Four faces of correctness: a Strategy Logic with no axiomatization, repairable epsilon-derivations, a refuted Bayesian convergence theorem, and what a proof must be
Four papers from 14–18 September 2026 on what a formal proof's checkability actually guarantees: Strategy Logic's missing axiomatization, epsilon-calculus repair for derivations that are wrong, a refuted convergence theorem from the Journal of Philosophical Logic, and the case for effective checking over polynomial-time checking.
Four papers published between 14 and 18 September 2026 take up one question from four directions: what does it take for a derivation to count as right, and what is left when it does not.
Nicholas J. J. Smith argues in The Review of Symbolic Logic that a formal proof is defined by the existence of an effective checking procedure, and that the polynomial-time version of that requirement, standard in the literature on propositional proof complexity, has no good argument behind it 1. Tikhon Pshenitsyn shows that satisfiability for Strategy Logic is computably isomorphic to true second-order arithmetic, which leaves that logic without a sound and complete effective axiomatization even with effectively defined ω-rules 2. Matthias Baaz and Mariami Gamsakhurdia build a repair procedure for incorrect and incomplete derivations out of Hilbert's ε-calculus, and prove that the extended first ε-theorem tolerates falsity 3. M. Ali Khan, Arthur Paul Pedersen and Maxwell B. Stinchcombe show that the principal theorem of a 2021 paper in the Journal of Philosophical Logic is false, name the step in its published proof where the argument fails, and construct a counterexample 4.
Read together, the four separate three things that a formal proof is routinely asked to deliver at once: that its correctness be checkable, that its conclusion be true, and that it survive being wrong. Smith defends the checkability requirement in its weakest form 1. Pshenitsyn exhibits a logic where no effective proof system exists, so that for Strategy Logic the weakest requirement has no instance at all 2. Baaz and Gamsakhurdia work the ground in between, computing the minimal assumptions under which a derivation with a broken step still yields a valid conclusion 3. Khan, Pedersen and Stinchcombe supply the case the other three are measured against: a theorem whose published proof fails, together with three further results in the same article whose published proofs fail with it 5. The four are grouped for what each adds to that question, and no citation count enters the grouping.
Four records, one question
| Paper and first-publication date | Authors and formal setting | Main result | Evidence status | Best fit |
|---|---|---|---|---|
| The Proof is in the Checking, published online 18 September 2026, The Review of Symbolic Logic 1 | Nicholas J. J. Smith, University of Sydney; the concept of a formal logical proof in general, and the two rival conceptions of what its checking procedure must be 1 | The core conception, on which a purported proof must be checkable by an effective or mechanical procedure, is defensible; the strengthened conception that demands polynomial-time checking is supported by no good argument 1 | Peer-reviewed article, 27 pages, 47 references deposited with the publisher; conceptual argument with no new theorem and no data 6 | Philosophy of logic, proof complexity, formalization |
| Exact Complexity of the Satisfiability Problem for Strategy Logic, first submitted 14 September 2026, arXiv:2609.16173 2 | Tikhon Pshenitsyn, no affiliation stated in the preprint; Strategy Logic over concurrent game structures, with strategies as first-class objects that can be quantified over 2 | The set of satisfiable Strategy Logic sentences is computably isomorphic to true second-order arithmetic, hence Π¹∞-complete, and the logic has no sound and complete effective axiomatization even with effective ω-rules 2 | Preprint, not peer-reviewed; full proofs, lower bound as an interpretation of second-order arithmetic, 10 references 2 | Logics of strategic reasoning, analytical hierarchy, non-axiomatizability |
| Towards a Proof-Theoretic Analysis of Incorrect/Incomplete Proofs, first submitted 16 September 2026, arXiv:2609.18486 3 | Matthias Baaz and Mariami Gamsakhurdia, Technische Universität Wien; Hilbert's ε-calculus as a framework for derivations that carry syntactic errors or missing steps 3 | Weakest preconditions recover the minimal assumption that makes an incorrect derivation valid; the ε-calculus is false-tolerant, and the extended first ε-theorem tolerates falsity 3 | Preprint submitted to LPAR 2026 Shorts, not peer-reviewed; proof-theoretic constructions and worked examples, no quantitative measures, 12 references 3 | Proof theory, epsilon calculus, error tolerance |
| Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times, first submitted 17 September 2026, arXiv:2609.20683 4 | M. Ali Khan, Johns Hopkins University; Arthur Paul Pedersen, City College of New York and the CUNY Graduate Center; Maxwell B. Stinchcombe, University of Texas at Austin; finitely additive Bayesian posteriors on Cantor space 4 | Nielsen's Theorem 2 in the Journal of Philosophical Logic is false: the two criteria it names are necessary but not jointly sufficient; the error is located in the published proof, and an explicit finitely additive counterexample realises it 4 | Preprint, not peer-reviewed; explicit construction with full proofs, 12 references, target journal not stated 4 | Formal epistemology, foundations of probability, topology of Cantor space |
The proof is in the checking
Paper and date
Nicholas J. J. Smith's The Proof is in the Checking was published online by Cambridge University Press on 18 September 2026 in The Review of Symbolic Logic, where it sits in the journal's collection on philosophical aspects of logic and foundations. The article runs to 27 pages and Smith is at the University of Sydney 1.
Formal object
The subject is the concept of a formal logical proof in general, as distinct from what a proof is inside this or that proof system. Smith sets out what he calls the core conception: a finite object counts as a proof of a formula only if some effective or mechanical procedure can check that it is one 1. He then states the strengthening that has become standard in the literature on propositional proof complexity, where a proof system is identified with a procedure that checks proofs in polynomial time. On the strengthened conception the checking procedure must be feasible, meaning its running time is bounded by a polynomial in the length of its input 1.
What it does
Smith defends the core conception and rejects the strengthening. He collects four arguments that have been offered for the polynomial-time requirement, drawing them from convenience, intuition, the absence of self-proof, and epistemology, and rejects each in turn. His conclusion is that for the purpose of analysing the idea of a formal logical proof there is no good reason to adopt the strengthened conception, and that the core conception should stand 1.
Evidence and scope
The support is conceptual. Smith argues from what a proof is for and from what the four arguments actually establish; the paper proves no new theorem and reports no data. Its scope is the concept of proof. The target is the claim that polynomial-time checkability belongs to the definition of formal proof; the ordinary use of polynomial-time bounds to compare how efficient proof systems are stands on its own. 1. The article is peer-reviewed, and the publisher's record deposits 47 references from it 6.
Why open it
Open it if the polynomial-time identification of proof systems has been a premise of your own work, or if you need the concept of formal proof stated cleanly for a philosophical argument. Read the two sections that state the conceptions first, then the four rejections in order, since each answers the previous one. The prerequisite is knowing what work the phrase "polynomial-time checkable" does in proof complexity.
Strategy Logic has no effective proof system
Paper and date
Tikhon Pshenitsyn's Exact Complexity of the Satisfiability Problem for Strategy Logic was first submitted to arXiv on 14 September 2026 under math.LO, with a cross-listing in logic in computer science. The preprint names Pshenitsyn as sole author and states no affiliation 2.
Formal object
Strategy Logic extends linear temporal logic by making strategies first-class objects that can be quantified over and assigned to agents. Pshenitsyn works with the multi-agent version introduced by Mogavero, Murano and Vardi for concurrent game structures (background reference) 7. A concurrent game structure consists of finite sets of propositions and agents, at most countable sets of actions and states, a transition function, a labelling function and an initial state; formulas add a next-time operator, until and release, strategy quantifiers ⟨⟨x⟩⟩ and [[x]], and an agent-binding construct. Satisfaction is defined relative to a complete assignment of strategies to agents, and the temporal clauses are read along the play that the assigned strategies generate 2. The logic subsumes CTL, CTL*, ATL and ATL*, and its model-checking problem is decidable with a non-elementary, TOWER-complete upper bound.
What it does
Pshenitsyn proves that the set of satisfiable Strategy Logic sentences is computably isomorphic to true second-order arithmetic, which places satisfiability in Π¹∞ on the analytical hierarchy and makes the problem Π¹∞-complete 2. The characterisation has an immediate corollary for proof: Strategy Logic admits no sound and complete effective axiomatization, even with ω-rules whose premises are defined by a computable procedure. The upper bound is the easy direction, since a Σ¹∞ truth predicate suffices to decide satisfiability; the work is in the lower bound, an interpretation of second-order arithmetic inside Strategy Logic that uses strategies as genuine second-order objects. The interpretation needs only the next-time operator among the temporal operators and obeys the Boolean-goal restriction, so a fragment that reaches just past the one-goal fragment already carries the full second-order power of strategy quantification 2.
The result settles a question left open by an earlier exchange. Mogavero, Murano, Perelli and Vardi had claimed Σ¹₁-hardness for Strategy Logic satisfiability through the recurrent tiling problem; Catta, Galimullin and Murano later observed a gap in that proof, after which only Π⁰₁-hardness survived, and Π⁰₁-hardness of satisfiability is consistent with recursive axiomatizability, so the question stayed open 2. Pshenitsyn closes it in the negative, and his stronger lower bound implies the Σ¹₁-hardness the earlier argument aimed at.
Evidence and scope
The preprint gives full proofs across its three sections, with the reduction of second-order arithmetic to Strategy Logic occupying the third. The statement is about the satisfiability problem; the model-checking complexity of the logic is unaffected. The one-goal fragment SL[1G], shown decidable and 2EXPTIME-complete earlier, keeps that status, since the reduction needs the next-time operator and the Boolean-goal relaxation. The work is a preprint and has not been peer-reviewed; it carries ten references and discloses that AI tools were used in its preparation 2.
Why open it
Open it if you work on logics of strategic reasoning, on the boundary between decidable and undecidable logical problems, or on what a proof system can be for a logic whose validities cannot be enumerated. The reduction in the third section is the part to read, with the two preliminary subsections on the Boolean-goal fragment and on computability and second-order theories supplying the definitions it uses 2. The prerequisites are basic temporal and modal logic, the analytical hierarchy, and computable reducibility.
The epsilon calculus repairs proofs that are wrong
Paper and date
Matthias Baaz and Mariami Gamsakhurdia's Towards a Proof-Theoretic Analysis of Incorrect/Incomplete Proofs was first submitted to arXiv on 16 September 2026 under cs.LO, and the authors note that it was submitted to LPAR 2026 Shorts. Both authors are at the Technische Universität Wien, and the work was funded by the Austrian Science Fund under grant P36571 3.
Formal object
The objects are derivations that contain syntactic errors or incomplete inferential steps and still preserve part of their semantic validity. The apparatus is Hilbert's ε-calculus, in which the term εx A(x) stands for an x satisfying A, if there is one, and quantifiers give way to terms 3. Against that background the authors define a weakest precondition for a formula B: a satisfiable formula A with A → B valid that is minimal in the implication order, with the falsum excluded because it would imply everything. In a fixed finite propositional language, weakest preconditions exist and can be read off a truth table, where each one is a conjunction of literals picking out a single line that makes B true 3.
What it does
The paper first shows why the notion has to be built rather than read off in general. For the first-order formula ∀x(A(p(x)) → A(x)) → A(0), every formula in the infinite list A(0), A(p(0)), A(p(p(0))), … is a valid precondition for the conclusion A(0), and no member of the list is minimal, so no weakest precondition exists at all. Reducing first-order logic to propositional logic through ε-terms is what makes the computation tractable again, and the elimination procedures of the ε-calculus turn out to retain information about where a non-valid ε-derivation went wrong 3.
The second contribution is the property the authors call false tolerance, which standard first-order logic lacks. Elimination in the ε-calculus can expose, preserve or remove specific propositional defects of an erroneous derivation, which makes the calculus a system of correction as well as existence. The paper's headline statement is that the extended first ε-theorem is false-tolerant 3. A worked example makes the mechanism concrete: a sequent derivation of (∀x(A(x) ∧ ∃yB(y)) → ∀xA(x)) ∧ ∃xC(x) contains one incorrect step, and the repair can be read as replacing ∃yB(y) with ∃yC(y), while other repairs, among them ¬∀xA(x) and C(a), are available as well 3.
Evidence and scope
The support is proof-theoretic: constructions, theorems and worked derivations, with no data and no quantitative measures. The propositional part is complete, since truth tables decide it; the first-order part depends throughout on the ε-calculus, and the paper's own framing is that this is an extension of Hilbert's programme in a new direction, from the conditions under which a correct proof guarantees truth to the conditions under which an incorrect one can still yield it 3. The motivating case is historical: Euler's derivation of ζ(2) = π²/6, which treated sin x / x as an infinite product by analogy with finite polynomials, and which the authors describe as an incomplete proof whose defective steps a later reconstruction inside Weierstrass's theory of entire functions turns into a valid theorem 3. The paper is a short conference submission, has not been peer-reviewed, and carries twelve references 3.
Why open it
Open it if you want a worked answer to the question of what a wrong derivation still carries, or if you want a reason to learn the ε-calculus that has nothing to do with Hilbert's programme as it is usually told. The weakest-precondition definition and the truth-table examples come first, and the ε-calculus section supplies the general machinery; reading them in that order shows why the first-order case needs the second. The prerequisites are sequent calculus, quantifier semantics, and enough familiarity with ε-terms to follow an elimination argument.
A published theorem in the Journal of Philosophical Logic is false
Paper and date
M. Ali Khan, Arthur Paul Pedersen and Maxwell B. Stinchcombe's Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times was first submitted to arXiv on 17 September 2026, filed primarily under mathematical probability with a cross-listing that includes mathematical logic. Khan is in the Department of Economics at Johns Hopkins University, Pedersen in the Department of Computer Science at the City College of New York and the CUNY Graduate Center, and Stinchcombe in the Department of Economics at the University of Texas at Austin 4.
Formal object
The paper works on Cantor space, the set of infinite binary sequences read as a sequence of observations, and studies Bayesian posteriors that are merely finitely additive. The target is Michael Nielsen's Convergence to the Truth Without Countable Additivity, published in the Journal of Philosophical Logic in 2021 and the background reference for this section 5. Nielsen distinguishes almost-sure convergence, on which the posterior probability of a hypothesis approaches its truth value outside an event of probability zero, from the stronger almost-uniform convergence, on which it does so uniformly outside an event of arbitrarily small probability. He then advances two criteria: an approximation property, requiring every event to be approximated by an event whose truth is settled after finitely many observations, and countable additivity on conditional hitting times, requiring the first time a posterior crosses a fixed threshold to have a countably additive distribution. Nielsen's Theorem 2 states that the two criteria are individually necessary and jointly sufficient for almost-uniform convergence to the truth 4.
What it does
Khan, Pedersen and Stinchcombe show that the two criteria are necessary and jointly insufficient, so the theorem is false, and they say where the published proof goes wrong. The sufficiency proof selects its exceptional event only after fixing both a probability budget and an accuracy threshold, while almost-uniform convergence requires one event, fixed first, that serves every threshold at once. Under countable additivity the mismatch does no harm, because summable budgets can be assigned to a sequence of thresholds and their countable union taken. Under finite additivity no bound constrains the probability of that union 4.
They then build the counterexample: a finitely additive probability function on Cantor space that gives every nonempty cylinder strictly positive probability, approximates every Borel event by a clopen event in symmetric difference, satisfies countable additivity on conditional hitting times for every event and every threshold, and yet has posteriors converging to the truth almost surely and not almost uniformly. The function is a mixture of a two-valued finitely additive probability that reads truth values along a sequence of paths converging to a limit path and a countably additive measure placing an atom on each of those paths 4. The obstruction lies in the diagonalisation step. For each fixed threshold r the error inequality holds on an ultrafilter-large set of indices, and that is countable additivity on hitting times doing its work; almost-uniform convergence needs one ultrafilter-large set of indices along which the errors go to zero, and none exists 4. Finite additivity emerges from the result intact: replacing the error sequence by one converging to zero in the ordinary sense restores almost-uniform convergence, and the resulting probability function is still merely finitely additive 4.
Four further results in the same article are drawn into the fallout. Theorem 3 asserts that uncountably many merely finitely additive probabilities converge almost uniformly; the extension argument in its published proof fails, because agreement on all open sets forces agreement on every Borel set, and the authors prove the statement another way. Theorem 4 uses the same extension claim and loses its proof, and they leave its standalone truth undecided. Corollary 1 draws almost-sure convergence from the two criteria and now has no published derivation; their counterexample does not refute it, so they record it as open in general and settle it affirmatively for the family containing the counterexample. The second claim of Theorem 5 is reproved directly, which restores one direction of the independence of Nielsen's two criteria 4. Separately, the authors show that almost-sure convergence of posteriors to the truth for every event does not characterise countable additivity, and they close by stating what a repaired characterisation of almost-uniform convergence would have to add 4.
Evidence and scope
The evidence is a construction checked line by line, together with proofs of the parts that survive: every finitely additive and ultrafilter notion the construction uses is defined in the paper, and the three countably additive results it invokes, Lévy's upward theorem, Egorov's theorem and regularity of Borel probabilities on compact metric spaces, are either quoted or proved for Cantor space. The paper keeps its distance from the philosophical dispute about countable additivity and says so: its stated contribution is to fix which convergence results hold when degrees of belief are merely finitely additive, on the grounds that the epistemological arguments built on those theorems depend on which guarantees survive the weakening 4. The preprint has not been peer-reviewed and does not name a target journal; it carries twelve references 4.
Why open it
Open it if you rely on a convergence-to-the-truth theorem in an argument about Bayesian learning, about scientific realism, or about whether Bayesianism commits anyone to a particular attitude toward countable additivity, since the first thing a reader needs is to know which theorems survive. The section that locates the error in the published proof is the place to start, and the construction and its verification come next. The prerequisites are measure-theoretic probability, finite additivity, ultrafilters and Egorov's theorem.
What the four papers share
Each paper sits on the same seam, where a derivation that can be mechanically checked comes apart from a conclusion that is true.
Smith's paper settles what the checking side can be asked to deliver at its weakest, and Pshenitsyn's answers what happens to a logic when that weakest condition cannot be met at all: Strategy Logic has no effective axiomatization, so no effective checking procedure reaches its validities 2. Baaz and Gamsakhurdia enter at the point where the check fails and ask what can still be recovered, with a calculus in which a defective derivation is projected onto the minimal assumption that would make it valid 3. Khan, Pedersen and Stinchcombe show the cost of not recovering it in time: an error in a published proof that no reader of the journal had a procedure for catching, propagated to three further results in the same article 4.
Three further papers from the same week show how routine the checking side has become, which is part of why these questions are live now. Pierre Senellart and Anton Gnatenko report a Lean library that formalises descriptive complexity, with 73 completeness results across 68 problems and 14 complexity classes, relations between classes established inside the logic rather than by simulating machines, and unconditional lower bounds among its results 8. MingKun Xiao and YiXuan Sun compute an explicit bound below ε₀ for the ordinal height of Escardó's dialogue trees over Gödel's System T, and formalise the proof in Agda over classical ordinals 9. A third submission reports the long-horizon autoformalization of a core theorem behind the MIP* = RE result 10. Three formalization reports in a single week is the rate at which the checking side now arrives; the four papers above are about what a check of that kind delivers.
Citation movement: what the counts can and cannot show
Every count in this section was read on 20 September 2026.
OpenAlex, which keeps a cited-by total for preprints with arXiv DOIs, records no citing work for the three arXiv preprints 111213. Crossref's record for the Review of Symbolic Logic article reports zero citing works, while the same record carries the article's 47 deposited references, so the reference side of the bibliographic record fills in well before the citation side 6. Neither of the two services that report citation movement for individual papers answered for any of the four: Semantic Scholar's graph endpoint returned HTTP 429 on every request, and Google Scholar's exact-record lookup returned HTTP 403 1415. This week's papers therefore carry no cited-by total from either service.
Where counts do exist is in the literature these papers build on, and there the trajectories are mixed rather than rising. Cook and Reckhow's The relative efficiency of propositional proof systems, the 1979 paper that fixed the framework Smith's argument takes apart, has 834 citing works in OpenAlex, distributed across recent years as 65 in 2019, 29 in 2020, 31 in 2021, 28 in 2022, 19 in 2023, 18 in 2024, 9 in 2025 and 11 so far in 2026 16. The multi-agent Strategy Logic paper by Mogavero, Murano and Vardi has 191 citing works, with 23 in 2019, 23 in 2020, 21 in 2021, 11 in 2022, 14 in 2023, 6 in 2024, 10 in 2025 and 4 so far in 2026 7. The count for the Nielsen article that this week's refutation takes apart is unusable: OpenAlex and Crossref both return zero citing works for it, a reading that is far more plausibly an indexing gap for that journal's articles than a statement about five years of engagement, so it is recorded here as a defect in the data rather than as a trend.
Three of the four papers are arXiv preprints; the fourth is a peer-reviewed journal article, and the window's journal sweep was run through Crossref metadata for each of the field's journals. It found new articles in Linguistics and Philosophy, the Journal of Semantics, Studia Logica and The Review of Symbolic Logic 1718, and the Journal of Philosophical Logic's most recent new articles before the window fall on 10 September 19.
A week is shorter than every indexing pipeline in this list, so the four papers can be read for structure this week and not for movement: what they show is which anchor literature each one enters, which is why the counts above are reported for the anchors rather than for the new papers.
Choose the next full read by question
- What must a formal logical proof be, at minimum? Read The Proof is in the Checking. Start with the two sections stating the core and strengthened conceptions, then the four rejected arguments in order.
- What happens to a logic that no effective proof system can reach? Read Exact Complexity of the Satisfiability Problem for Strategy Logic. The reduction of second-order arithmetic in the third section is the technical core.
- What can be recovered from a derivation that is wrong? Read Towards a Proof-Theoretic Analysis of Incorrect/Incomplete Proofs. The weakest-precondition definition and the truth-table examples come before the ε-calculus machinery that generalises them.
- How does a false theorem survive a published proof? Read Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times. The section that locates the quantifier-order defect and the construction that realises it are the parts to compare against the 2021 original.
References
- 1THE PROOF IS IN THE CHECKING
doi.org
- 2
- 3
- 4
- 5
- 6Smith reference list
api.crossref.org
- 7Reasoning About Strategies
api.openalex.org
- 8
- 9
- 10
- 11OpenAlex record for Pshenitsyn
api.openalex.org
- 12OpenAlex record for Baaz and Gamsakhurdia
api.openalex.org
- 13OpenAlex record for Khan, Pedersen and Stinchcombe
api.openalex.org
- 14Semantic Scholar graph record for arXiv:2609.20683
api.semanticscholar.org
- 15Google Scholar lookup for arXiv:2609.16173
scholar.google.com
- 16OpenAlex record for Cook and Reckhow
api.openalex.org
- 17
- 18Crossref listing for Studia Logica, 12–18 September 2026
api.crossref.org
- 19
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