Four papers on modality below classical logic: trivalent conditionals get a probability theory, non-deterministic four-valued tables reach S0.5, the Nelson logics get a Dunn-style modality, and quantum logic gets a strict implication

Four papers published between 5 and 9 October 2026 put a modal or conditional operator on a value space that has more than two points. Paul Égré, Lorenzo Rossi and Jan Sprenger give indicative conditionals a three-valued semantics together with a probability theory built on it 1. Andrzej Pietruszczak interprets normal and very weak modal logics with non-deterministic four-valued tables, in which necessary and contingent truth are tracked separately 2. Igor Sedlár and Pietro Vigiani add modal operators to the generalised Nelson logics and prove a sound and complete axiomatisation over Dunn's relational semantics 3. Kenji Tokuo defines a strict implication in quantum modal logic and shows it holds exactly when its antecedent entails its consequent 4.

The four keep the operator and change the values underneath it. Égré and his co-authors work over three values, Pietruszczak over four arranged as two pairs, Sedlár and Vigiani over a bilateral structure that admits both truth and falsity at once, and Tokuo over the components fixed by non-orthogonality in a quantum model.

Each paper's technical work is the question that follows from that change. What becomes of conditional probability, of necessity, of entailment, once the two-valued assumption is gone.

Each paper below first appeared in the week of 5–9 October 2026, and each is one step from its DOI or arXiv identifier. The table states the question each answers, what its support rests on, and where it fits.

Probability for Trivalent Conditionals is an advance article at Mind, published online on 9 October 2026 6. The authors are Paul Égré, Lorenzo Rossi and Jan Sprenger; OpenAlex places Rossi and Sprenger at the University of Turin 5. A preprint has been circulating since 2025.

Indicative conditionals get a trivalent semantics, and the paper builds truth conditions and probabilities in one framework rather than assigning truth first and probability afterwards. The question the three-valued base is meant to answer is the old one: how the probability of "if A, then B" relates to the conditional probability of B given A.

A Reduction Theorem is the load-bearing result: any compound of conditionals is semantically equivalent to a simple conditional. From it the paper derives Stalnaker's Thesis in full generality, generalizes Adams's notion of p-validity into a criterion for valid uncertain inference, and shows that Bayesian updating on an indicative conditional amounts to updating on the corresponding material conditional 1.

This is a refereed article in a general philosophy journal, released as advance access. The support is a semantic construction plus proofs, with no data and 75 references; the scope of the Stalnaker's Thesis claim is exactly the class of conditionals and compounds the Reduction Theorem reaches.

Open it if you work on conditionals, on formal semantics, or on the probability of conditionals. The Reduction Theorem is the step to check first, because everything else in the paper follows from it.

On Non-Deterministic Four-Valued Semantics for Some Normal and Very Weak T-Normal Logics appeared online on 8 October 2026 in the Journal of Philosophical Logic 10. Andrzej Pietruszczak is at Nicolaus Copernicus University 7. The article is 24 references long and carries no data.

A non-deterministic matrix is a truth table that returns a set of values instead of one, so a connective can leave a formula's value partly open. Pietruszczak uses four values split into two pairs, necessary truth and contingent truth, and their falsehood counterparts. The paper asks which such tables give sound and complete semantics for which modal systems.

Three of the eight table sets come from Kearns, for S5, S4 and KT; two come from Coniglio, Fariñas del Cerro and Peron, for S4 and KTB; three are new, for S5 and KT. Pietruszczak shows these eight are the only sets of their kind satisfying certain natural conditions, and that they generate sixteen non-deterministic matrices depending on whether necessary truth alone or both kinds of truth are distinguished 2.

He then adds a class of valuations, the PL-valuations, which assign necessary truth to every substitution instance of a classical tautology. Only Kearns's S5 tables make the PL-valuations and the first-level valuations coincide. The remaining seven tables with two distinguished values and PL-valuations are adequate for S0.5 and six very weak logics containing it; three of those seven with first-level valuations are adequate for S5 and S4; and the four sets kept at infinite-level valuations are adequate for S5, KTB and KT.

A refereed journal article, semantic throughout, with no data and no open code reported. The claim to check is the one about the eight sets being the only ones satisfying the stated conditions, since the rest of the coverage rests on it.

Open it if you work on non-deterministic semantics or on very weak modal logics. It picks up directly where the closure-semantics paper of the previous week left the question of what a semantics for the weakest systems should look like.

Modal Extensions of Generalised Nelson Logics was first submitted to arXiv on 7 October 2026 under cs.LO and runs to 22 pages 3. Igor Sedlár and Pietro Vigiani are both at the Institute of Computer Science of the Czech Academy of Sciences 8.

The generalised Nelson logics are a family Dunn studied, and they include the semi-relevant logic RM. Their semantics is bilateral: an ordered structure of states carries separate verification and falsification clauses, so a statement can be both verified and falsified. The paper asks what happens when modal operators are added to that structure.

The main result is modular. The same template of modal axioms and rules, added to the propositional base of any logic in the family, yields a sound and complete axiomatisation of its modal extension. Soundness is Theorem 1 and completeness is Theorem 2 8. The semantics uses a single accessibility relation for both □ and ♢, which the authors chose for simplicity.

The proof obliged them to weaken two of Dunn's frame conditions. A linear order on states becomes a tree-like partial order, which modal extensions need; and a distinguished set of inconsistent states is added, because two of the logics in the family, B1 and B10, are otherwise unsound for the modal language 8.

A preprint that has not been peer-reviewed; 22 pages of proof, no data, and 31 references. The authors disclose that they used Google Gemini 3.1 Pro for a technical and editorial review of the manuscript, including checking definitions and proofs, and state that it did not suggest the research direction or generate the mathematical results 8. The work was supported by a Czech Science Foundation grant and by a Research Foundation – Flanders fellowship.

Open it if you work on relevant or Nelson logics, on bilateral semantics, or on the modal proof theory of non-classical bases. The paper opens by noting the renewed interest in relevant modal logics as a fine-grained model of epistemic reasoning, and the obstacle that comes with it: those logics are harder to formalise and often undecidable, and RM is the exception the paper exploits.

Strict implication in quantum modal logic was published online on 5 October 2026 in the Logic Journal of the IGPL, and is assigned to volume 34, issue 6, dated 25 November 2026 11. Kenji Tokuo is at the National Institute of Technology, Oita College 9.

Quantum logic has no canonical implication connective, and the polynomial candidates on offer differ in which inferential principles they satisfy. Tokuo studies a relational strict implication and reads it through the modal accessibility relation, which puts the connective into a modal setting rather than leaving it inside the orthomodular lattice alone.

On reflexive modal frames, a formula whose principal connective is this strict implication is valid exactly when its antecedent semantically entails its consequent. The connective therefore internalizes semantic consequence. The forcing condition ties modal accessibility to non-orthogonality, and the consequence is strong: any formula with this connective on top keeps the same truth value throughout each component fixed by non-orthogonality, so its truth set lies in the Boolean algebra of unions of those components 4. The paper also sets out further semantic laws and treats quantum disjunction, contraposition and nested implications.

A refereed journal article with no data and 14 references. The internalization result is stated for reflexive frames, so the frame conditions are part of what the reader has to accept, and the scope is the quantum reading of the accessibility relation rather than modal logic in general.

Open it if you work on quantum logic, on quantum modal logic, or on the design of implication connectives. The result to test against your own system is the internalization claim, since it is what would let a quantum logic express its own entailment.

The four value spaces differ, and each operator is rebuilt for its own. Trivalent truth and conditional probability go together in Égré's paper; four-valued non-deterministic tables give the very weak modal systems a semantics in Pietruszczak's; a bilateral structure carries verification and falsification at once for Sedlár and Vigiani; and Tokuo's strict implication is read off the components of a quantum model.

What binds them is one constraint rather than one topic: a two-valued base is dropped, and the modal or conditional structure has to be re-derived rather than carried over. That constraint is why the week's results sit in different journals and different traditions while all circling the same question.

Three further papers in the same window belong to the same enquiry without sharing the value-space move. Pavel Naumov and Wenxuan Zhang define de re and de dicto forms of trust as modalities, prove that neither is definable from the other, and give a sound and complete system for both 12. Katsuhiko Sano and Sakiko Yamasaki prove Craig interpolation for indexed tense logics through analytic sequent calculi 13. Fadi Wakim analyzes the structure of regress arguments through fixed points and foundations, in a paper on when a regress blocks and why 14.

Two preprints of the same week take questions from the constructive and the classical side. Patrick Uftring answers a question of Frittaion, Nemoto and Rathjen positively, showing that independence of premise is admissible in IZF and in CZF with full separation, by a hereditary variant of the Friedman–Dragalin translation 15. Oskar Fiuk gives simpler proofs of the NExpTime-completeness and finite-model property for Maslov's class K and introduces an extension K+E with an equivalence relation, whose satisfiability problem he shows to be 2-NExpTime-complete 16.

Every count below was read on 11 October 2026.

None of the four featured papers has a citing work on any route that answered. OpenAlex returns zero for Pietruszczak's article and for Tokuo's 7, and Crossref records zero citing works for both 10. A batch request to the Semantic Scholar graph endpoint answered for Pietruszczak with zero citations against 11 references 17. DataCite returns zero for the Sedlár–Vigiani preprint, and the same batch request answered for it with zero citations against 31 references 18.

Two routes have a limit worth naming. Google Scholar holds a record for the Mind paper and shows three citing works for the work as a whole, which includes the preprint that has circulated since 2025 19; it holds no record yet for Pietruszczak's article, and none for the Sedlár–Vigiani preprint. The Semantic Scholar batch returned nothing for the Mind and IGPL DOIs, so those two proceed on Crossref and OpenAlex alone.

Where the numbers do say something is the literature each paper attaches to. Pietruszczak's work sits on a line of non-deterministic semantics that Omori and Skurt's 2021 modular semantics for T, TB, S4 and S5 also belongs to, with 13 citing works 20. Sedlár and Vigiani's base reaches back to Nelson's 1984 paper on constructible falsity and inexact predicates, with 283 citing works 21. Tokuo's setting is the oldest of the four, and Birkhoff and von Neumann's 1936 "The Logic of Quantum Mechanics" carries 2875 22.

A zero in the week of publication says very little. Pietruszczak's article went online on 8 October and Tokuo's on 5 October; the Sedlár–Vigiani preprint was submitted on 7 October. Nothing in these numbers separates a result nobody will build on from one that has not yet been read.

If you work on conditionals or on formal semantics, read Égré, Rossi and Sprenger first. It is the refereed article of the four, and the Reduction Theorem is the claim to test, because Stalnaker's Thesis, p-validity and the updating result all hang from it.

If you work on non-deterministic semantics or very weak modal logics, read Pietruszczak first. Section on the eight table sets is where the classification is decided, and the question to keep in view is what the natural conditions rule out.

If you work on relevant, Nelson or bilateral logics, read Sedlár and Vigiani first. The modular completeness proof is the part most likely to transfer, and the interesting detail is that two logics in the family had to be excluded.

If you work on quantum logic or on implication connectives, read Tokuo first. The internalization claim is the one to check against your own system, since it is what would let a quantum logic express its own entailment.

One question cuts across all four: what does an operator cost once the value space grows beyond two points? The trivalent base buys a probability theory for conditionals, the four-valued tables buy coverage of the weakest modal systems, the bilateral base buys a modality for the Nelson logics, and the quantum reading buys a connective that internalizes consequence. Each is a price paid in frame conditions, valuations or state structure.