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Axioms
Special Issues
Modal Logic and Logical Geometry
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Modal Logic and Logical Geometry

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Logic".

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 10071

Special Issue Editors

Prof. Dr. Lorenz Demey

E-Mail Website
Guest Editor
Institute of Philosophy, KU Leuven, 3000 Leuven, Belgium
Interests: logical geometry; modal logic; history of logic; epistemic logic
Dr. Stef Frijters

E-Mail Website
Guest Editor
Institute of Philosophy, KU Leuven, 3000 Leuven, Belgium
Interests: deontic logic; logical geometry; modal logic

Special Issue Information

Dear Colleagues,

The area of modal logic and the toolkit of Aristotelian diagrams (e.g., the square of opposition) have a long history in philosophy, and nowadays they are also intensively studied in computer science, mathematics, linguistics, etc. Modal logic has traditionally been concerned with modal notions such as necessity, possibility and contingency, and today, it encompasses various other notions, such as knowledge, obligation, action, etc. Aristotelian diagrams, such as the square of opposition, were traditionally used as a mnemotechnic tool to teach syllogistics, and today, they are frequently used in various areas of logic research. Over the past decade, it has become increasingly clear that these diagrams can be fruitfully studied as objects of independent interest, which has given rise to the burgeoning research program of logical geometry.

Modal logic and logical geometry enjoy a deep and many-faceted connection. On the one hand, modal logic has yielded some of the most interesting Aristotelian diagrams known to date, e.g., John Buridan’s 14th-century octagon of opposition for de re modal statements. On the other hand, many of the relations studied in logical geometry are themselves thoroughly modal in nature, e.g., the condition that two statements cannot be true together, which is used to define the relations of contrariety and contradiction.

This Special Issue aims to provide a platform for further exploration along these lines. It will be devoted to state-of-the-art research on modal logic, logical geometry, and especially, their interaction. Both ‘modal logic’ and ‘logical geometry’ are understood in a broad sense, as should be clear from the following non-exhaustive list of topics:

modal logic; epistemic logic; doxastic logic; STIT logic; dynamic logic; temporal logic; hybrid logic; term-modal logic; relational semantics; topological semantics; logical geometry; square of opposition; hexagon of opposition; Aristotelian diagram; historical case studies; bitstring semantics; Aristotelian isomorphism; category theory; existential import; combinatorial semantics; logic-sensitivity.

Contributions may be submitted on a continuous basis before the deadline. After peer review, submissions will be selected for publication based on their quality and relevance.

Prof. Dr. Lorenz Demey
Dr. Stef Frijters
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • logical geometry
  • modal logic
  • square of opposition
  • bitstring semantics
  • negation

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Published Papers (6 papers)

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Research

9 pages, 1044 KiB  
Article
Display Conventions for Octagons of Opposition
by David Makinson
Axioms 2024, 13(5), 287; https://doi.org/10.3390/axioms13050287 - 24 Apr 2024
Viewed by 1034
Abstract
As usually presented, octagons of opposition are rather complex objects and can be difficult to assimilate at a glance. We show how, under suitable conditions that are satisfied by most historical examples, different display conventions can simplify the diagrams, making them easier for [...] Read more.
As usually presented, octagons of opposition are rather complex objects and can be difficult to assimilate at a glance. We show how, under suitable conditions that are satisfied by most historical examples, different display conventions can simplify the diagrams, making them easier for readers to grasp without the loss of information. Moreover, those conditions help reveal the conceptual structure behind the visual display. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
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19 pages, 374 KiB  
Article
Cut-Free Gentzen Sequent Calculi for Tense Logics
by Zhe Lin and Minghui Ma
Axioms 2023, 12(7), 620; https://doi.org/10.3390/axioms12070620 - 21 Jun 2023
Viewed by 1456
Abstract
The cut-free single-succedent Gentzen sequent calculus GKt for the minimal tense logic Kt is introduced. This sequent calculus satisfies the displaying property. The proof proceeds in terms of a Kolmogorov translation and three intermediate sequent systems. Finally, we show that tense [...] Read more.
The cut-free single-succedent Gentzen sequent calculus GKt for the minimal tense logic Kt is introduced. This sequent calculus satisfies the displaying property. The proof proceeds in terms of a Kolmogorov translation and three intermediate sequent systems. Finally, we show that tense logics axiomatized by strictly positive implication have cut-free Gentzen sequent calculi uniformly. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
16 pages, 960 KiB  
Article
Aristotelian Fragments and Subdiagrams for the Boolean Algebra B5
by Koen Roelandt and Hans Smessaert
Axioms 2023, 12(6), 604; https://doi.org/10.3390/axioms12060604 - 18 Jun 2023
Viewed by 1025
Abstract
On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra B5, i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification [...] Read more.
On a descriptive level, this paper presents a number of logical fragments which require the Boolean algebra B5, i.e., bitstrings of length five, for their semantic analysis. Two categories from the realm of natural language quantification are considered, namely, proportional quantification with fractions and percentages—as in two thirds/66 percent of the children are asleep—and normative quantification—as in not enough/too many children are asleep. On a more theoretical level, we study two distinct Aristotelian subdiagrams in B5, which are the result of moving from B5 to B4 either by collapsing bit positions or by deleting bit positions. These two operations are also argued to shed a new light on earlier results from Logical Geometry, in which the collapsing or deletion of bit positions triggers a shift from B4 to B3. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
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26 pages, 520 KiB  
Article
The Modal Logic of Aristotelian Diagrams
by Stef Frijters and Lorenz Demey
Axioms 2023, 12(5), 471; https://doi.org/10.3390/axioms12050471 - 13 May 2023
Cited by 4 | Viewed by 1510
Abstract
In this paper, we introduce and study AD-logic, i.e., a system of (hybrid) modal logic that can be used to reason about Aristotelian diagrams. The language of AD-logic, LAD, is interpreted on a kind of birelational Kripke frames, which we call [...] Read more.
In this paper, we introduce and study AD-logic, i.e., a system of (hybrid) modal logic that can be used to reason about Aristotelian diagrams. The language of AD-logic, LAD, is interpreted on a kind of birelational Kripke frames, which we call “AD-frames”. We establish a sound and strongly complete axiomatization for AD-logic, and prove that there exists a bijection between finite Aristotelian diagrams (up to Aristotelian isomorphism) and finite AD-frames (up to modal isomorphism). We then show how AD-logic can express several major insights about Aristotelian diagrams; for example, for every well-known Aristotelian family A, we exhibit a formula χA∈LAD and show that an Aristotelian diagram D belongs to the family A iff χA is validated by D (when the latter is viewed as an AD-frame). Finally, we show that AD-logic itself gives rise to new and interesting Aristotelian diagrams, and we reflect on their profoundly peculiar status. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
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22 pages, 800 KiB  
Article
A Relational Semantics for Ockham’s Modalities
by Davide Falessi and Fabien Schang
Axioms 2023, 12(5), 445; https://doi.org/10.3390/axioms12050445 - 30 Apr 2023
Cited by 2 | Viewed by 1396
Abstract
This article aims at providing some extension of the modal square of opposition in the light of Ockham’s account of modal operators. Moreover, we set forth some significant remarks on the de re–de dicto distinction and on the modal operator of contingency by [...] Read more.
This article aims at providing some extension of the modal square of opposition in the light of Ockham’s account of modal operators. Moreover, we set forth some significant remarks on the de re–de dicto distinction and on the modal operator of contingency by means of a set-theoretic algebra called numbering semantics. This generalization starting from Ockham’s account of modalities will allow us to take into consideration whether Ockham’s account holds water or not, and in which case it should be changed. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
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19 pages, 2201 KiB  
Article
Diagrammatic and Modal Dimensions of the Syllogisms of Hegel and Peirce
by Paul Redding
Axioms 2022, 11(12), 702; https://doi.org/10.3390/axioms11120702 - 7 Dec 2022
Cited by 2 | Viewed by 2283
Abstract
While in his Science of Logic, Hegel employed neither diagrams nor formulae, his reinterpretation of Aristotle’s syllogistic logic in the “Subjective Logic” of Book III strongly suggests a diagrammatic dimension. Significantly, an early diagram depicting a “triangle of triangles” found among his [...] Read more.
While in his Science of Logic, Hegel employed neither diagrams nor formulae, his reinterpretation of Aristotle’s syllogistic logic in the “Subjective Logic” of Book III strongly suggests a diagrammatic dimension. Significantly, an early diagram depicting a “triangle of triangles” found among his papers after his death captures the organization of categories to be found in The Science of Logic. Features of this diagram help us understand Hegel’s logical project as an attempt to retrieve features of Plato’s thinking that are implicit within Aristotle’s syllogistic logic. It is argued that parallels between Hegel’s modification of Aristotle’s syllogistic figures and Peirce’s functional alignment of those syllogistic figures with his three inference forms—deduction, induction, and abduction—suggest modifications of the traditional “square of opposition” into a logical hexagon as found in recent discussions. However, Hegel had conceived of Aristotle’s syllogism as a distorted version of the “syllogism” thought by Plato to bind the parts of the cosmos into a unity as described in the dialogue Timaeus. In accord with this, it is argued that seen in the light of Hegel’s platonistic reconstruction of Aristotle’s logic, such logical hexagons should be understood as two-dimensional projections of a logical polyhedron. Full article
(This article belongs to the Special Issue Modal Logic and Logical Geometry)
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