter.

What carries the argument

MPos-hyperformalism: closure under non-uniform substitutions that take modal nesting into account.

If this is right

  • B-Box is closed under MPos substitutions.
  • Several extensions of B-Box are closed under specified classes of non-uniform substitutions.
  • Refined versions of the variable sharing property hold for B-Box and its extensions.
  • K-MPos is sound and complete.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The MPos restriction may allow modal relevant logics to preserve relevance more precisely when necessity and possibility operators interact with propositional atoms.
  • The same substitution classes could be checked for other base relevant modal systems.
  • K-MPos may serve as a benchmark for measuring how much classical modal reasoning can be retained while keeping MPos-hyperformalism.

Load-bearing premise

The syntactic definition of MPos-hyperformalism correctly extends non-modal hyperformalism to the modal relevant setting without introducing inconsistencies.

What would settle it

A concrete substitution respecting modal nesting under which B-Box fails to be closed, or a strictly larger MPos-hyperformal sublogic of K than K-MPos.

Figures

Figures reproduced from arXiv: 2606.31872 by Shay Allen Logan (Kansas State University), Thomas Macaulay Ferguson (Rensselaer Polytechnic Institute).

Figure 1
Figure 1. Figure 1: CPos-substitutions on the left; LRCNPos-substitutions on the right. the extent that they can be replaced by different formulas with no worry about coordinating between the different parts of this replacement. It turns out, as first shown in [10], that B is CPos-hyperformal. This fact is the first of several hyperformalism results one can now find in the literature (see [21, 13, 5] for other such results). … view at source ↗
Figure 2
Figure 2. Figure 2: LRCNPos-assignments LRCNPos-hyperformalism is interesting in other ways too. To see this, let an LRCNPos￾assignment be a function α : LRCNPos ×At → {0,1}. LRCNPos-assignments extend to all of LRCNPos ×L by the recursive clauses given in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: MPos-substitutions. σxnny(p) = σxy(p) for all {x,y} ⊆ MPos and p ∈ At. We extend MPos-substitutions to act on all of L by the recursive clauses in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tableau rules; we assume y ̸= ε and ν is fresh. the ω-index is drawn from some set Ω = {ωi} ∞ i=1 used to track worlds, A is a formula and i ∈ {0,1}. R-expressions, on the other hand, are used to track accessibility facts. Tableaux are governed by the rules in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

The property of hyperformalism has proven to be a powerful tool in the analysis of relevant logics, revealing that increasingly weak relevant logics are closed under increasingly strong classes of non-uniform substitutions. In such substitutions, two instances of the same atom may be treated independently in virtue of syntactic features of their appearances in a complex. In this work, we extend the scope of hyperformalism to relevant modal logics by considering MPos-hyperformalism, that is, a property in which relevant modal logics are closed under substitutions in which nesting within the scope of modal operators is taken into account. We prove that the weak relevant modal logic B-Box is MPos-hyperformal and investigate the classes of non-uniform substitutions under which several extensions are closed. We then consider corresponding refinements of the variable sharing property that hold of such logics. We conclude by introducing a modal logic K-MPos that constitutes the largest MPos-hyperformal sublogic of the classical modal logic K and provide soundness and completeness results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends hyperformalism to relevant modal logics via the notion of MPos-hyperformalism (non-uniform substitutions that account for nesting inside modal operators). It proves that the weak relevant modal logic B-Box is closed under MPos-substitutions, studies the substitution classes for several extensions of B-Box, derives corresponding refinements of the variable-sharing property, and defines K-MPos as the largest MPos-hyperformal sublogic of classical modal logic K, for which soundness and completeness are established.

Significance. If the proofs hold, the work supplies a technically natural extension of the non-modal hyperformalism framework into the modal relevant setting, identifies a maximal logic inside K, and furnishes soundness/completeness results. These contributions are load-bearing for the central claim that MPos-hyperformalism is a well-behaved and useful property for relevant modal logics; the explicit provision of soundness and completeness theorems is a clear strength.

minor comments (3)
  1. The definition of MPos-substitution (presumably in §2 or §3) would benefit from an explicit side-by-side comparison with the non-modal hyperformalism substitution class to make the modal nesting clause immediately legible.
  2. A short table or diagram summarizing which logics are closed under which substitution classes (B-Box, its extensions, K-MPos) would improve readability and help the reader track the maximality claim for K-MPos.
  3. The variable-sharing refinements are mentioned in the abstract and conclusion; a dedicated subsection collecting the precise statements would make the connection to the substitution results easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary of our work extending hyperformalism to relevant modal logics via MPos-hyperformalism, including the results on B-Box, substitution classes, variable-sharing refinements, and the definition of K-MPos with soundness and completeness. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims consist of syntactic definitions of MPos-hyperformalism (extending non-modal hyperformalism via a modal nesting clause in substitutions) followed by explicit proofs that B-Box is closed under this class, that K-MPos is the largest such sublogic of K, and that soundness/completeness hold. No equations, fitted parameters, or self-referential definitions appear; the derivation chain relies on stated closure properties and completeness results rather than reducing any prediction or uniqueness claim to a prior self-citation or ansatz. Self-citations, if present for background on non-modal hyperformalism, are not load-bearing for the modal extension or maximality results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the work relies on standard definitions of relevant modal logics and substitution classes.

pith-pipeline@v0.9.1-grok · 5705 in / 1022 out tokens · 36095 ms · 2026-07-01T02:58:39.924084+00:00 · methodology

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Reference graph

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