Pablo Cobreros
Title: Global Spring Sprouts
Abstract: This talk is about metainferences and global validity (oh no, Pablo; again??!!) In recent work Kortenbach predicts the beginning of the global spring in the literature on metainferences –-the local winter is over. The global spring comes with Kortenbach's defense of schematic global by showing how to understand this notion in relation to:
1. The hierarchy of metainferences (in a global-global way),
2. Multiple conclusions,
3. Hybrid metainferences,
4. Transfinite levels
In this talk I'd like to promote the arrival of global spring exploring how these items can be addressed for the P-global case (my preferred version of global validity). I have nothing to say about transfinite levels —as I have no idea about the topic— but I think P-global works well for items 1 and 2, perhaps for 3 and adding a 5th: the addition of modalities. As a bonus, although the initial motivation for P-global validity comes from the semantics, it doesn't look difficult to formulate labelled calculi for it.
Francesco Paoli
Title: TOPIC-TRANSFORMATIVE CONNECTIVES VIA ACTIONS ON SEMILATTICES
Abstract: Within the Yablovian approach to topicality (Yablo 2014), a distinction
is drawn between thin propositions—individuated solely by their truth
conditions, for instance as sets of possible worlds—and thick
propositions—individuated by both their truth conditions and their
subject matter. In the same framework, logical connectives are generally
taken to be topic-transparent: the subject matter of a sentential
compound is simply the mereological fusion of the subject matters of its
component subsentences. Fusion is in turn modelled as a semilattice
join.
In 2C-semantics (Berto 2022, Hawke et al. 2024) thick propositions are
represented as members of direct products of an algebra of thin
propositions (typically, a Boolean algebra) and a semilattice of topics.
Other approaches achieve greater flexibility by using fibred systems of
algebras of thin propositions over an indexing semilattice of topics
(Goodman 2019, Tedder 2026). In all these frameworks, the
topic-transparent character of connectives is taken to be a fundamental
feature of the model.
However, it has been argued (Angell 1989, Fine 2016, Ferguson 2023,
Randriamahazaka 2024, Paoli et al. in press) that at least some logical
connectives (also in substructural logics) are topic-transformative in
nature. Accommodating this feature calls for a different framework. In
this talk, we investigate monoidal actions on the semilattice of topics,
which induce actions on the aforementioned fibred systems of algebras of
thin propositions. On this approach, topic-transformative connectives
are not on a par with topic-transparent ones: rather, they correspond to
external “perturbations” of algebras of propositional contents whose
topical component is built via mereological fusion only. As an
illustration, we focus on three phenomena: topical polarity, topic
ampliativity, and topic cancellation.
References
Angell R.B. (1989), “Deducibility, entailment and analytic containment”.
In: J. Norman, R. Sylvan (Eds.), Directions in Relevant Logic, Kluwer,
Dordrecht, pp. 119–144.
Berto F. (2022), Topics of Thought: The Logic of Knowledge, Belief,
Imagination, Oxford Academic, Oxford.
Ferguson T.M. (2023), “Subject-matter and intensional operators I:
Conditional agnostic analytic implication”, Philosophical Studies, 180,
7, pp. 1849-1879.
Fine K. (2016), “Angellic content”, Journal of Philosophical Logic 45,
2, pp. 199–226.
Goodman J. (2019), “Agglomerative algebras”, Journal of Philosophical
Logic, 48, pp. 631–648.
Hawke P., Hornischer L., Berto F. (2024), “Truth, topicality, and
transparency: One component versus two-component semantics”, Linguistics
and Philosophy, 47, pp. 481–503.
Paoli F., Szmuc D., Zirattu M. (in press), “The algebra of analytic
containment”, Journal of Logic, Language, and Information.
Randriamahazaka T. (2024), “De Morgan-Płonka sums”, Studia Logica, 112,
pp. 1343–1371.
Tedder A. (2026), “Topics in relevant logic: A semantic perspective”,
Erkenntnis, 91, pp. 23-51.
Yablo S. (2014), Aboutness, Princeton University Press, Oxford.
Andrea Iacona
Title: Alethic Pluralism and Weak Kleene Logic
Abstract: This paper argues that there is a close connection between alethic pluralism and Weak Kleene logic, or Kw3 , the logic associated with Weak Kleene trivalent semantics. Given some plausible assumptions, grounded in structural features of alethic pluralism and compatible with a purely bivalent framework, Kw3 emerges as the most natural logic for alethic pluralism.
Agustina Borzi
Title: Sequent calculi for some containment logics
Abstract: This talk touches on the proof theory of variable inclusion (or containment) logics. In particular, it introduces new sequent calculi for right and left variable inclusion companions of Classical logic, broadly understood as proper subsystems of Classical logic in which the variables of the conclusions are contained in those of the premises, and vice versa. The starting point are sequent calculi for Paracomplete and Paraconsistent Weak Kleene, two systems closely related to containment logics in that they impose variable inclusion constraints on valid inferences, albeit only in the absence of (anti)theorems. Two strategies are then proposed to obtain subsystems of them that can be regarded as genuine containment logics: modifying the rules of negation, which yields calculi sound and complete for the pure companions of Classical logic, both paracomplete and paraconsistent logics, and restricting Weakening, which produces non-monotonic systems sound and complete for the analytic and synthetic companions of Classical logic.
Bogdan Dicher
Title: Cutless cuts
Abstract: I investigate rules for the conditional in a sequent calculus and their adequacy for expressing the ST-conbditional. I show that the number of Cut applications required to derive a rule for the conditional from the modus ponens sequent is too coarse a measure of its admissibility in ST. Instead, the decisive factor is whether a rule's internalised composition remains governed by the principal conditional after analytic decomposition, or whether it detaches the intermediate formula and composes across the strict–tolerant asymmetry without semantic backing.
Robin Martinot
Title: Syntax-rich proof formalisms and semantic pollution
Abstract: Syntax-rich formalisms are increasingly popular for logics that do not admit ‘traditional’ proof systems with satisfactory properties. Consider display calculi, that have been constructed as a way of combining modal, temporal, substructural and other logics; labelled calculi, that are successful for modal and intuitionistic logics but also for relevant logics and substructural logics; and bilateral calculi, that are used to characterize classical but also inconsistency-tolerant logics. Surrounding such syntax-rich formalisms, the phenomenon of semantic pollution has gained recent attention in the philosophy of proof theory (Read, 2015; De Martin Polo, 2024; and more). Semantic pollution is primarily
considered to occur in labelled proof systems, which can be seen as internalizing Kripke semantics in the syntax of the proof system. The behavior of semantic pollution in more general proof-theoretic syntax extensions is underexplored. Based on a formal measure of semantic pollution (M., 2026), we will compare the kinds of semantic pollution occurring in labelled calculi, display calculi and bilateral calculi. Our case studies include modal logics and inconsistency-tolerant logics such as FDE, N4 and Abelian logic (and their negation-free fragments). The results show different levels of semantic pollution, depending on the types of syntax used in the proof system, but also on the semantic principles used.
References
De Martin Polo, Fabio (2024). “Beyond Semantic Pollution: Towards a Practice-Based Philosophical Analysis of Labelled Calculi”. In: Erkenntnis, pp. 1–30.
Martinot, Robin (2026). "A Formal Characterization of Semantic Pollution of Modal Proof Systems". In: Journal of Philosophical Logic, forthcoming.
Read, Stephen (2015). “Semantic pollution and syntactic purity”. In: The Review of Symbolic Logic 8.4, pp. 649–661.
Martina Zirattu
Title: Topical Recapture for Content Inclusion Logics
Abstract: Content inclusion logics form a family of systems whose consequence relation admits a Yablovian reading (Yablo, 2014), on which content inclusion amounts to implication plus preservation of topic. Accordingly, the notion of validity proper to these systems is commonly labeled topic-sensitive, in opposition to the standard topic-neutral conception, on which logic preserves truth irrespective of topic distinctions. In this work I propose a procedure for "recapturing" topic-neutrality within such topic-sensitive consequence relations – that is, for recovering topic-neutral inferences within a topic-sensitive implication. The strategy consists in enriching the language of content inclusion logics with topicality operators, which allow one to express whether a formula is about a given topic.
Davide Botticchio
Title: A Non-binary Naïve Validity Theory
Abstract: In this paper, by adding a validity predicate Val to a propositional language, we develop a validity
theory that fully and correctly captures its metalinguistic validity. More precisely, for any inference,
metainference, and higher-order metainference, they are valid (invalid) iff the object language V al-
sentences that express their validity (invalidity) are true in every model. We precisely formalize
these desiderata with the so-called V -schemata. For example, for inferences, an inference is valid
(Γ ⊨ ∆) iff the V al-sentence that expresses this fact is true in every model (⊨ V al(⌜Γ⌝, ⌜∆⌝)), and
an inference is invalid (Γ ⊭ ∆) iff the negated V al-sentence that expresses this fact is true in every
model (⊨ ¬V al(⌜Γ⌝, ⌜∆⌝)). We generalize this schemata for higher level n-metainferences, for every
n. We say that a validity theory is na ̈ıve iff it satisfies all the V -schemata. This project faces two
challenges. First, weak validity and logical principles are inconsistent because of the well-known
Validity Paradox (Beall and Murzi [4]). Second, all existing validity theories, even those that avoid
the paradox, get some features of validity wrong. More precisely, we show that, in all validity theories
based on a logic with a bivalent validity (i.e. in which inferences and n-metainferences are either
valid or invalid, and not both), either the V -schema for valid inferences or V -schema for invalid
inferences must fail. To overcome this limitation, we build a validity theory based on non-binary
logics, logics in which inferences and n-metainferences can be both valid and invalid, or neither of
the two (Pailos [8], Barrio et al. [1]). We define a logic, which we call L̸⊨⊨ / ⊨ ̸⊨st/ts , that has some valid and invalid inferences and n-metainferences, for every n.
We show that L̸⊨⊨ / ⊨ ̸⊨st/ts has same validities as the logic STω, and so the same validities as classical logic (Barrio et al. [3], Pailos [6]),
and the same invalidities as TSω, and so no non-invalidities in the language without propositional
constants (Pailos [7], Barrio and Pailos [2]). We build a fixed-point interpretation ΦV al for the
validity predicate taking inspiration from Meadows [5]. We define the validity theory LV ̸⊨⊨ / ⊨ ̸⊨
st/ts by taking the Strong Kleene valuations vV al of the fixed point ΦV al and evaluating inferences and
n-metainferences according to the logic L̸⊨⊨ / ⊨ ̸⊨st/ts . We prove that this non-binary validity theory is
consistent and na ̈ıve: it avoids paradoxes and satisfies all the V -schemata. We argue that L̸⊨⊨ / ⊨ ̸⊨st/ts
shares some features with other non-classical logics. It is analogous to STω in its capability to
deal with paradoxes while maintaining classical validities. The relation between its validities and
invalidities mirrors the relation between truths and falsities in LP. It shares some features with T S
in the analysis of what goes wrong in paradoxical reasoning.
References
[1] Eduardo Barrio, Camillo Fiore, and Federico Pailos. Non-bivalent validity: E. barrio et al. Studia
Logica, pages 1–35, 2026.
[2] Eduardo Barrio and Federico Pailos. Validities, antivalidities and contingencies: A multi-
standard approach. Journal of Philosophical Logic, 51(1):75–98, 2022.
[3] Eduardo Alejandro Barrio, Federico Pailos, and Damian Szmuc. A hierarchy of classical and
paraconsistent logics. Journal of Philosophical Logic, 49(1):93–120, 2020.
[4] Jeffrey Charles Beall and Julien Murzi. Two flavors of curry’s paradox. The Journal of Philos-
ophy, 110(3):143–165, 2013.
[5] Toby Meadows. Fixed points for consequence relations. Logique et Analyse, 227:333–357, 2014.
[6] Federico Pailos. A fully classical truth theory characterized by substructural means. The Review
of Symbolic Logic, 13(2):249–268, 2020.
[7] Federico Pailos. Empty logics. Journal of Philosophical Logic, 51(6):1387–1415, 2022.
[8] Federico Pailos. Na ̈ıve non-substructural solutions to the validity paradox. Synthese, 206(1):39,
2025.
Shuwen Wu
Title: Substructural Theories of Truth: Fine-Grained Distinctions and Classical Validity
Abstract: Truth is a central topic in philosophy and logic. Since Alfred Tarski in troduced formal theories of truth ([1]), a wide range of formal accounts have been developed. The study of truth and semantic paradoxes has also motivated various non-classical approaches that aim to accommodate a transparent truth predicate while preserving classical reasoning (e.g. [2]; see also [3]). In recent years, substructural theories of truth have emerged as an approach to semantic paradoxes (e.g. [4–6]). The core idea is that paradoxical reasoning depends on certain structural rules; by restricting these rules, one can block paradox without trivializing logic. More specifically, substructural approaches restrict rules such as contraction, transitivity, or reflexivity, developing along different lines. This paper offers a systematic review of these substructural approaches and argues that, in addition to blocking paradoxical derivations, they provide new ways of understanding logical consequence. On the one hand, they shift atten tion from truth values to patterns of inference. As a result, they make visible distinctions that remain unnoticed in more familiar non-classical frameworks, allowing us to distinguish between semantic notions that would otherwise co incide in standard settings. At the same time, we examine the claim that non-transitive theories of truth preserve classical validity. While such preservation is established in prominent strict–tolerant frameworks (cf. [7]), we argue that this claim requires qualifica tion. In particular, there exists a class of sentences for which some classically valid inferences fail in strict–tolerant theories of truth (STTT; see [5, 8]), with out generating paradox. This result is particularly striking in light of the fact that, in the absence of a truth predicate, the underlying strict–tolerant logic coincides with classical con sequence (cf. [7, 9]). The failure therefore does not stem from non-transitivity alone, but from its interaction with the expressive resources introduced by the truth predicate. In this sense, it is precisely the introduction of the truth predicate that gives rise to these failures of classical validity. The main conclusion is that substructural approaches offer a fruitful frame work for addressing semantic paradoxes while preserving important aspects of classical reasoning. At the same time, their behavior in certain non-paradoxical cases shows that the preservation of classical validity is more limited than is sometimes assumed. Taken together, these results explain the enduring appeal of formal theories of truth, while also showing that the interaction between structural rules and the truth predicate remains in need of further investigation.
References
[1] Tarski, A. (1933). The concept of truth in formalized languages. Logic, Semantics, Metamathematics, 152–278. [2] Field, H. (2008). Saving Truth from Paradox. New York: Oxford University Press. [3] Leitgeb, H. (2007). What theories of truth should be like (but cannot be). Philosophy Compass, 2(2), 276–290. [4] Zardini, E. (2011). Truth without contra(di)ction. Review of Symbolic Logic, 4(4), 498–535. [5] Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching trans parent truth. Mind, 122(488), 841–866. [6] French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, 3. [7] Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378. [8] Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 2(13), 299–328. [9] Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, clas sical, strict. Journal of Philosophical Logic, 41(2), 347–385
Gideon Noß
Title: Theoretical virtues of substructural validity
Abstract: Semantic paradoxes arise in every sufficiently expressible language. They question our beliefs about rationality and valid reasoning, as they seem to lead from acceptable premises to unacceptable conclusions with otherwise acceptable reasoning. Substructural approaches to semantic paradox provide explanations of the paradox based on the denial of the validity of certain structural rules of reasoning such as transitivity, contraction, monotonicity or reflexivity. Opposing views claim that these principles cannot simply be denied as they constitute necessary properties of logical consequence. Further, as there is no consensus as to which structural rules are at fault to lead to paradox, abandoning some simply for the sake of avoiding paradox appears to be an ad hoc theory choice. In contrast, proponents of substructural logic might highlight the unificatory strength of their approaches. I take these schematic abductive arguments as my starting point and argue as they stand, neither one is satisfactory. In order to see the value of substructural approaches to semantic paradox, I analyze the applicability of a more comprehensive set of theoretical virtues such as empirical accuracy, consistency, explanatory scope or simplicity. It will become clear that these general criteria still are not sufficient to draw a conclusion. I propose that in the case of logic or semantic paradox, the set of theoretical virtues should be expanded by more theory specific virtues, aiming at a more conclusive inference to the best explanation. As an example set I consider several desiderata suggested by Hannes Leitgeb. The aim is to both consider suitable theoretical virtues for theory choice in logic, and evaluate how some substructural approaches fare among them. While the set of relevant theoretical virtues might not yet be complete, the idea is that an evaluation of a broader and more logic specific set might highlight the value of substructural validity
Matthew McClure
Title: Humble connexivity on the Bochum Plan
Abstract: To a first approximation, a connexive logic is one which finds valid
(Aristotle) ¬(𝐴 → ¬𝐴)
(Boethius) (𝐴 → 𝐵) → ¬(𝐴 → ¬𝐵)
and related principles. A very natural connexive logic is 𝐂 (Wansing 2005). It has a Kripke-style semantics based on first degree entailment logic. Where ‖− is verification and −‖ falsification (by a point), the clauses for → are:
• 𝑤‖− 𝐴→𝐵iff,forall 𝑢 with 𝑤𝑅𝑢, if 𝑢 ‖− 𝐴 then 𝑢 ‖− 𝐵
• 𝑤−‖ 𝐴→𝐵iff,forall 𝑢 with 𝑤𝑅𝑢, if 𝑢 ‖− 𝐴 then 𝑢 −‖ 𝐵
where 𝑅 is a partial order with:
• if 𝑤 ‖− 𝑝 and 𝑤𝑅𝑢 then 𝑢 ‖− 𝑝
• if 𝑤 −‖ 𝑝 and 𝑤𝑅𝑢 then 𝑢 −‖ 𝑝
and ¬,∧,∨ are as in 𝐅𝐃𝐄. 𝐂 is the flagship of the Bochum Plan, on which connexivity is achieved by a novel falsification clause for →. We might think that (Aristotle) and (Boethius) don’t by themselves capture con nexivity. Kapsner (2012) suggests that the underlying intuitions are, roughly, that (AInt) 𝐴 can’t imply ¬𝐴. (BInt) If 𝐴 implies 𝐵 then it can’t imply ¬𝐵. So we require also the strongly connexive principles
(ASat) 𝐴 → ¬𝐴isunsatisfiable.
(BSat) 𝐴 → 𝐵 and𝐴 →¬𝐵
are jointly unsatisfiable. Where ⊥ is an absurdity constant, we can write:
(A⊥) (𝐴→¬𝐴)→⊥
(B⊥) (𝐴 →𝐵) →((𝐴→¬𝐵)→⊥)
A logic validating (A⊥) and (B⊥) we might call ⊥-connexive (Kapsner & Omori 2022). The Bochum Plan logics are not strongly connexive. Neither are their extensions by ⊥ (clauses: 𝑤 /‖− ⊥, 𝑤 −‖ ⊥) ⊥-connexive. But strong/⊥-connexivity is too strong. Consider: 𝐴∧¬𝐴→𝐴 𝐴∧¬𝐴→¬𝐴 Since the Bochum Plan logics validate these (very) attractive principles and the con flicting (Aristotle) and (Boethius), they end up inconsistent. But on ⊥-connexivity, we can’t have it both ways—given modus ponens, the tension yields absurdity. Kapsner (2019) suggests we thus restrict the strong and ⊥-connexive principles to instances where certain subformulae are possible or consistent; this he calls humble connexivity. I argue that we can achieve humble connexivity on the Bochum Plan. Two flavours of the humble idea are salient. One is that we insist on the strong prin ciples only when the salient formulae are satisfiable, so (BSat) would become
(BSatH) 𝐴 → 𝐵and𝐴 →¬𝐵
are jointly unsatisfiable whenever 𝐴 is satisfiable. Another is that we do so only when they don’t take a glutty truth-value. This glutty reading we can express using the object language operator C𝐴 ∶= 𝐴 ∧ ¬𝐴 → ⊥, yielding a glutty-humble form of (B⊥) in the sequent
(BCH) C𝐴,C𝐵,C(𝐴 → 𝐵) ≻ (𝐴 →𝐵) →((𝐴 →¬𝐵)→⊥)
which is valid in the ⊥-extension of 𝐂, 𝐂⊥. Given the inconsistency of Bochum Plan logics, C(𝐴 → 𝐵) in (BCH) is quite limit ing; it’s more natural, I think, to phrase the restriction solely in terms of 𝐴 and 𝐵. By blending the satisfaction and glutty flavour of humility, we can get what we need. The suggestion is this. If we think of a formula as being satisfiable with respect to some model as its being verified by some point in the model, we can, I claim, ade quately express this in the object language by means of the operator S:
• 𝑤‖− S𝐴 iff some 𝑢 with 𝑤𝑅𝑢 has 𝑢 ‖− 𝐴
• 𝑤−‖ S𝐴iff, for all 𝑢 with 𝑤𝑅𝑢, 𝑢 −‖ 𝐴
giving us 𝐂⊥S, which delivers
(BSCH) S𝐴,C𝐵 ≻ (𝐴 →𝐵)→((𝐴→¬𝐵)→⊥)
(ASCH) S𝐴,C𝐴 ≻ (𝐴 →¬𝐴)→⊥
where S doesn’t occur in 𝐴,𝐵. This form of humble connexivity in 𝐂⊥S sheds some interesting light on the bar bershop paradox (Carroll 1894). If I have time, I want to tentatively suggest that the moral of the paradox is that humble connexivity might be too strong too.
References
Carroll, L. (1894). A logical paradox. Mind, 3(11), 436–438.
Kapsner, A. (2012). Strong connexivity. Thought: A Journal of Philosophy, 1(2), 141 145.
Kapsner, A. (2019). Humble connexivity. Logic and Logical Philosophy, 28(3), 513–536.
Kapsner, A., & Omori, H. (2022). Superconnexivity reconsidered. Proceedings of the NCL 2022, 160–173.
Wansing, H. (2005). Connexive modal logic. In R. Schmidt, I. Pratt-Hartmann, M. Reynolds, & H. Wansing (Eds.), Advances in modal logic (pp. 387–399, Vol. 5). College Publications.