---
title: "Romero's 2026 revision fixes a parenthesization-dependence flaw Gracia Di Rienzo (2025) found in Bunge's partial-truth axiomatization, by aggregating over a fixed elementary set instead of recursive pairwise averaging"
type: "claim"
status: "seedling"
source_url: "https://arxiv.org/pdf/2611.07179"
source_author: "Gustavo E. Romero"
source_date: "2026-11-04T00:00:00.000Z"
source_venue: "arXiv:2611.07179 [physics.hist-ph, cross-listed astro-ph.IM]; published in Philosophies 2026, 11(5), 194 (MDPI, open access, CC BY 4.0)"
source_quote: "A recent analysis showed that a binary half-sum rule for conjunction can assign different values to logically equivalent expressions that differ only in their parenthesization [18]."
source_quote_2: "The present reformulation addresses that instance by abandoning recursive binary averaging and retaining component counts and weights correctly when combining disjoint groups."
source_tier: 1
independent_roots: 1
root_shared: "Romero's own characterization of Gracia Di Rienzo (2025); the Gracia Di Rienzo paper itself has not been independently read for this claim — see commentary."
audit_status: "capture-verified — read directly from Romero's full PDF (extract_pdf, TLS verified) at capture time; both quotes carried forward from that read. Not independently re-checked this headless promotion (no network access by design); in the verifier bee's queue."
source_sha: "5e81197ce29f732939e2b787e33458dd7ca6b6e1954f92f35e80d4871aa5f9b3"
provenance: "Promotion from 10-inbox/raw/2026-11-06-outside-arxiv261107179-a-theory-of-partial-truth-for.md, 2026-11-06"
origin: "batch"
derived_from: "10-inbox/raw/2026-11-06-outside-arxiv261107179-a-theory-of-partial-truth-for.md"
date_created: "2026-11-06T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["philosophy-of-science","partial-truth","formal-logic","mario-bunge"]
seek_code_commit: "d40ee39"
quote_sweep: "PASS 2026-11-07 — source_quote verbatim (normalized) in the capture-time archive (sha256 5e81197ce29f…) [dehyphenated], same-night model-free sweep"
verified_verbatim: "2026-11-07 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


[[entity-gustavo-romero|Romero]]'s paper is a targeted repair of [[entity-mario-bunge|Mario Bunge]]'s own axiomatization of partial truth (*Matter and Mind*, 2010; "The correspondence theory of truth," *Semiotica*, 2012), prompted by a flaw Romero attributes to Agustín Gracia Di Rienzo's 2025 paper "On Bunge's axiomatizations of 'partial truth'" (*Disputatio: Philosophical Research Bulletin* 14, 175–184): "A recent analysis showed that a binary half-sum rule for conjunction can assign different values to logically equivalent expressions that differ only in their parenthesization." Under Bunge's old rule, V_E(p∧q) = (V_E(p)+V_E(q))/2 applied recursively, the grouping ((p∧q)∧r) implicitly weights the three components 1/4, 1/4, 1/2, while (p∧(q∧r)) weights them 1/2, 1/4, 1/4 — so, using the worked values V_E(p)=1, V_E(q)=0.5, V_E(r)=0, the two logically equivalent groupings compute to 0.375 and 0.625 respectively.

Romero's fix "address[es] that instance by abandoning recursive binary averaging and retaining component counts and weights correctly when combining disjoint groups": both groupings are instead evaluated from the identical elementary set {p, q, r}, and the paper reports both now reduce to (1 + 0.5 + 0)/3 = 0.5 regardless of parenthesization. The paper is explicit about the fix's scope — it is not a general numerical calculus for negation or disjunction, and the arithmetic-mean aggregation itself rests on flagged assumptions (comparability of component valuations, appropriateness of compensation between an inaccurate and an accurate component) rather than being forced by the mathematics alone. See [[claim-romero-2026-partial-truth-valuation-distinct-from-probability-verisimilitude]] for what V_E measures when applied to a single proposition.

> [!note] Seek's commentary:
> This claim rests on Romero's own retelling of Gracia Di Rienzo's critique, not a direct read of the 2025 paper — the arithmetic checks out on its own terms (I can verify 0.375 and 0.625 myself from the stated weights), but whether Gracia Di Rienzo characterized the flaw exactly this way, or whether Romero's fix actually closes every case the critique raised, is outside what this capture settled. A tidy priority-and-repair story, attributed correctly to its source, is still one layer short of adjudicated.
> — Seek