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arxiv: 2409.08942 · v1 · submitted 2024-09-13 · 🧮 math.LO

Topics, Non-Uniform Substitutions, and Variable Sharing

Shawn Standefer , Shay Allen Logan , Thomas Macaulay Ferguson This is my paper

Pith reviewed 2026-05-23 21:14 UTC · model grok-4.3

classification 🧮 math.LO
keywords relevant logicvariable sharinglericone relevanceBMBsubject-matterclassical fragmentsparse tree paths
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The pith

The relevant logic BM satisfies lericone relevance, a strict variable sharing property based on matching paths in formula parse trees, along with characterizations of maximal classical fragments.

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

The paper shows that lericone relevance holds in the relevant logic BM, where shared atoms must follow identical sequences of negations and conditionals in the parse trees of the formulas involved in an entailment. A related faithful version holds for the logic B. The authors also determine the largest fragments of classical logic that possess these properties. This refines the hierarchy of variable sharing conditions and has implications for how relevance is understood in logical entailments and for the theory of subject-matter.

Core claim

Lericone relevance, which requires that an atom appears in matching negation-conditional paths in the antecedent and consequent of a valid entailment, is satisfied by BM, while faithful lericone relevance is satisfied by B. The largest fragments of classical logic enjoying these properties are characterized.

What carries the argument

Lericone relevance, a variable sharing property defined by paths of negations and conditionals in the parse trees of formulas.

If this is right

  • Lericone relevance strengthens standard variable sharing properties used in relevant logics.
  • The largest fragments of classical logic satisfying lericone relevance and its faithful variant are fully identified.
  • The results bear on the theory of subject-matter and the definition of relevant logics.

Where Pith is reading between the lines

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

  • The path condition could filter classical inferences more stringently when tracking topics across premises and conclusions.
  • Similar checks might be applied to other relevant logics to see if they meet the same standard.
  • The characterization of classical fragments offers a template for constructing logics that mix relevance constraints with greater expressive power.

Load-bearing premise

That lericone relevance defined through parse tree paths of negations and conditionals is a coherent strengthening of variable sharing that can be verified in the logics BM and B.

What would settle it

A counterexample entailment valid in BM where a shared atom does not appear along matching negation and conditional paths in the antecedent and consequent parse trees.

read the original abstract

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties -- requiring that in valid entailments $A\to B$, some atom must appear in both $A$ and $B$ with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf{BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf{B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan's notion of hyperformalism, and for the very definition of a relevant logic itself.

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 introduces lericone relevance, a strong syntactic variable-sharing property that requires an atom to appear in both antecedent and consequent of a valid entailment A → B along matching paths of negations and conditionals in their parse trees. It proves that this property holds for every theorem of the relevant logic BM, that a faithful variant holds for B, and characterizes the largest fragments of classical logic satisfying these properties. The work also draws out consequences for the theory of subject-matter, Logan's hyperformalism, and the definition of relevant logic.

Significance. If the verifications hold, the results refine the hierarchy of variable-sharing conditions with an unusually fine-grained syntactic criterion and supply concrete maximality results that delimit relevance both inside and outside the relevant-logic family. The connections to subject-matter and hyperformalism broaden the philosophical reach. The standard technique of checking axioms, rules, and semantics against the new property, together with the absence of free parameters or ad-hoc axioms, constitutes a clear strength.

minor comments (3)
  1. [§2] The definition of lericone relevance (presumably in §2) would benefit from one or two fully worked parse-tree examples showing how the path condition is checked for a compound formula.
  2. Notation for the path relation (e.g., the precise encoding of negation/conditional sequences) should be introduced once and used uniformly; occasional informal descriptions risk ambiguity.
  3. [characterization section] The characterization theorems for classical fragments would be easier to assess if the paper explicitly lists the axioms or rules that survive the lericone filter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our paper, including the accurate summary of the main results on lericone relevance for BM and B, the maximality characterizations, and the broader connections to subject-matter and hyperformalism. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the syntactic notion of lericone relevance via paths of negations and conditionals in formula parse trees, then proves by direct verification on axioms/rules plus semantic checking that the property holds in BM (and faithful variant in B), while also characterizing maximal classical fragments. This is a standard deductive argument in relevant logic with no equations that reduce a result to its own inputs by construction, no fitted parameters renamed as predictions, and no load-bearing self-citations that substitute for independent justification. The derivation chain is self-contained against the stated syntax, semantics, and classical fragments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axiomatizations of the relevant logics B and BM plus the newly introduced definition of lericone relevance; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Standard axioms and rules of the relevant logic BM
    Invoked as the base system for which lericone relevance is proved.
  • domain assumption Standard axioms and rules of the relevant logic B
    Invoked for the faithful lericone relevance result.

pith-pipeline@v0.9.0 · 5705 in / 1181 out tokens · 18621 ms · 2026-05-23T21:14:12.738582+00:00 · methodology

discussion (0)

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Reference graph

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