Avicenna on Possibility and Necessity

Smessaert

2017

J Philos Logic

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. Keywords bitstrings • combinatorial semantics • Aristotelian diagram • Boolean algebra • existential import • public announcement logic

Section: Logical Preliminariesmentioning

confidence: 99%

Combinatorial Bitstring Semantics for Arbitrary Logical Fragments

Smessaert

2017

J Philos Logic

“…However, from the twelfth century onwards, philosophers started to make use of these diagrams to explicate their theorizing on modalities as well [2][3][4][5]. Furthermore, historical scholarship has shown that Aristotelian diagrams for modal logic can be reconstructed from the works of many earlier authors, such as Theophrastus [6,7], Chrysippus [8,9], and Avicenna [10,11]. Today, Aristotelian diagrams not only appear in well-known textbooks on modal logic [12,13], but they are also used in applications of modal logic to a variety of philosophical and logical topics, such as paraconsistency [14,15], logic-sensitivity [16][17][18], and theories of truth [19,20].…”

Section: Introductionmentioning

confidence: 99%

The Modal Logic of Aristotelian Diagrams

Frijters

2023

Axioms

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.

“…It has a very rich tradition, going back-together with the discipline of logic itself-to the works of Aristotle. Over the centuries, authors such as Avicenna, William of Sherwood, John Buridan, Boole and Frege have studied the square and other, larger Aristotelian diagrams [15,16,40,48,63,67]. Since the beginning of the 21st century, logicians have started studying Aristotelian diagrams Parts of this paper were presented at CLMPS 15 (Helsinki).…”

Section: Introductionmentioning

confidence: 99%

Metalogical Decorations of Logical Diagrams

Smessaert

2016

Log. Univers.

Abstract. 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).Mathematics Subject Classification (2010). Primary 03A10, 03B45; Secondary 03B65, 03B80.