Lorenz Demey

Structures of Oppositions in Public Announcement Logic

In this paper we study dynamic epistemic logic, and in particular public announcement logic, using the tools of n-opposition theory. Dynamic epistemic logic is a contemporary development of epistemic logic, which takes into account changes of knowledge through time. It studies, for example, public announcements and the influence they have on the agents’ knowledge. n-Opposition theory is a systematic generalization of the traditional squares of oppositions. In the paper we provide introductions to the intuitions behind and the formal details of both of these disciplines. We construct a square and a hexagon of oppositions for the structural properties of public announcement. We then generalize this to opposition structures for any partially functional process, and show how this generalization can be used to support the structuralist philosophy surrounding n-opposition theory. Next, we focus on the epistemic properties of public announcement, and construct an octagon and a (three-dimensional) rhombic dodecahedron of oppositions. Many of the techniques applied in this paper were originally developed by Smessaert (Logica Univers. 3:303–332, 2009); they are thus shown to be very powerful and widely applicable. Summing up: we establish a strong connection between public announcement logic and n-opposition theory, and show that this connection has definite advantages for both of the disciplines involved.

The Relationship between Aristotelian and Hasse Diagrams

The aim of this paper is to study the relationship between two important families of diagrams that are used in logic, viz. Aristotelian diagrams (such as the well-known ‘square of oppositions’) and Hasse diagrams. We discuss some obvious similarities and dissimilarities between both types of diagrams, and argue that they are in line with general cognitive principles of diagram design. Next, we show that a much deeper connection can be established for Aristotelian/Hasse diagrams that are closed under the Boolean operators. We consider the Boolean algebra \(\mathbb{B}_n\) with 2n elements, whose Hasse diagram can be drawn as an n-dimensional hypercube. Both the Aristotelian and the Hasse diagram for \(\mathbb{B}_n\) can be seen as (n − 1)-dimensional vertex-first projections of this hypercube; whether the diagram is Aristotelian or Hasse depends on the projection axis. We show how this account provides a unified explanation of the (dis)similarities between both types of diagrams, and illustrate it with some well-known Aristotelian/Hasse diagrams for \(\mathbb{B}_3\) and \(\mathbb{B}_4\).

Some remarks on the model theory of epistemic plausibility models

The aim of this paper is to initiate a systematic exploration of the model theory of epistemic plausibility models (EPMs). There are two subtly different definitions in the literature: one by van Benthem and one by Baltag and Smets. Because van Benthems notion is the most general, most of the paper is dedicated to this notion. We focus on the notion of bisimulation, and show that the most natural generalization of bisimulation to van Benthem-type EPMs fails. We then introduce parametrized bisimulations, and prove various bisimulationimplies- equivalence theorems, a Hennessy-Milner theorem, and several (un)definability results. We discuss the problems arising from the fact that these bisimulations are syntax-dependent (and thus not fully structural), and we present and compare two different ways of coping with this issue: adding a modality to the language, and putting extra constraints on the models. We argue that the most successful solution involves restricting to uniform and locally connected (van Benthem-type) EPMs: for this subclass the intuitively most natural notion of bisimulation and the technically sound notion coincide. Such EPMs turn out to correspond exactly with Baltag/Smets-type EPMs, which can be interpreted as constituting a methodological argument, favoring Baltag and Smetss definition of EPM over that of van Benthem.

Metalogical Decorations of Logical Diagrams

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams (viz. Aristotelian and duality diagrams) and four kinds of metalogical decorations (viz. those based on the opposition, implication, Aristotelian and duality relations).

Logical and Geometrical Complementarities between Aristotelian Diagrams

This paper concerns the Aristotelian relations of contradiction, contrariety, subcontrariety and subalternation between 14 contingent formulae, which can get a 2D or 3D visual representation by means of Aristotelian diagrams. The overall 3D diagram representing these Aristotelian relations is the rhombic dodecahedron (RDH), a polyhedron consisting of 14 vertices and 12 rhombic faces (Section 2). The ultimate aim is to study the various complementarities between Aristotelian diagrams inside the RDH. The crucial notions are therefore those of subdiagram and of nesting or embedding smaller diagrams into bigger ones. Three types of Aristotelian squares are characterised in terms of which types of contradictory diagonals they contain (Section 3). Secondly, any Aristotelian hexagon contains 3 squares (Section 4), and any Aristotelian octagon contains 4 hexagons (Section 5), so that different types of bigger diagrams can be distinguished in terms of which types of subdiagrams they contain. In a final part, the logical complementarities between 6 and 8 formulae are related to the geometrical complementarities between the 3D embeddings of hexagons and octagons inside the RDH (Section 6).

Combinatorial Bitstring Semantics for Arbitrary Logical Fragments

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before.

Algebraic aspects of duality diagrams

Duality phenomena are widespread in logic and language; their behavior is visualized using square diagrams. This paper shows how our recent algebraic account of duality can be fruitfully used to study these diagrams. A duality cube is constructed, and it is shown that 14 duality squares can be embedded into this cube (two of which were hitherto unknown). This number is also an upper bound.

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

This paper studies the logical context-sensitivity of Aristotelian diagrams. I propose a new account of measuring this type of context-sensitivity, and illustrate it by means of a small-scale example. Next, I turn toward a more large-scale case study, based on Aristotelian diagrams for the categorical statements with subject negation. On the practical side, I describe an interactive application that can help to explain and illustrate the phenomenon of context-sensitivity in this particular case study. On the theoretical side, I show that applying the proposed measure of context-sensitivity leads to a number of precise yet highly intuitive results.

The Unreasonable Effectiveness of Bitstrings in Logical Geometry

This paper presents a unified account of bitstrings—i.e. sequences of bits (0/1) that serve as compact semantic representations—for the analysis of Aristotelian relations and provides an overview of their effectiveness in three key areas of the Logical Geometry research programme. As for logical effectiveness, bitstrings allow a precise and positive characterisation of the notion of logical independence or unconnectedness, as well as a straightforward computation—in terms of bitstring length and level—of the number and type of Aristotelian relations that a particular formula may enter into. As for diagrammatic effectiveness, bitstrings play a crucial role in studying the subdiagrams of the Aristotelian rhombic dodecahedron, and different types of Aristotelian hexagons turn out to require bitstrings of different lengths. The linguistic and cognitive effectiveness of bitstring analysis relates to the scalar structure underlying the bitstrings, and to the difference between linear and non-linear bitstrings.

Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers

The aim of this paper is to discuss and extend some of Beziau’s (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few. After reviewing some of the basic notions of logical geometry, we discuss Beziau’s work on visualising the Aristotelian relations in S5 by means of two- and three-dimensional diagrams, such as hexagons and a stellar rhombic dodecahedron. We then argue that Beziau’s analysis is incomplete, and show that it can be completed by considering another three-dimensional Aristotelian diagram, viz. a rhombic dodecahedron. Next, we discuss Beziau’s proposal to transpose his results on the logical geometry of the modal logic S5 to that of the subjective quantifiers many and few. Finally, we propose an alternative analysis of many and few, and compare it with that of Beziau’s. While the two analyses seem to fare equally well from a strictly logical perspective, we argue that the new analysis is more in line with certain linguistic desiderata.