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arxiv: 2605.20769 · v1 · pith:Y7X2BEZYnew · submitted 2026-05-20 · 🧮 math.LO

Very weak subintuitionistic logics

Taishi Kurahashi , Mashu Noguchi This is my paper

Pith reviewed 2026-05-21 02:33 UTC · model grok-4.3

classification 🧮 math.LO
keywords subintuitionistic logicvery weak logicrelational semanticssoundness and completenessdisjunction propertyfinite frame propertymodal companionsGödel translation
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The pith

A new propositional logic VF is defined by adapting necessitation semantics and proven strictly weaker than the weak subintuitionistic logic WF.

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

The paper defines a new logic called very weak subintuitionistic logic VF by taking the relational semantics for the pure logic of necessitation and applying it to the propositional case. Soundness and completeness are shown for VF and its closed negative extensions with respect to those frames. The logic is proven to have the disjunction property and the finite frame property. VF is shown to be strictly weaker than the existing weak subintuitionistic logic WF. Modal companions are examined using a modified version of the Gödel translation.

Core claim

The paper introduces very weak subintuitionistic logic VF by adapting the relational semantics of Fitting, Marek, and Truszczyński for the pure logic of necessitation N to the propositional setting. VF and its closed negative extensions are sound and complete with respect to this semantics. They have the disjunction property and the finite frame property. VF is strictly weaker than the weak subintuitionistic logic WF. Modal companions of VF and its extensions are studied via Corsi's modified Gödel translation.

What carries the argument

The adapted relational semantics from the pure logic of necessitation, which supplies the frames and validity conditions for propositional formulas in VF.

If this is right

  • VF and its closed negative extensions are sound and complete for the adapted relational semantics.
  • VF and its extensions have the disjunction property.
  • VF and its extensions have the finite frame property.
  • Modal companions for VF and its extensions exist via the modified Gödel translation.

Where Pith is reading between the lines

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

  • Even weaker subintuitionistic logics than previously studied can retain standard metalogical properties under this kind of semantic adaptation.
  • Similar direct adaptations from modal semantics could be used to define weak versions of other propositional logics.
  • The finite frame property opens the possibility of effective decision procedures for VF that may be simpler than those for stronger logics in the same family.

Load-bearing premise

The relational semantics for the modal logic of necessitation can be transferred directly to a propositional setting while preserving soundness, completeness, and the intended weakness without extra constraints.

What would settle it

Exhibit a formula provable in VF but invalid on some frame of the adapted semantics, or valid on all such frames yet unprovable in VF; or derive in WF a formula that fails in VF.

Figures

Figures reproduced from arXiv: 2605.20769 by Mashu Noguchi, Taishi Kurahashi.

Figure 1
Figure 1. Figure 1: The two specific relations of M Since M, 2 ⊮ p∧q and 1R⊤→(p∧q)2, we have M, 1 ⊮ ⊤ → (p∧q). On the other hand, since there is no x such that 1R⊤→(q∧p)x, we have M, 1 ⊩ ⊤ → (q ∧ p). Therefore, since 0 is a root of M, we obtain M, 0 ⊮ (⊤ → (q ∧ p)) → (⊤ → (p ∧ q)). Thus M, 0 ⊮ (⊤ → (p ∧ q)) ↔ (⊤ → (q ∧ p)). By soundness, VF does not prove this formula. 4.2 Frame conditions of closed negative axioms In this su… view at source ↗
read the original abstract

We introduce a new propositional logic, called very weak subintuitionistic logic $\mathbf{VF}$, by adapting the relational semantics of Fitting, Marek, and Truszczy\'nski for the pure logic of necessitation $\mathbf{N}$ to the propositional setting. We prove that $\mathbf{VF}$ and its closed negative extensions are sound and complete with respect to this semantics, and that they have the disjunction property and the finite frame property. We also prove that $\mathbf{VF}$ is strictly weaker than the weak subintuitionistic logic $\mathbf{WF}$ of Shirmohammadzadeh Maleki and de Jongh. Finally, we study modal companions of $\mathbf{VF}$ and its closed negative extensions via Corsi's modified G\"odel translation.

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 / 2 minor

Summary. The manuscript introduces a new propositional logic VF by adapting the relational semantics of Fitting, Marek, and Truszczyński for the pure logic of necessitation N to the propositional setting. It proves that VF and its closed negative extensions are sound and complete with respect to this semantics, establishes the disjunction property and finite frame property, shows that VF is strictly weaker than the weak subintuitionistic logic WF of Shirmohammadzadeh Maleki and de Jongh via an explicit countermodel or formula, and studies modal companions of VF and its extensions using Corsi's modified Gödel translation.

Significance. If the central derivations hold, the work supplies a natural and parameter-free extension of existing relational semantics to define a very weak subintuitionistic logic with the disjunction property and finite frame property. The explicit strict weakness result over WF and the modal companions via a standard translation technique provide clear connections to the literature on subintuitionistic and modal logics. These are load-bearing strengths that advance the classification of weak intuitionistic variants.

minor comments (2)
  1. [§2] §2 (semantics definition): the adaptation of the Fitting-Marek-Truszczyński frames to the propositional case is presented clearly, but a short remark on why no additional frame conditions are imposed would improve readability for readers familiar with subintuitionistic Kripke semantics.
  2. [§5] §5 (modal companions): the application of Corsi's modified Gödel translation is appropriate, yet the statement of the companion theorem could explicitly list the image of the connectives to avoid any ambiguity in the translation clauses.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces VF via direct adaptation of the Fitting-Marek-Truszczyński relational semantics for N to the propositional case, then establishes soundness, completeness, disjunction property, and finite frame property by explicit proofs inside the new semantics. Strict weakness relative to WF is witnessed by a concrete countermodel or formula, and modal companions are obtained via the external Corsi modified Gödel translation. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; all derivations are independent constructions and verifications against the adapted frames.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central definitions rest on the adaptation of an existing relational semantics for necessitation logic; no free parameters, new axioms beyond standard propositional ones, or invented entities are mentioned in the abstract.

pith-pipeline@v0.9.0 · 5651 in / 1126 out tokens · 32840 ms · 2026-05-21T02:33:04.291635+00:00 · methodology

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Reference graph

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