Tho Modal Logic of Minimal Upper Bounds

S{\o}ren Brinck Knudstorp

REVIEW 2 major objections 4 minor 15 references

Tho Modal Logic of Minimal Upper Bounds

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Pith's one-line read This paper proves that the modal logic of minimal upper bounds (MIN) coincides exactly with the modal logic of least upper bounds (MIL): no formula distinguishes the two interpretations, and MIN inherits the complete axiomatization and…

desk verdict A clever p-morphic construction shows minimal and least upper bounds are modally indistinguishable; the proof is essentially correct with fillable valuation gaps. read the letter →

arxiv 2411.15940 v1 pith:7LZO3HVK submitted 2024-11-24 math.LO

S{o}ren Brinck Knudstorp This is my paper · ORCID

classification math.LO MSC 03B4503B25

keywords modallogicinformationminimalupperboundleastaxiomatizationdecidabilitypartialordersp-morphism

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that two competing semantics for the same modal language—least upper bounds (informational sums as unique suprema) and minimal upper bounds (informational sums as possibly multiple incomparable fusions)—cannot be told apart by the logic. The main theorem states MIN = MIL: every formula valid under the minimal-upper-bound semantics is valid under the least-upper-bound semantics, and conversely. The paper also proves the same equality for the logics enriched with the residual implication and for preorder frames. If the result is right, the minimal-upper-bound logic inherits a complete axiomatization and decidability from the already-studied least-upper-bound logic.

What carries the argument

The load-bearing mechanism is the distinction between the relations $s \in \operatorname{mub}\{t,u\}$ and $s = \sup\{t,u\}$ in a partial order, together with a representation lemma that forces the two to coincide. Lemma 1 takes a poset containing a minimal upper bound $s$ that is not a supremum, forms the disjoint union $P \sqcup \downarrow s$, and positions a duplicate of the whole downset $\downarrow s$ directly below $s$, so that $(s,1) \leq' (s,0)$ and $s$ ceases to be a minimal upper bound of $\{t,u\}$. A set $G(t,u)$, defined as the least downset containing $t,u$ and closed under binary suprema, ensures that all existing suprema are preserved; iterating the construction over an enumeration of triples yields a poset satisfying $s' \in \operatorname{mub}\{t',u'\}$ iff $s' = \sup\{t',u'\}$. A supremum p-morphism—a function preserving and reflecting suprema—then transfers validity from the constructed poset back to the original one.

What would settle it

Test the formula $(\langle P\rangle p \land \langle P\rangle q) \to \langle P\rangle(\langle mub\rangle pq)$ on infinite posets formed by adding an infinite descending chain to a configuration with two incomparable upper bounds; the paper's argument predicts the formula stays valid under both semantics on these infinite posets, so finding an infinite model where it holds under MIN but fails under MIL would refute Theorem 3.

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Extended reading notes

Core claim

The central claim is that the modal language cannot distinguish minimal from least upper bounds. Writing MIN for the logic whose modality says 's is a minimal upper bound of two states' and MIL for the logic whose modality says 's is their least upper bound,' the paper proves MIN = MIL at the level of validities and consequences, and extends the equality to the residual-implication extensions and to preorder frames. The difficult inclusion is obtained by a representation construction: every poset is shown to be the p-morphic image of a poset in which every minimal upper bound is a least upper bound, and p-morphic images preserve the least-upper-bound semantics. Because MIL is known to be completely axiomatized and decidable, the equality transfers those properties to MIN.

Load-bearing premise

The central proof assumes that the representation construction can be lifted from posets to models: the duplicated downset must be given a valuation so that every formula satisfied at an original point is also satisfied at its duplicate, and the paper states this preservation without explicitly constructing the valuation.

Editorial extensions

If this is right

MIN and MIL have the same valid formulas and the same consequence relation, so no modal principle separates the minimal-upper-bound reading from the least-upper-bound reading.
MIN is completely axiomatized by the axioms (Re.), (4), (Co.), and (Dk.) and is decidable, because these properties hold for MIL.
The residual-implication extensions of MIN and MIL also coincide and are decidable.
The poset and preorder variants of the four logics form one uniform family: MINPos = MINPre = MILPre = MILPos.
On finite frames the two semantics genuinely differ, and the paper exhibits a formula that separates them there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the equality holds, the philosophical contrast between unique and multiple information fusions is not expressible in ordinary modal validities; it would show up only in finite-frame behavior or in model-level definitions.
The construction depends essentially on infinite descending chains, so restricting to well-founded or finite posets is the natural place to look for a logic genuinely different from MIL; the paper's finite-frame witness is a starting point.
A testable extension is to add a device that can express uniqueness of a minimal upper bound; such an enrichment would likely produce formulas that separate MIN from MIL.
The same duplicating-downset technique may apply to other binary order-theoretic connectives, such as maximal lower bounds, possibly yielding analogous collapse theorems for their modal logics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper proves something I didn't expect: the minimal-upper-bound semantics and the least-upper-bound semantics are modally indistinguishable. The proof that MIN = MIL is the real result. The technique — duplicating downsets to replace incomparable minimal upper bounds by infinite descending chains — is a genuine contribution, and the way it forces suprema to be preserved is clever.

What's new: MIN is a new logic, the equality with MIL is not in the literature, and the finite-frame distinguishing formula (valid on finite frames for MIN, not MIL) is a nice observation. The paper does a good job of showing MIL ⊆ MIN by verifying the known MIL axioms under the mub clause, and the principal lemma (Lemma 1) is worked out in detail.

Where the soft spots are, in proportion: the biggest one is the one the reader flagged. Lemma 1 is stated for frames, but the proof of Theorem 3 needs preservation of satisfaction at the level of models with valuations. The paper never defines the valuation on the duplicated copy ↓s. This is a real rigor gap. But it is not a mathematical problem: defining V'(x,i) = V(x) on the disjoint copy gives you the pullback, and with that valuation the p-morphism conditions proved in Lemma 1 are exactly the forth/back conditions needed for the induction on formulas. So the central claim survives; the paper just needs to state the valuation and the model-level preservation lemma explicitly.

Two smaller gaps: the transfinite recursion for uncountable posets is dismissed as 'straightforward' — it is, but a line or two would help — and the proof of Theorem 5 (the residual implication) says the back condition suffices and then does the case analysis, but the preservation argument for formulas with '\' is only sketched. Again, standard, but worth tightening.

The citation pattern is fine: the axiomatization and decidability come from the author's published prior work on MIL, which is independent support. Self-citation here is legitimate.

Overall: this is a solid, interesting paper. The main theorem appears correct, the proof method is novel, and the gaps are fillable without changing the argument. I'd send it to a serious referee. It will be of value to anyone working on modal logics over order-based information semantics, and the equality result will be cited.

Referee Report

2 major / 4 minor

Summary. The paper defines MIN, the modal logic of minimal upper bounds on posets, and proves that MIN coincides with van Benthem's MIL, the modal logic of least upper bounds. The proof uses a step-by-step p-morphic representation that eliminates failures of the implication mub ⇒ sup by duplicating downsets and adding infinite descending chains of upper bounds. From this equality and the author's earlier axiomatization and decidability result for MIL, the paper derives an axiomatization and decidability for MIN; it then extends the result to preorder frames and to the residual-implication extensions MIN(\) and MIL(\).

Significance. The result is genuinely interesting: although the two semantic clauses differ at the model level, the paper claims that no modal formula separates them, so the modal language cannot distinguish least from minimal upper bounds at the level of validities. The central construction is explicit, the iterative representation is natural, and the corollaries concerning axiomatization and decidability are a real payoff. If the proof gaps noted below are closed, the paper makes a solid contribution to modal information logic; the finite-frame remark also gives a useful boundary for the main result.

major comments (2)

[§3, Lemma 1 and Proposition 1] The construction is carried out for frames only: Lemma 1 defines P', ≤', and f, but never defines the valuation V' on the duplicated copy ↓s. The Proof Strategy paragraph announces a model M'=(P',≤',V'), and Theorem 3 requires transferring refutations of formulas containing proposition letters. The natural pullback V'(x,i)=V(x) for all (x,i)∈P' should be stated explicitly, and the standard p-morphic preservation induction showing M',(x,i) ⊩_S φ iff M,f(x,i) ⊩_S φ for all φ should be written out. As the text stands, the model-level preservation claim is asserted rather than proved, so the converse inclusion in Theorem 3 is not fully justified.
[§4, Theorem 5] In the third case of the \-back verification, the proof asserts 'Thus, v /∈ ↓s, hence we must have i = 0' without giving the needed argument. The reader must infer that x∉↓s and x≤v imply v∉↓s, since if v∈↓s then x≤v≤s contradicts x∉↓s; then i=1 would force v∈↓s by the definition of ≤'. This is a small but real omission in the proof of the residual implication case, and it should be supplied explicitly.

minor comments (4)

[§3, Lemma 1] The notation P' := P ⊔ ↓s = {(x,0),(y,1) | x∈P, y∈↓s} is a slight abuse of notation for a disjoint union; condition 1, which says P⊆P', should be read under the identification x ↦ (x,0). This identification deserves to be stated once explicitly.
[§3, Proposition 1] Footnote 7 suppresses the transfinite recursion needed for uncountable posets. Since Theorem 3 is claimed for all posets, a brief indication of the limit step would make Proposition 1 completely rigorous.
[§2, Theorem 2] The soundness of the axioms (Re.), (4), (Co.), and (Dk.) with respect to MIN is dismissed as a routine check. In particular, a sentence showing how (Dk.) is verified under the mub semantics would help the reader confirm the inclusion MIL ⊆ MIN.
[§2, Definition 3] The strict inequality 'x ⁄< s' should be written as the explicit condition 'it is not the case that x < s' to avoid ambiguity, especially since the paper uses '≤' and '<' informally in the surrounding discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No definitional circularity: the MIN=MIL proof is a self-contained representation argument, with [11] used only as external prior work.

full rationale

The central derivation is self-contained and none of its load-bearing steps reduces to its own inputs. The inclusion MIN ⊆ MIL is proved via an explicit p-morphic representation: Lemma 1 constructs a poset P' in which the offending minimal upper bound is no longer minimal, and Proposition 1 iterates the construction until m.u.b. and sup coincide. Preservation of ⊩_S is asserted in footnote 5; the valuation on the duplicated copy is not written out, but the intended pullback V'(x,i)=V(x) makes the standard induction go through. This is a rigor gap, not circularity. The converse inclusion MIL ⊆ MIN is a soundness check of the [11] axioms under the m.u.b. clause; it does not presuppose MIN = MIL. The axiomatization and decidability corollaries inherit [11]'s completeness and decidability results for MIL, but [11] is a separate published paper, not the target claim of this paper, so the self-citation is legitimate external support rather than a definitional loop. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to forbid alternatives, and no known result is merely relabeled. The preorder and residual-implication extensions are direct adaptations of the same lemmas. Overall, the equality MIN = MIL is not forced by definition or by self-citation; the proof has independent mathematical content.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central result is a pure mathematical theorem; no constants are fitted to data. The paper relies on standard set theory and modal logic, on the frame classes (posets and preorders), and on the standard but unproved p-morphic preservation lemma for the supremum semantics, which is the main implicit assumption flagged in the report.

assumptions (3)

standard math ZFC set theory and standard modal logic background.
All proofs presume classical set theory and standard facts about modal logic and p-morphisms.
domain assumption Frames are posets (or preorders in Section 4) and the semantics of the modality is the mub (or sup) clause of Definitions 4 and 5.
The definition of MIN and MIL rests on this modeling choice.
domain assumption P-morphic images preserve the supremum semantics at the level of pointed models with a pullback valuation.
Invoked in the Proof Strategy before Lemma 1; the paper asserts this is readily verified but does not define the valuation on the constructed duplicate copy.

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Cite this review

Pith. "Pith review of Tho Modal Logic of Minimal Upper Bounds." pith.science (2026). https://pith.science/paper/7LZO3HVK

@misc{pith2026241115940,
  author       = {Pith},
  title        = {Pith review of: Tho Modal Logic of Minimal Upper Bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LZO3HVK}},
  note         = {Machine review of arXiv:2411.15940}
}

read the original abstract

To formalize patterns of information increase and decrease, Van Benthem (1996) proposed modal information logic (MIL), a modal logic over partial orders. In MIL, points are interpreted as information states and least upper bounds, when existent, as informational sums. A natural counterpart to this logic is the modal logic of minimal upper bounds (MIN), interpreting minimal, rather than least, upper bounds as informational sums. This paper presents the logic MIN, and in the main result, it is shown that the modal language cannot distinguish the two interpretations: a formula is valid in MIN if and only if it is valid in MIL. Leveraging the work of [11], as corollaries, an axiomatization of MIN and a proof of decidability are obtained.

Figures

Figures reproduced from arXiv: 2411.15940 by the authors.

**Figure 1.** x is an upper bound of {y, z} and minimal in this regard, as no other upper bound is below x. However, x is not least in this regard, as x ′ is another upper bound that is not above x. Thus, {y, z} has minimal upper bounds, namely x and x ′ , but no least upper bound. This example also illustrates that multiple minimal upper bounds may exist, unlike least upper bounds which, when existent, are unique due to antisym… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Semantics and Prag- matics 15(5), 1–60 (Jun 2022)

Aloni, M.: Logic and conversation: The case of free choice . Semantics and Prag- matics 15(5), 1–60 (Jun 2022)

work page 2022

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Anderson, A.R., Belnap, N.D.: Entailment: The Logic of Re levance and Necessity, Vol. I. Princeton University Press (1975)

work page 1975

[3] [3]

In : Copeland, J

van Benthem, J.: Modal logic as a theory of information. In : Copeland, J. (ed.) Logic and Reality. Essays on the Legacy of Arthur Prior, pp. 1 35–168. Clarendon Press, Oxford (1996)

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J ournal of Philosophical Logic 48(3), 571–601 (2019)

van Benthem, J.: Implicit and explicit stances in logic. J ournal of Philosophical Logic 48(3), 571–601 (2019). https://doi.org/10.1007/s10992-018-9485-y

work page doi:10.1007/s10992-018-9485-y 2019

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In: Weiss, Y., Birman, R

van Benthem, J.: Relational patterns, partiality, and se t lifting in modal seman- tics. In: Weiss, Y., Birman, R. (eds.) ‘Saul Kripke on Modal L ogic’ in the series ‘Outstanding Contributions to Logic’. Springer Cham. Preprint available at: https://eprints.illc.uva.nl/id/ eprint/1773/ (Forthcoming)

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doi:10.1007/978-3-030-63787-3_1 2021

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In: Hale, B., Wright, C., Miller, A

Fine, K.: Truthmaker semantics. In: Hale, B., Wright, C., Miller, A. (eds.) A Com- panion to the Philosophy of Language, chap. 22, pp. 556–577. John Wiley & Sons, Ltd (2017). https://doi.org/https://doi.org/10.1002/9781118972090.ch22

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The Journal of Philosophy 66, 477–487 (1969)

van Fraassen, B.C.: Facts and tautological entailments. The Journal of Philosophy 66, 477–487 (1969)

work page 1969