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Fundamental Propositional Logic with Preconditional: Strong Completeness, Finite Model Property, and Modal Translations

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Pith's one-line read The paper claims that adding Holliday's preconditional to fundamental logic yields a consequence relation F that is strongly complete over reflexive, pseudo-symmetric frames, has the finite model property (with countermodels of at most 4^{|

desk verdict Solid extension of fundamental logic with a preconditional; internal completeness proofs are sound, but the modal full-and-faithful claims inherit load-bearing unproved results from [HM26]. read the letter →

arxiv 2607.20221 v1 pith:YRDTF3C5 submitted 2026-07-22 math.LO

Zhicheng Chen This is my paper · ORCID
classification math.LO MSC 03B4503B2003G1003F03
keywords fundamentallogicpreconditionalrelationalsemanticsstrongcompletenessfinitemodelpropertyortho-S4intuitionisticKTBmodaltranslation
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 claims that a single operation—Holliday's preconditional—can be added to fundamental logic to produce a logic F that is exactly the logic of reflexive, pseudo-symmetric openness frames with a specific relational clause for the conditional. It proves strong completeness for F (and its weaker companion T) by constructing a canonical model whose points are pairs of theories and counter-theories, and it shows every non-derivable consequence has a finite countermodel with at most 4^{|Σ|} states, making F decidable. The paper further claims that F embeds fully and faithfully into two well-studied modal logics—ortho-S4 and intuitionistic KTB—via translations whose conditional clauses are a boxed Sasaki hook and a strict intuitionistic conditional. If these claims are right, fundamental logic with a preconditional occupies the same mediating role among conditional logics that fundamental logic occupies among negation-based logics.

What carries the argument

The key mechanism is the relational preconditional operation on propositions, A →◁ B = □◁(−A ∪ ♢▷(A∩B)), together with the canonical model built from (theory, counter-theory) pairs where y◁x iff Γ_x ∩ Δ_y = ∅ and the counter-theory is →-closed relative to the theory. The further ingredient is the factoring-condition framework: frames that are balanced (prefactoring and postfactoring) or strongly factoring give rise to same-carrier modal companions G1 and G2, and the canonical model is proven to satisfy these conditions, enabling the round-trip constructions with no extra conditions. The translations use a boxed Sasaki hook on the ortho side and a strict conditional on the intuitionistic side

What would settle it

Consult the cited results [HM26, Lemmas 3.14, 3.16, 4.14] and check whether the balanced/strongly-factoring conditions they assume match exactly the conditions proved for the canonical model in Theorem 7.9; any discrepancy (e.g., a hidden normality or totality requirement) would break the faithfulness direction of Theorems 8.14 and 9.14.

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Extended reading notes

Core claim

The central discovery is that the consequence relation ⊢_F—fundamental logic augmented by Holliday's preconditional—is both strongly complete and finite-model-definable over the class of reflexive, pseudo-symmetric frames, with the preconditional interpreted as A →◁ B = {x | ∀y◁x (y ∈ A ⇒ ∃z (y◁z ∧ z ∈ A ∩ B))}. This clause simultaneously generalizes Heyting implication and the Sasaki hook. Moreover, the logic is exactly what is obtained by translating into ortho-S4 with a boxed Sasaki hook, or into intuitionistic KTB with a strict clause classically equivalent to the Goldblatt translation of that hook: φ ⊢_F ψ iff φ^I ⊢_{OS4} ψ^I iff φ^O ⊢_{ITB} ψ^O. The proof of these embeddings rests on a

Load-bearing premise

The faithfulness of the two modal embeddings (Theorems 8.14 and 9.14) inherits unproved-in-this-paper results of Holliday and Massas—OS4 completeness for ortho-S4 frames, ITB completeness for FSTB frames, and the companion-frame lemmas that balanced or strongly-factoring fundamental frames produce the needed modal companions; if any of those cited results carries an additional frame condition not covered by the factoring conditions proved here, the round-trip constructions wo

Editorial extensions

If this is right

  • F is decidable: a non-derivable consequence φ ⊬ ψ has a countermodel with at most 4^{|Σ|} states, where |Σ| grows linearly with the subformulas of φ and ψ.
  • A single conditional covers a wide range of existing conditionals: Heyting implication and the Sasaki hook are both instances of the preconditional clause, so theorems proved for F apply to both.
  • Reasoning about preconditionals can be systematically translated into ortho-S4 and intuitionistic KTB, and conversely; the two modal logics become conservative computational mirrors of F.
  • Strong completeness holds for arbitrary premise sets, not just finite ones, because the canonical model is built from infinite theories and counter-theories.

Reading between the lines

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

  • The 4^{|Σ|} bound is likely not tight; a cut-free sequent calculus in the spirit of the decidability proof for fundamental logic could give a much better complexity bound.
  • The factoring conditions may carve out exactly the class of frames that are reducts of OS4 and FSTB frames, so the same technique could be reused for other extended languages (e.g., quantifiers, strict implication).
  • The two modal readings of the preconditional—boxed Sasaki hook vs. strict intuitionistic conditional—could correspond to two natural-language readings of conditionals (strict vs. variably strict), giving a precise modal map between them.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper introduces and studies a family of propositional consequence systems K ⊆ T ⊆ F obtained by adding Holliday's preconditional axioms to a basic ∧,∨,⊥,⊤ consequence relation, with negation defined as φ→⊥. K is shown to be algebraically complete for bounded lattices with a preconditional; T adds conditional identity and semicomplementation of the defined negation; F adds double-negation introduction and is a conservative extension of fundamental logic. The paper proves strong completeness of T and F with respect to purely relational frames — reflexive, and reflexive-plus-pseudo-symmetric, respectively — using a canonical model whose points are pairs (theory, counter-theory), and proves the finite model property with an explicit 4^|Σ| bound. In the second half, the paper adapts the Holliday–Massas modal-embedding framework: the GMT-style translation with clause (α→β)^I = □(α^I →_s β^I) is claimed to be a full and faithful embedding of F into ortho-S4, and the Goldblatt-style translation with clause (α→β)^O = □(α^O → ♢(α^O ∧ β^O)) is claimed to embed F fully and faithfully into intuitionistic KTB. The frame-level technical core is the reduct-and-companion construction, with the preconditional-specific semantic transfer calculations carried out in Sections 8 and 9.

Significance. If the cited [HM26] results hold, this is a substantial contribution. It gives a syntactic presentation of Holliday's preconditional base, a clean relational completeness proof for two natural extensions, an explicit FMP bound, and two nontrivial modal embeddings that connect fundamental logic with a conditional to well-studied modal targets. A particular strength is that the canonical-model and FMP arguments are explicit and rule-by-rule; the closure conditions on counter-theories, the Σ-restricted existence lemmas, and the truth lemmas are carefully checked. The semantic transfer calculations for the preconditional clauses are the genuinely new technical work and are internally coherent. The algebraic completeness proof for K, T, and F is clean and parameter-free. I found no internal gap in Sections 2–7 or in the frame-level calculations of Sections 8–9.

major comments (2)
  1. [§8.5, §9.5 (Theorems 8.14 and 9.14)] The faithfulness directions of the two main modal theorems are load-bearing and are not self-contained. Theorem 8.14(⇐) uses Theorem 8.13, which delegates the construction of the OS4-model to [HM26, Lem. 3.14, 3.16(1)]; Theorem 9.14(⇐) similarly delegates the FSTB-model construction to [HM26, Lem. 4.14]. In addition, the completeness of the target logics — Theorems 8.4 and 9.4 — is cited from [HM26, Thm. 3.3 and Lem. 4.21]. Since [HM26] is an arXiv preprint and these lemmas carry the target-side frame conditions, the full/faithful conclusions are conditional on external results. Please either state and prove the needed companion lemmas in the present notation, or cite a published version of [HM26] and explicitly confirm that the definitions of frame, validity, and companion coincide with Definitions 8.2 and 9.2.
  2. [Definitions 8.2/9.2 and Theorems 8.13/9.12] The paper verifies that the canonical F-model satisfies the factoring conditions (Theorem 7.9), but it does not verify that these conditions are exactly what the cited [HM26] companion lemmas require. Definition 8.2 uses the interaction condition (int) in the algebraic form '□_≤ sends ≬-propositions to ≬-propositions', while Definition 9.2 uses (fs). The proof of Theorem 8.13(1) says only that '[HM26, Lem. 3.14] shows' the frame is an OS4-frame, and Theorem 9.12(1) does the same with [HM26, Lem. 4.14]. If a cited lemma carries an extra frame condition not implied by balancedness or strong factoring, the equalities F1(G1(N))=N and F2(G2(N))=N would no longer suffice for the modal-frame conclusion. Please spell out the alignment step in the notation of this paper, or include proofs of the companion lemmas.
minor comments (3)
  1. [§2.5] The notation is confusing because the same letter F is used for fundamental logic and for the system F in Proposition 2.6 and Corollary 4.8. Please distinguish the two, for example by using a different symbol for fundamental logic.
  2. [§4.1/Notation 4.2] The predecessor-style conventions for ◁ and the modal operators are explained, but the reader must constantly translate between y◁x and forward accessibility. A small table displaying the forward-relation reading of each operator identity would improve readability.
  3. [§8–9] The references to [HM26] are precise as to lemma numbers, but for resilience the paper should state the exact inclusion FP(c◁) ⊆ FP(c◁▷) used for OS4-valuations and the exact FSTB-frame condition used in Theorem 9.12, even if the proofs are cited. This would also make it easier for a reader to check the definitional alignment raised in the major comments.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: core derivation is self-contained; modal faithfulness relies on external [HM26] lemmas, not on the paper's own equations.

full rationale

The core results—algebraic completeness (Thm 3.10), relational strong completeness (Thm 5.12), and the finite model property (Thm 6.10)—are proved from explicit rule systems and canonical/filtration constructions; no fitted parameter or target result is reinserted as an input. The system K is presented as a syntactic counterpart of Holliday's preconditional axioms, and Theorem 3.10 checks the correspondence; this is an axiomatization, not a disguised prediction. The modal translations are proved by explicit frame constructions and semantic transfer calculations (Thms 8.11, 8.13, 9.10, 9.12), with the canonical model shown to satisfy the balanced/strongly factoring conditions (Thm 7.9, Cor 7.11). The non-trivial external inputs are [HM26]'s OS4/ITB completeness theorems (8.4/9.4) and companion-frame lemmas (HM26 Lemmas 3.14, 3.16, 4.14), used in the converse directions of Theorems 8.13 and 9.12; these are load-bearing dependencies on another preprint, but they are not self-citations and do not reduce to the paper's own definitions. The only self-citation [Che26] in §1 is explicitly comparative and is never used as a premise. No uniqueness theorem from the author's own prior work is invoked, and no known empirical pattern is merely renamed. Thus there is no circular step; the score 2 reflects one minor non-load-bearing self-citation, with the external dependency risk noted as a correctness concern rather than circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper fits no data and introduces no new postulated entities. Its assumptions are standard background theorems from the cited literature, chiefly Holliday's preconditional representation facts and Holliday–Massas frame-companion and completeness results. The only genuinely ad hoc-looking construction, the W_e points in Lemma 7.8, is a proof device rather than a free parameter or new postulate.

assumptions (5)
  • domain assumption Holliday's preconditionals P1–P5 and the fact that the frame operation U→◁V is a preconditional on the propositions of every relational frame (Fact 4.3, from [Hol25]).
    The relational semantics for → rests on Fact 4.3. If it failed, the truth set of the conditional would not be a proposition and the soundness proof would break.
  • domain assumption OS4 is sound and complete for its ortho-S4 relational semantics, and ITB is sound and complete for FSTB models (Theorems 8.4 and 9.4, cited from [HM26]).
    The faithfulness directions of the two modal translations use these completeness theorems; they are cited, not proved in the present paper.
  • domain assumption Companion construction lemmas: balanced fundamental frames produce OS4-frames via G1, and strongly factoring fundamental frames produce FSTB-frames via G2 (HM26 Lemmas 3.14, 3.16, 4.14).
    Theorems 8.13 and 9.12 depend verbatim on these companion lemmas, which are outside the present paper's own proofs.
  • domain assumption Fundamental logic is complete with respect to reflexive pseudo-symmetric frames (used in Corollary 4.8).
    Conservativity of F over fundamental logic imports this completeness result from [Hol23]/[HM26, Thm 2.15] rather than proving it here.
  • standard math Standard set-theoretic closure properties for forming theories, counter-theories, Lindenbaum quotients, and finite closure sets.
    The canonical model and FMP constructions presuppose ordinary ZFC-style reasoning about sets of formulas and quotient structures.

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Pith. "Pith review of Fundamental Propositional Logic with Preconditional: Strong Completeness, Finite Model Property, and Modal Translations." pith.science (2026). https://pith.science/paper/YRDTF3C5

@misc{pith2026260720221,
  author       = {Pith},
  title        = {Pith review of: Fundamental Propositional Logic with Preconditional: Strong Completeness, Finite Model Property, and Modal Translations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRDTF3C5}},
  note         = {Machine review of arXiv:2607.20221}
}
abstract

Fundamental logic (Holliday 2023) is a non-classical logic based only on the introduction and elimination rules for conjunction, disjunction, and negation in a Fitch-style natural deduction system, while a preconditional (Holliday 2025) is a binary operation on a bounded lattice satisfying five natural axioms and subsuming Heyting implication, the Sasaki hook on ortholattices, and Lewis-Stalnaker-style conditionals satisfying flattening. We combine the two by giving a consequence-relation presentation $\mathsf{K}$ whose algebras are exactly Holliday's bounded lattices with a preconditional, and then studying two natural extensions, $\mathsf{T}$ and $\mathsf{F}$, the latter being fundamental propositional logic with a preconditional. For $\mathsf{T}$ and $\mathsf{F}$, we prove strong completeness with respect to a purely relational semantics, using a canonical model whose points are pairs of theories, and establish the finite model property and hence decidability. Finally, following Holliday and Massas (2026), we adapt their GMT- and Goldblatt-style embeddings to fundamental logic with a preconditional. The resulting translations are full and faithful into ortho-$\mathsf{S4}$ and intuitionistic $\mathsf{KTB}$, respectively. The new conditional clauses send the preconditional to a boxed Sasaki hook on the former side and to a strict intuitionistic conditional on the latter whose classical $\mathsf{KTB}$ reading is equivalent to the Goldblatt translation of the Sasaki hook. The frame constructions follow the reduct-and-companion pattern of Holliday and Massas; the essential additional ingredient is the semantic transfer calculation for the preconditional.

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