Existence and Nonexistence in the History of Logic: Afterword

This two-part special issue^[1] on Existence and Nonexistence in the History of Logic collects the proceedings of the Second Pan-American Symposium on the History of Logic: Existence and Nonexistence.^[2] In spirit and aims, this event followed the Inaugural Pan-American Symposium: Validity throughout History, ^[3] bringing together historians and philosophers of logic with expertise on chronologically, geographically, linguistically, and culturally diverse traditions.

The Pan-American Symposia are recurring meetings, each focusing on a core technical notion in the history of logic. The intention is to provide an opportunity for historians of logic to come together in the Americas and entertain conversations on the history of logic qua logic, without any restrictions on a particular historical period or geographical region. We aim to promote a global and comparative approach to logic’s past, taking this to be valuable for its own sake and also for a better understanding of logic’s present and future. A cross-traditional awareness of the historical developments shaping logical notions, theories, and techniques can widen the often too-narrow views about what counts as logic. At the same time, such an awareness helps mitigate anachronistic or ahistorical interpretations of past views while demonstrating their theoretical relevance today.

These are ambitious aims that may not be achievable all at once, and we don’t presume to have fully achieved them ourselves. But we do hope to have contributed to a collective enterprise to promote meaningful conversations among experts. The various contributions address a wide range of periods, traditions, and methodologies, but they all ultimately address similar fundamental notions and questions within the discipline of logic and its history.

For the second Pan-American Symposium, the common theme uniting all contributions was “existence and nonexistence,” which encompasses subtopics such as existential import, empty reference, logical and linguistic puzzles involving fictional/imaginary/impossible beings, modality and existence, nominalism, quantification, and contradiction. This theme addresses the most fundamental building blocks of semantics – how to refer to what is, what is not, what can be, or what cannot be – thereby exploring the intersection of logic and ontology in various traditions.

The event was inspired by, and these proceedings are dedicated to, the legacy of Professor Terence (Terry) Parsons (1939–2022).^[4] Terry’s interests and contributions were very wide-ranging. He was, inter alia, a champion parachutist and author of the 1987 Canopy Relative Work for Skydivers. His published academic work (there was much that circulated but he never bothered to publish) included numerous articles and several books in semantics and metaphysics, including Events in the Semantics of English and Indeterminate Identity. Issues about existence were already central to his work in his 1966 PhD dissertation and in published work dating back to 1967. Terry pioneered new directions in thinking about the relations among existence, predication, quantification, and reference. His 1980 book Nonexistent Objects is a fundamental treatment of the topic.

The articles in this two-part special issue reflect this latter focus. Some take up Terry’s views explicitly and in detail. Both Graham Priest (in Volume 7, 2024) and Ed Zalta (in Volume 8, 2025) treat in detail the distinction between “nuclear” and “non-nuclear” properties introduced by Alexis Meinong and Ernst Mally and revived after nearly a century of neglect by Richard Routley and Terry. Both find it limited (in different ways) but both insist on the fundamental importance of Terry’s work in the area. As Ed Zalta puts it (and he would be speaking for many of us), “Parsons’ work thereby opened up a vista onto axiomatic metaphysics and cleared an interesting path for the systematic study of fundamental philosophical issues related to the problem of existence and nonexistence.”

In the late 1990s, Terry began to work more extensively in the history of logic, especially medieval logic. He had for some time been interested in historical explorations of the liar paradox and the inexpressible, but his interest deepened and became more central. He began to collaborate with others on translations of medieval logical texts (first an anonymous Tractatus de proprietatibus Sermonum published in Topoi in 1997) and at nearly the same time to explore two issues which became increasingly important to him: what to make of the traditional square of opposition, with its thesis that affirmative sentences had existential import while negative ones did not, and whether Augustus De Morgan and others were right to see medieval logic as deductively and expressively impoverished in ways more recent logics were not. These issues led to nearly a dozen articles, collaboration on the translation of arguably the most widely used medieval logic text, Peter of Spain’s Tractatus, and to his groundbreaking Articulating Medieval Logic, in which he shows that medieval logic had the resources of contemporary quantification theory and the power to express the Peano axioms for arithmetic.

Many of these interests of Terry’s are reflected in this special issue, where we find, in addition to those mentioned above, essays on Aristotle’s treatment of existential import (by Byeong-uk Yi (Volume 8)), Avicenna’s treatment of nonexistent objects (by Allan Bäck (Volume 8) – which not only lays out Avicenna’s picture in detail but also compares it closely with Terry’s), Boethius of Dacia’s treatment of tense and its existential force (by Mary Sirridge (Volume 7) – again in close dialogue with Terry’s), John Mair’s treatment of existential import (by Guido Alt (Volume 8)) and Jens Lemanski’s (Volume 7) discussion of these issues in visual reasoning. Even where Terry’s immediate influence is less obvious – as in the articles by Bo Mou (Volume 8) and Fenrong Liu and Zhiqiang Sun (Volume 8) on Ancient Chinese Logic, Heine Hanson’s (Volume 7) discussion of Peter Abelard’s Logic, and Dave Beisecker’s (Volume 7) account of Lewis Carroll’s treatment of the Sorites – one can sense Terry’s interests not far below the surface.

A personal note: Terry was not only a path-breaking philosopher, but also a generous mentor, teacher, and colleague. Whether through his inspiring example (in Graziana’s case), his patient instruction (in Milo’s), or his impactful collaboration (in Calvin’s), we three each had the benefit of his warmth.

In what follows, we provide brief synopses of each of the eleven articles within this two-part special issue, presenting them here in more or less chronological order:

Fenrong Liu and Zhiqiang Sun (Volume 8) examine the term zhou (周) in Mozi’s Xiaogu chapter (c. fourth- to third-century BCE), a foundational text of Mohist logic. The study argues that zhou, when preceding verb–object constructions, functions as a universal quantifier, while you (有) serves as an existential quantifier. This interpretation challenges prior scholarship that equated zhou with the Western logical concept of the term “distribution,” demonstrating that such analogies fail to account for Classical Chinese syntax and Mohist textual specifics. The authors analyze zhou’s linguistic usage across the Mozi and its role in resolving the validity of shi er ran (是而然) versus shi er bu ran (是而不然) reasoning patterns. In doing so, they provide a systematic account of quantification in mou (侔) reasoning. Their application of monotonicity principles from natural logic offers a unified framework to explain why inferences succeed or fail based on quantifier scope, without resorting to ad hoc intentional analyses. This approach clarifies how Mohist logic navigates context-dependent quantification, grounding interpretations in both textual evidence and formal logical analysis rather than relying on cross-cultural conceptual mappings.

Byeong-uk Yi (Volume 8) re-examines the treatment of existential import in Aristotle’s logic. His reading challenges the two prevalent interpretations, namely, the affirmative-negative doctrine (affirmatives imply subject existence, negatives do not) and the non-vacuity interpretation (Aristotle’s syllogistic excludes empty terms). Instead, Yi argues that these frameworks misalign with Aristotle’s texts. He proposes a no-import interpretation, asserting that Aristotle ascribed existential import neither to singular/general affirmatives (e.g., “Homer is a poet”) nor to negatives. Central to Yi’s argument is textual evidence showing Aristotle’s acceptance of obversion and contraposition (e.g., “No A is B” as equivalent to “Every A is non-B”), which undermines the claims of the affirmative-negative doctrine. Yi further distinguishes logical implications from semantic entailments, clarifying that Aristotle’s discussions (e.g., Categories 10 vs De Interpretatione 10–11) address context-specific semantic features (e.g., “sick” implying existence) rather than universal logical rules. This reinterprets Aristotle’s logic as systematically rejecting existential import across categoricals, offering an alternative to traditional and modern readings. Yi’s analysis reframes key Aristotelian theses, challenging common interpretations about vacuous terms and the square of opposition in ancient logic and beyond.

Bo Mou (Volume 8) examines three semantic paradoxes – the White-Horse-Not-Horse, Ultimate-Unspeakable, and Liar paradoxes – within the philosophy of language and logic, bridging classical Chinese philosophy and modern Western analytic logic. Mou’s primary contribution lies in proposing a unified referential framework to address these paradoxes, arguing they share a common structure rooted in the “double-reference” feature of language: primary reference to an object as a whole and secondary reference to specific attributes. Besides offering cross-traditional insights, Mou reframes paradox resolution by prioritizing referential over predicative contradictions, preserving the principle of non-contradiction through layered reference. For example, the White-Horse paradox is dissolved via complementary identity aspects, the Ultimate-Unspeakable via contradictory dimensions of the Dao, and the Liar by separating truth/falsity attributes. In contrast with fragmented solutions, Mou provides a unified account of paradoxes typically treated in isolation. Diverging from most dialetheist approaches, this treatment considers contradictions to be referential and resolvable through analysis, rather than ontological.

Allan Bäck (Volume 8) analyzes the interplay between Avicenna’s medieval Islamic metaphysics (eleventh–twelfth century) and Terry Parsons’s formal semantics. Focusing on Avicenna’s threefold distinction of quiddity (in se, in re, in intellectu) and Parsons’s theory of nonexistent objects, the study challenges the notion that these frameworks represent incommensurate paradigms (per Kuhn), instead situating them within a shared research tradition (per Laudan). Bäck argues that Avicenna’s and Parsons’s approaches structurally align. Avicenna’s metaphysics emphasizes semantic grounding (e.g., truth-makers for propositions about nonexistents via ampliated temporal reference) while Parsons prioritizes a syntactic approach. Yet, Bäck argues that quiddities in intellectu parallel Parsons’s domain of nonexistent referents, and Avicenna’s modalities (logical vs physical necessity) map onto Parson’s nuclear/extranuclear distinction. In this way, Bäck aims to resolve ambiguities in Avicenna’s modal theory by categorizing his impossible objects (e.g., anquā, goatstag) as either logically grounded in quiddities in se or hypothetically impossible under contingent in re conditions. Bäck’s reading clarifies how Avicenna’s contingent nonexistents (e.g., unactualized potentials) align with Parsons’s incomplete objects and shows that Avicenna’s integration of metaphysical truth-makers would address gaps in Parsons’s more abstract system. Through a cross-temporal dialogue in metaphysics and logic, Bäck reframes the differences as complementary, with Avicenna offering richer semantics for existential claims and Parsons providing systematic syntax.

Heine Hansen (Volume 7) contributes new evidence from twelfth-century sources – primarily unedited logical texts, including the Notae Abrincenses and commentaries by Abelard’s followers (Nominales) and his rival Alberic’s school (Albricani) – to resolve debates about Abelard’s ontology of forms. While scholars like Peter King (defending a deflationary view) and Chris Martin (emphasizing robust realism) have disagreed on whether forms (e.g., redness, sitting) are reducible to substances or distinct entities, Hansen shows that Abelard’s later position was nuanced: only specific qualities (e.g., colors, virtues) were considered entitatively and numerically distinct entities, while others (e.g., sitting, relations) were modes of substances with no independent ontological standing. Crucially, the Notae Abrincenses and Nominales commentaries reveal criteria for distinguishing “entity” status: forms must not depend on extrinsic factors, part arrangements, or substance additions. Meanwhile, critiques from the Albricani highlight contemporary resistance to Abelard’s partial realism. Hansen’s analysis of these overlooked texts positions Abelard as a precursor to Ockham’s ontology, grounding metaphysics in substances and select qualities. This evidence clarifies Abelard’s middle path between full reductionism and full realism, advancing scholarship by contextualizing his views within immediate philosophical debates rather than relying solely on his extant works.

Mary Sirridge (Volume 7) compares the medieval modist grammar of Boethius of Dacia to Terry Parsons’s contribution to modern semantics, analyzing their approaches to verbs, tense, and ontological commitments. Boethius, one of the thirteenth-century modistae, links grammar to metaphysics: verbs signify actions/states (res verbi) through modi significandi (modes of signifying) grounded in real-world properties (modi essendi). Tense and voice reflect temporal and relational aspects of actions, with past, present, and future verb-times anchored to an implicit “now.” Parsons, in Events in the Semantics of English, employs predicate logic to analyze verbs as denoting event types, with tense indicating temporal relations (e.g., “Past I < now”). Both posit events as fundamental entities: Boethius treats acts as numerically identical across active/passive constructions, while Parsons uses existential quantification over events to explain syntactic structures. Both recognize events as particulars and the indexicality of tense, but Boethius ties grammar to a strong metaphysical grounding, assuming universal grammar rooted in the structure of reality. Instead, Parsons avoids metaphysical claims and acknowledges potential gaps between linguistic and physical events. Yet both theories model how linguistic structures encode temporal reality, bridging medieval and modern analyses of verb semantics.

Guido Alt (Volume 8) examines John Mair’s (1467–1550) semantics of impossible objects within sixteenth-century nominalist logic, focusing on how Mair, influenced by John Buridan and Marsilius of Inghen, integrates impossibilia (e.g., chimeras) into a nominalist semantic theory committed to ontological parsimony. Alt contrasts Buridan’s rejection of signification for impossibilia (reducing them to fictional composites of possible concepts) with Marsilius’s allowance for ampliated reference to imaginabilia (impossible yet conceivable objects). Mair synthesized these approaches: he adopted Marsilius’s ampliation framework but grounded impossibilia in complex concepts formed by combining possible components via intellectual and volitional activity. Thus, this study clarifies Mair’s distinction between per se and per accidens conceptual representation, resolving ambiguities about nominalist adherence to parsimony, and showing that Mair’s semantics allowed reference to impossibilia without ontological commitment by treating them as mere mental constructs.

Jens Lemanski (Volume 7) traces the evolution of existential commitments in visual reasoning, focusing on the interplay between nominalism and realism in logic diagrams from the eighteenth to twentieth centuries. Initially, Euler’s circle diagrams (eighteenth century) embodied strict nominalism: they represented abstract concepts without existential import, avoiding ontological claims about individuals. Logic was a formal, geometric tool, divorced from metaphysics. The nineteenth century saw a realist shift. Lambert’s line diagrams introduced existential assumptions by treating lines as collections of points qua individuals. Later, Venn formalized realism via shading to denote empty classes, embedding existence/nonexistence into diagrammatic logic. Figures like Bachmann and Meinong further solidified existential commitments, interpreting compartments in Euler/Venn diagrams as populated by individuals. By the late nineteenth century, Boolean logic and Venn’s combinatorial methods entrenched realism, aligning diagrams with symbolic logic’s existential quantifiers. Modern systems (e.g., Peirce’s) balance nominalist formalism with realist conventions, prioritizing technical efficacy over philosophical alignment. Lemanski underscores how visual reasoning’s evolution reflects broader ontological debates, transitioning from Euler’s abstraction to Venn’s existential precision. Today’s diagrammatic systems, while rooted in this history, prioritize logical functionality, often sidelining the metaphysical motivations that once drove nominalist-realist divides.

Dave Beisecker (Volume 7) examines Lewis Carroll’s late nineteenth-century contributions to logic, focusing on his “Method of Trees” for resolving and constructing multi-literal sorites puzzles. This study analyzes Carroll’s method within the history of logical diagramming, contrasting it with modern tableau systems. Beisecker provides a detailed analysis of Carroll’s peculiar approach, particularly his use of a register to identify logical terms (retinends and eliminands) and his method’s reliance on hypothetical entities to derive contradictions, differing from tableau’s premise decomposition. Beisecker shows how Carroll’s method, though resembling tableau in branching structures, applies to solving complex sorites by strategically narrowing deductive paths, avoiding combinatorial explosions inherent in modern systems. Additionally, Carroll’s technique could generate new puzzles through controlled branching and premise selection, a process that Beisecker illustrates via reconstructed examples. By linking Carroll’s work to earlier logical traditions (e.g., Peirce’s existential graphs and Ladd-Franklin’s nullities), the study clarifies Carroll’s overlooked innovations in logical diagramming, challenging dismissals of his contributions as merely recreational.

Graham Priest (Volume 7) addresses the Neo-Meinongian solutions of Terry Parsons and Richard (Sylvain) Routley to the Characterization Principle (CP) – the idea that an object characterized in a certain way possesses those properties. Both philosophers were pivotal in reviving noneism and challenging the Russell/Quine orthodoxy, thus rekindling the interest and the debate on nonexistent objects. Parsons’s and Routley’s approaches are similar. They both distinguish nuclear (characterizing) from non-nuclear properties, restricting the CP to the former. Priest challenges this approach, arguing that their distinction lacks a coherent theoretical foundation, rendering their solutions ad hoc and incomplete. Priest further highlights unresolved issues in their frameworks, such as accounting for intentional relations (e.g., imagining nonexistent objects) and relational predicates involving existent and nonexistent entities. Priest rejects their proposed fixes (e.g., Parsons’s “plugging-up” of relations) as linguistically ungrounded. In contrast, Priest advocates a possible and impossible worlds approach, inspired by Medieval logic’s treatment of nonexistence. By situating the CP within a framework of possible and impossible worlds, nonexistent objects retain their characterizing properties in some world, circumventing the need for nuclear/non-nuclear divides. This aligns with historical noneist traditions while offering a more robust solution.

Edward Zalta (Volume 8) reimagines the foundations of object theory by integrating two modes of predication – exemplification and encoding – into a second-order modal logic framework. This synthesis yields new treatments of longstanding philosophical problems. Individuals exist by exemplifying properties (e.g., Socrates exists by being human), while relations exist through encoding (e.g., “redness” exists if some abstract object encodes it). This approach bypasses traditional reliance on identity, grounding existence directly in predication. Zalta introduces hyperintensional identity for properties and relations, where sameness requires necessary co-encoding by all objects. This distinguishes necessarily equivalent properties (e.g., triangularity vs trilaterality), resolving Quinean skepticism about intensional entities. From these definitions, the necessity of identity for individuals and relations can be derived, by contrast with Kripke’s treatment of this necessity as primitive. Free logic principles are incorporated by linking term denotation to existence constraints. For essential properties, Zalta emphasizes the definable natures of abstracta versus the modal dependencies of ordinary objects: abstracta (e.g., mathematical entities) derive essence from encoded properties, while ordinary objects (e.g., Socrates) depend on modal conditions. Truth can be formulated as a tautology – propositions (modeled as 0-ary relations) satisfy [λφ] ≡ φ – mirroring Tarski’s schema but without semantic ascent. By unifying encoding, hyperintensionality, and modal logic, Zalta extends Parson’s Meinongian framework into a cohesive system addressing existence, identity, essence, and truth – bridging logic and metaphysics while solving classical puzzles.

This collection, large and wide-ranging as it is, has some limits.

First, some traditions are more represented than others. In particular, some traditions (e.g., classical Indian logic) that were present in the initial conference program are not represented in the proceedings.^[5] While we achieved a more balanced representation here than we did in the inaugural symposium’s proceedings, there is still ample room for improvement. We will strive to make these cross-traditional and cross-historical conversations genuinely global and more evenly distributed in future iterations of the Pan-American Symposia on the History of Logic.

Second, these essays are in no way an exhaustive compilation of the most historically influential or theoretically valuable discussions of existence and nonexistence throughout the history of logic up to this day. We simply offer a small but significant sample. The studies collected here exhibit a variety that is unusual for historically-focused collections. We deliberately sought this variety, deeming it an essential feature of the enterprise we envisioned while organizing the Symposia.

Even more, this variety matches the breadth of Terry Parson’s contributions to contemporary semantics, linguistics, philosophy of logic, and the history of medieval logic. It is the common reflection on existence, nonexistence, and Terry’s intellectual legacy that ties these studies together into what we hope is received as both a diverse and coherent collection.

Funding information: Graziana Ciola’s editorial work on this special issue was funded by the European Union (ERC Starting Grant, i ² , 101078785). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.