Workshop on Logic and Its Engagement

	Wednesday (24 June)	Thursday (25 June)	Friday (26 June)
10:00 – 11:00	Thomas Ågotnes University of Bergen & Shanxi University Weakening Distribution over Conjunction in Modal Logics, with Two Applications in Epistemic Logic	Da Fan Shandong University Background Intervention and Modality	Graham Priest CUNY, UoM, RUB, SDU & Tohoku U. Two Objections to a Dialetheic Solution to the Semantic Paradoxes
	Coffee Break	Coffee Break
11:30 – 12:30	Kunhuan Li & Liwu Rong Shandong University Weak Denial in Layered Discourse Representation Theory	Sara Ayhan Tohoku University Constructive Negation Is not Constructive	Ching Hui Su Shandong University Conditionals as Question-Answer Pairs
		Lunch Break
13:30 – 14:30		Takako Nemoto Tohoku University Finiteness and Infiniteness in Constructive Mathematics	Hitoshi Omori Tohoku University Gentzen’s Idea in View of Non-Deterministic Semantics
	Coffee Break
15:00 – 16:00	Lun Zhang Shandong University The Metaphysical Difficulties of Mathematical Structuralism	Yì N. Wáng Shandong University Resource-Bounded PDL

A Note on Infinitesimal and Vagueness

Fei Liang

Shandong University

This note identifies a structural isomorphism between the concept of the infinitesimal in Zeno‑type paradoxes and the phenomenon of vagueness in sorites‑style paradoxes: both involve a series of imperceptibly small changes that, when accumulated, produce an abrupt qualitative shift. Just as an infinite sum of infinitesimals converges to a finite limit, the gradual removal of single grains or hairs eventually flips the truth of a vague predicate. This parallel suggests that mathematical techniques developed to handle infinitesimals—such as non‑standard analysis, limit arguments, and convergence criteria—can be transposed to model and resolve vagueness. By treating vague boundaries as hyperreal thresholds or continuous membership functions, the same logical machinery that tames the infinitely small offers a unified dissolution of the sorites paradox.

If Something Is Borderline, It Is Borderline Borderline – On Bobzien's idea of Columnar Higher Order Vagueness

Long Chen

Beijing Normal University

Traditional theories of vagueness for solving the sorites paradox often generate so-called higher order vagueness paradoxes. By distinguishing two notions of higher order vagueness, the hierarchical and the columnar one, Bobzien (2015,2024) has proposed an influential theory which doesn’t require one to relinquish classical logic or bivalent semantics to solve the sorites paradox, and more importantly, obviates all higher order vagueness paradoxes. In the modal logical system for dealing with columnar higher order vagueness S4M (or its first order extension QS4M+Fin), it is shown that something is borderline precisely when it is higher order borderline, and that a predicate is vague precisely when it is higher-order vague. In this talk we will re-examine the notion of columnar higher order vagueness, discuss its connection with the intuitionistic solution to the sorites and see whether it is more explanatory compared with other (such as the dynamic) notions of higher order vagueness.

Finiteness and Infiniteness in Constructive Mathematics

Takako Nemoto

Tohoku University

For a set A of natural numbers, there are at least five notions of finiteness:
FIN1: There are k and m_0,,,,m_{k-1} such that A={m_0,…,m_{k-1}};
FIN2: There is an upper bound for A;
FIN3: There is m such that for all B ⊆ A (|B| y such that x ∈ A;
INF5: For all m, there is B ⊆ A such that (|B|=m).
We show that none of the above are equivalent in constructive first order arithmetic with weak induction. As an example, we see the usage of finiteness and infiniteness in recursion theory.

Language Use and the Source of Vagueness

Boxuan Zheng

Shandong University

The dominant conception of vagueness treats it as a form of semantic under-determination. On this view, vague expressions fail to determine a unique boundary of application, and vagueness arises from the existence of multiple admissible precisifications. However, such accounts face significant difficulties. Philosophically, they threaten to undermine the contentfulness of ordinary language. Formally, they often struggle to preserve a fully compositional treatment of vague expressions.

Meanwhile, approaches that treat vagueness as a form of semantic over-determination have received comparatively little attention. On this kind of view, vagueness arises not because language determines too little, but because it determines too much. Very roughly, a vague predicate is associated with multiple admissible patterns of application rather than a single uniquely privileged one. Yet this approach has not been developed into a systematic account.

This paper develops a use-sensitive, bivalent theory of vagueness grounded in semantic over-determination. Drawing on a Wittgensteinian conception of language use, I argue that vagueness arises from the coexistence of multiple equally legitimate patterns of use. These patterns largely converge on paradigm cases while diverging at the boundaries, thereby generating multiple admissible boundaries of application. Because these patterns substantially overlap in their application and belong to a shared linguistic practice, the resulting phenomenon is vagueness rather than ambiguity.

Formally, the paper introduces an additional parameter U representing use patterns. This yields a compositional semantic framework capable of modelling the proposed account of vagueness. The framework explains the Sorites paradox without abandoning bivalence. I conclude that semantic over-determination offers a more satisfactory explanation of vagueness than traditional under-determination approaches.

Background Intervention and Modality

Da Fan

Shandong University

This paper examines whether a causal-modeling semantics for counterfactuals can also yield a plausible semantics for metaphysical modality. Standard interventionist causal models are too limited in language and too permissive in admissible interventions to explain necessity and possibility. This paper develops a modified interventionist framework in which modal facts depend not on every formally specifiable alteration, but on the range of admissible interventions grounded in the background structure of a model. On this view, necessity is understood as the absence of an admissible intervention realizing the contrary. The resulting framework preserves the connection between counterfactuals and modality while resisting an overly liberal recombination of possibilities.

The Metaphysical Difficulties of Mathematical Structuralism

Lun Zhang

Shandong University

This talk examines the metaphysical difficulties of realist mathematical structuralism. Starting from Benacerraf’s paper, it argues that mathematical objects are positions within structures rather than isolated entities. Although realist structuralism fits mathematical practice and helps explain the applicability of mathematics, it faces metaphysical problems of existence and identity, especially in symmetric cases such as i & −i. The talk considers responses from structural embedding, grounding, object theory, and category theory, suggesting that a categorical framework offers a more natural account of mathematical objects as determined by relations. In addition, the talk discusses a world structure proposal.

Resource-Bounded PDL

Yì N. Wáng

Shandong University

We introduce a propositional dynamic logic (PDL) designed to reason about scenarios involving resource-bounded program executions, such as program verification under strict resource constraints. This presentation is based on ongoing joint work with Xiaolong Liang.

Gentzen’s Idea in View of Non-Deterministic Semantics

Hitoshi Omori

Tohoku University

In the dissertation “Investigations into Logical Deduction”, Gentzen introduced systems for logic based on ‘introduction’ and ‘elimination’ rules, and left a remark: “introductions represent the ‘definitions’ of the symbols concerned, and eliminations are no more than the consequences of these definitions.” This idea of Gentzen was developed further by Prawitz who added that the introduction rule “may now be taken as the ‘meaning’ of the logical constant,” and also explicitly acknowledged that the question arises how to make Gentzen’s idea precise since the idea is quite vague. Against this background, the aim of this paper is to have a fresh look at Gentzen’s idea from the perspective of non-deterministic semantics, an elegant and natural generalization of many-valued semantics. This is a project with a strong exploratory nature, with interesting implications for both proof-theoretic and model-theoretic semanticists, and I will try to present them without assuming any familiarity with non-deterministic semantics. [Ongoing joint work with Sara Ayhan]

Constructive negation is not constructive

Sara Ayhan

Tohoku University

The intuitionistic negation has been frequently criticized for its non-constructive flavor. From a bilateralist point of view it seems to make sense to consider not only the notion of proof but also the notion of refutation if you work in a constructive setting. Such a basis is usually agreed upon to give rise to the so-called ‘strong negation’, introduced by David Nelson for his constructive logics N3 and N4. In this talk I want to share some thoughts on why I do not agree that this negation can be considered constructive.

Starting with the Forced March Sorites Paradox

Wen-fang Wang

Shandong University

I would like to promote the view that borderline sentences of a vague predicate are neither true nor false, but there are many objections to this view. I want to claim that all of them are not very plausible. However, to make my presentation simple, I will only consider the question of why many people take the considerations about the forced-march sorites paradox to be good reasons to reject my view. In particular, I will consider what Williamson, Raffman, and Priest take the lesson of the forced-march sorites paradox to be and argue that their views are not plausible.

Ontological Commitments of Composition as Identity and Irreducibly Plural Existents

Oleh Bondar

Shandong University

Composition as Identity (CAI) is usually taken to be a form of many-one identity: if some plurality yy is fused by a singular entity x, then x is identical to yy. Yet, as witnessed in the contemporary literature, the view that something can be both one and many is highly problematic. First, according to Yi (2014), “being one” and “being many” are incompatible notions. Second, assuming that the standard principle of plural logic, Plural Comprehension, is unrestrictedly valid, it appears very hard to say to exactly which plurality a given singular entity is identical. Third, even if we follow Sider’s (2014) proposal to restrict Plural Comprehension (and thus exclude pluralities of a certain form from our ontological picture), most of the existing theories that rely on the restricted form of Plural Comprehension seem defective in the following respect: they either fail to deliver the highly plausible view that x and yy are “the same portion of reality,” or they are committed to rejecting some principles of plural logic. However, there is a different way to express CAI that does not require restricting Plural Comprehension: we may take the notion of fusion to be irreducibly plural and express CAI in terms of the many, thus avoiding the problem of many-one identity.

Set-Theoretic Mereology As A Foundation Of Mathematics?

Speaker: Joel David Hamkins

Professor of Logic, University of Notre Dame
Guest Chair Professor, Peking University

Abstract: Mereology, the study of the relation of part to whole, is often contrasted with set theory and its membership relation, the relation of element to set. Whereas set theory has found success in the foundation of mathematics, since the time of Cantor, Zermelo and Hilbert, nevertheless mereology has been strangely absent. Why is this? In this talk, I shall introduce and discuss set-theoretic mereology, a form of mereology based upon the set-theoretic inclusion relation ⊆ rather than the element-of relation ∈. In particular, we shall investigate the role to be played by set-theoretic mereology in the foundations of mathematics, and come perhaps to an explanation of why it has been absent.

Modal Approaches towards Mereology

Speaker: Dazhu Li

Institute of Philosophy, Chinese Academy of Sciences
Department of Philosophy, University of Chinese Academy of Sciences

Abstract: Mereology in its formal guise is usually couched in a predicate-logical language whose signature contains a primitive binary predicate symbol representing the “part of” relation. In this talk, by employing a propositional modal language, we capture the properties of various mereological theories. We show that all the usual mereological theories can be captured by formulas in our language via frame correspondence. Also, we put forward an approach to mereology that uses the mereological sum as its primitive notion, and demonstrate that it is definitionally equivalent to the standard parthood-based theory of mereological structures, which lays a foundation for different modal characterizations of mereology with other semantic proposals.

Weakening distribution over conjunction in modal logics, with two applications in epistemic logic

Thomas Ågotnes

University of Bergen & Shanxi University

In normal modal logics the box modality distributes over conjunction. Weakening the equivalence in one direction we get non-monotonic modal logics. In the talk we look at a particular weakening of monotonicity by a rule we call the interpolation (or convexity) rule. We demonstrate an application in epistemic logic: the interpolation rule characterises the logic of “secretly-knowing”. Weakening the equivalence in the other direction we get non-adjunctive modal logics. We look at a particular weakening of conjunctive closure – “weak” conjunctive closure. Again, we show an application in epistemic logic: weak conjunctive closure characterises the notion of “somebody-knows”. In both cases we present completeness and other meta-logical results. The talk is based on joint works with Zuojun Xiong and with Yi N. Wang.

Dynamic Structuralism: To Be is to Be (Flexibly) Connected

Patrick Greenough

University of St Andrews

(Joint work with Krisztian Kos, Cambridge.)

Given Moderate Structuralism, an essentially networked entity (an ENE) is an entity that is defined in whole or part by its place within a constitutive network. Where such entities are subject to change, and so where this network is dynamic, a troubling problem emerges. Namely, The Problem of Over-Connection: a (structural or non-structural) change to one node in the network will impact all nodes. Why? Because the constitutive connections that hold between neighbouring nodes can be chained together to connect distant nodes. Hence, an ENE ends up being hypersensitive to all network change—no matter how small and distant. To resolve this problem, we deploy an insight from a debate regarding the phenomenal sorites paradox (due originally to Jackson (1973), Raffman (2000), and Graff (2000)). This is the insight that the relation “is phenomenally identical to” is, contrary to orthodoxy, a transitive relation. Upshot: there are no phenomenal continua (in the classical sense). Likewise, we argue that the relation “is constitutively connected to” is also a transitive relation; and, that there is a shifting cut-off point in a long series of putatively connected ENE. This shifting cut-off represents the point at which the network gives way in the face of two opposing forces: a force of autonomy (don’t be constrained by other things!); and a force of heteronomy (do be constrained by other things!). The result is an ever-restless, ever-stressed, dynamical system which never settles into a privileged equilibrium. A kind of Sturm und Drang Structuralism. (Cf. the Dynamic Structuralism of Hegel.) (I will try to illustrate this kind of Structuralism via an analogy with spin glass magnets.)

The Paradoxes of Meaning Change and Flexible Wholes

Speaker: Patrick Greenough

Affiliation: University of St Andrews

Abstract: Meaning Change is a ubiquitous feature of natural language. On the Orthodox View of meaning change, such change is numerical in that a word changes in meaning via having distinct meanings over time. This widespread view is threatened by a family of paradoxes which deploy the plausible claim that what is said by the utterance of, e.g., a sentence “Chocolate is healthy” is static over time. Thus, the conventional meaning of this sentence is static too. However, since it is also (initially) plausible that “Chocolate” has changed in meaning over time then we have a paradox (cf. Dorr and Hawthorne 2011). Furthermore, “Chocolate” is a mass term and there has scant work on how mass terms change in meaning. Ditto for species names such as “Canus Familaris” and group names such as “The Celts”, “The Conservatives”, “The French”. In this talk, I propose to deploy intensional mereology to make sense of meaning change for such terms. This proposal addresses the paradoxes by rejecting the premise that meaning change is numerical meaning change. Rather, “Canus Familiaris” has one and the same meaning over time—it is just that this meaning is an evolving, flexible entity. That is because the meaning of “Canus Familaris” just is an intensional, evolving object. This allows us to draw an important distinction between very different kinds of meaning change: (1) Use-First Meaning Change: in virtue of a change in the use of a word, and (2) Reality-First Meaning Change: in virtue of a change in the intensional object being picked out by a word. I close by considering how these forces of change may interact and loop.

Conditionals as Question-Answer Pairs

Ching Hui SU

Shandong University

At the current stage of the development of conditionals, in both philosophy and logic, divergences remain unsettled. Various accounts have been proposed, ranging from strict implication (C. I. Lewis, 1912) and conditional probability (E. Adams, 1965) to variably strict conditionals (R. Stalnaker, 1968; D. Lewis, 1973), premise semantics (A. Kratzer, 1977), and interventionism (D. Galles and J. Pearl, 1998). However, none have silenced the others. The present paper aims to provide a promising account that can supersede the others. We treat conditionals as question-answer pairs (or issue-resolution pairs), where the antecedent (taken as a salient supposition of the speaker) and background knowledge are used to pose a question (issue) to the hearer, with the consequent providing the speaker’s answer (resolution). We demonstrate that this approach is more general in its ability to express the connections between the antecedents and consequents than many other approaches. Furthermore, this approach can explain (at least) two puzzles surrounding conditionals: Inconsistent Consequents pairs and Missing Link Conditionals. Our approach provides a straightforward solution: when the question raised by some IC pairs is a polar question, the law of conditional excluded middle is taken as true. However, when the question raised by other IC pairs is a wh-question, the intuition to accept multiple answers comes from the fact that wh-questions allow for this. Regarding the missing link conditionals, we conclude that either the answer provided by the consequent is not a qualified answer to the speaker’s question (e.g., “1 + 1 = 2” is not an adequate response to “who is going to the party?”), or the supposition offered by the antecedent plays no part in determining the truth value of the conditional (e.g. the fact that raccoons have no wing is irrelevant to whether they can breathe under water or not). Finally, we explore an issue concerning the ambiguity of conditional sentences.

Weak Denial in Layered Discourse Representation Theory

Kunhuan Li & Liwu Rong

Shandong University

Abstract: In this paper, the treatment of denial in layered discourse representation theory (LDRT) is revisited. In LDRT, denial reduces contextual content, whereas multilateralists identify weak denial, a speech act that blocks the addition of content without removing any. We show that LDRT undercounts such cases: new content is blocked from becoming a common belief, yet it cannot be easily removed when reasons remain unspecified. Drawing on multilateralist insights, we recast the contextual effect of denial in LDRT as update-blocking and treat the star condition as its formal marker, ensuring that weak denial is represented as having occurred even before any withdrawal. This preserves LDRT’s representational virtues while closing its explanatory gap.

Keywords: Discourse Representation Theory; Denial; Speech Acts; Informal Reasoning

Ibn Sı̄nā’s Burhān Al-Ṣidiqīn

Speaker: Graham Priest

Affiliations: City University of New York, University of Melbourne, Ruhr University Bochum, Shandong University & Tohoku University

Abstract: One of the most important of Ibn Sīnā’s innovations in the history of Western philosophy was an argument for the existence of a necessary existent (wajīb al-wajūd). The argument is known as the burhān al-ṣiddiqīn (demonstration of the truthful). In this talk I analyse and evaluate the burhān using the tools of contemporary logic and mereology.

Two Objections to a Dialetheic Solution to the Semantic Paradoxes

Graham Priest

City University of New York, University of Melbourne, Ruhr University Bochum, Shandong University & Tohoku University

Saul Kripke’s 1975 paper on truth marked a turning point in research on the semantic paradoxes towards languages which contain their own truth predicate. One such approach is a dialetheic one. In my experience, the two most common objections to this (other than a blanket rejection of dialetheism) concern (a) Curry’s paradox, and (b) the “truth-only” problem. I find neither of these persuasive. In this talk I will explain why.

Preliminary Program