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arxiv: 2606.31891 · v1 · pith:5ZINM6FLnew · submitted 2026-06-30 · 💻 cs.LO

Knowing-Value Logic with Successor Arithmetic

Hongyi Wang (Department of Philosophy , Peking University) This is my paper

Pith reviewed 2026-07-01 02:19 UTC · model grok-4.3

classification 💻 cs.LO
keywords knowing-value logicconditional epistemic logicsuccessor arithmeticfinite model propertystrong completenessweak completenesspublic announcementlogic puzzle
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The pith

Knowing-value logic extended with equality and successor arithmetic has the finite model property and an axiomatization that is strongly complete for non-standard models.

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

The paper enriches conditional knowing-value logic by adding equality and the successor function so that epistemic reasoning about known values can incorporate arithmetic steps. This is done to capture natural language and puzzle cases where value knowledge interacts with numbers. Compactness fails for standard models, so non-standard models are introduced to carry out the analysis. The results establish the finite model property together with an axiomatization that is strongly complete over non-standard models and weakly complete over standard models. The system is then enlarged with public announcement operators and used to formalize and solve the consecutive numbers puzzle.

Core claim

The central claim is that the language of conditional knowing-value logic, augmented with equality and the successor function, admits the finite model property and possesses an axiomatization that is strongly complete with respect to the class of non-standard models and weakly complete with respect to the class of standard models.

What carries the argument

Non-standard models of the successor-enriched knowing-value logic, introduced because compactness fails over standard models, which carry the proofs of the finite model property and the two completeness results.

If this is right

  • The extended language can express knowing-value statements that involve the successor operation on numbers.
  • Public announcement operators added to the logic permit formalization of information updates in arithmetic epistemic settings.
  • The consecutive numbers puzzle receives a formal solution inside the public-announcement extension.
  • Sound and complete reasoning becomes available for both the non-standard and standard semantics of the enriched logic.

Where Pith is reading between the lines

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

  • The same technique of adding non-standard models might apply when other arithmetic functions beyond successor are incorporated.
  • The weak completeness result could support algorithmic verification procedures restricted to standard models despite the technical workaround.
  • Similar combinations of epistemic operators with arithmetic might be used to analyze other number-based logic puzzles.

Load-bearing premise

Non-standard models can be introduced to facilitate technical analysis when compactness fails over the class of standard models.

What would settle it

A formula valid in every standard model but not derivable from the given axioms would falsify the weak completeness result for standard models.

read the original abstract

In their prior work, Wang and Fan proposed conditional knowing-value logic and provided a complete axiomatization. However, in natural language scenarios and logic puzzles, knowing-value reasoning often appears together with arithmetic operations, which motivates us to enrich knowing-value logic with arithmetic function symbols. In this paper, we extend the language of conditional knowing-value logic with equality and the successor function. Due to the failure of compactness over the class of standard models, we additionally introduce non-standard models to facilitate the technical analysis. Our main results establish the finite model property and provide an axiomatization that is strongly complete with respect to the class of non-standard models and weakly complete with respect to the class of standard models. Furthermore, we extend our logic with public announcement operators and use the resulting system to formalize and solve the "Consecutive Numbers" puzzle. This work provides a novel framework for integrating epistemic logic with arithmetic.

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

1 major / 2 minor

Summary. The paper extends conditional knowing-value logic with equality and the successor function. Due to compactness failure over standard models, non-standard models are introduced. Main results are the finite model property, an axiomatization that is strongly complete w.r.t. non-standard models and weakly complete w.r.t. standard models, plus an extension with public announcements used to formalize and solve the Consecutive Numbers puzzle.

Significance. If the completeness and finite model property results hold, the work supplies a technically useful bridge between epistemic logic and arithmetic, enabling formal treatment of value-knowing reasoning in arithmetic contexts such as puzzles. The non-standard model technique to recover strong completeness is a standard device in the field and, when paired with the announced application, gives the contribution a clear niche.

major comments (1)
  1. [Abstract, §3] Abstract and §3 (non-standard semantics): the claim that non-standard models recover compactness (and thereby strong completeness) is load-bearing for the central axiomatization result, yet the precise interpretation of the successor function and the equality predicate in those models is not exhibited; without an explicit construction it is impossible to verify that the models remain conservative extensions of the standard ones.
minor comments (2)
  1. [Introduction] The reference to Wang and Fan's prior work on conditional knowing-value logic should appear with a full citation in the introduction rather than only in the abstract.
  2. [§5] The formalization of the Consecutive Numbers puzzle in the public-announcement extension would benefit from an explicit derivation showing which announcement axioms are used at each step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (non-standard semantics): the claim that non-standard models recover compactness (and thereby strong completeness) is load-bearing for the central axiomatization result, yet the precise interpretation of the successor function and the equality predicate in those models is not exhibited; without an explicit construction it is impossible to verify that the models remain conservative extensions of the standard ones.

    Authors: We agree that an explicit construction of the non-standard models is needed to make the claim fully verifiable. In the revised manuscript we will expand Section 3 with a precise definition: a non-standard model is a structure whose domain is any model of the first-order theory of successor arithmetic (i.e., the axioms for a unary successor function that is injective and has no cycles), with the successor symbol interpreted by the successor operation of that arithmetic model and equality interpreted as the identity relation on the domain. Standard models appear as the special case in which the domain is isomorphic to the natural numbers. This construction is conservative by design and restores compactness via the usual compactness theorem for first-order logic, thereby supporting strong completeness. We will also add a short remark confirming that every standard model is a non-standard model under this definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends conditional knowing-value logic by adding equality and successor, introduces non-standard models to recover compactness (a standard move when standard models fail compactness), and proves finite model property plus strong/weak completeness for the respective classes. These are independent technical results resting on explicit semantic definitions and axiomatic derivations rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation chain. The prior Wang-Fan work is cited only for the base language; the new completeness theorems are presented as fresh contributions. No equation or claim reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the language extension with successor and equality plus the ad-hoc introduction of non-standard models to bypass compactness failure; these are not drawn from prior literature but postulated for this paper.

axioms (2)
  • domain assumption Axioms of conditional knowing-value logic from prior work by Wang and Fan
    The paper builds directly on the complete axiomatization in the cited prior work.
  • ad hoc to paper Non-standard models introduced to handle failure of compactness
    Explicitly added in the abstract to enable the technical analysis and completeness proofs.

pith-pipeline@v0.9.1-grok · 5676 in / 1140 out tokens · 41496 ms · 2026-07-01T02:19:59.682333+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

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