Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.

The Mystery of the Fifth Logical Notion (Alice in the Wonderful Land of Logical Notions)

We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski-Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a nonlogical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first-order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. In the seventh part, we present an enigma: is variety formalizable in first-order logic without equality? There follow recollections concerning Jan Woleński. This paper is dedicated to his 80th birthday. We end with the bibliography, giving some precise references for those wanting to know more about the topic.

The Analytic Model of Consent and The Square of Opposition

Modelling consent is a process prior to any discussion about it, be it theoretical or practical. Here, after examining consent, I shall attempt to present a “logical generator” that produces all different cases of consent (and/or of non-consent), so that afterwards we may articulate a two-dimensional model which will enable us to coherently demonstrate all possible types of consent. The resulting model will be combined with Aristotle’s square of opposition, offering us even greater insight. I shall claim that full(y) informed consent is an archetype, not realized in most cases; it is just one case out of hundreds more. I shall conclude with an educational model for consent, the principle of specificity, arguing that if we wish to both understanding consent and become more adept in exercising it, we need a targeted educational system – not just “better education” in general.

Logic and Metaphysics in Vilnius during 16th–18th Centuries: The Most Important Sources of Vilnius Libraries

The aim of the article is to present the results of research conducted as part of the project Polonica Philosophica Orientalia: namely, to give an overview of the most important logical and metaphysical treatises (both manuscripts and printed editions) written in Vilnius between the sixteenth and eighteenth centuries that are currently accessible in some of the Vilnius libraries. Although the research focused primarily on the Vilnius University Library and its resources, some interesting results were also obtained while researching the Wróblewski Library of the Lithuanian Academy of Sciences. Most sources containing information about the state and the nature of metaphysical and logical investigations conducted in Vilnius during the period discussed are the manuscripts of lecture courses delivered for the students of Vilnius University and other schools. The analysis of those manuscripts has revealed a strong and long-lasting influence of scholastic Aristotelianism in both logic and metaphysics. Namely, up to the mid-eighteenth century, the main questions discussed in logic were Aristotelian syllogism, the square of opposition and the medieval topics of semantical antinomies, modalities and supposition theory, while investigations into metaphysical issues were limited to commentaries on Aristotle’s theoretical treatises without seriously questioning their main theses. In the second part of the eighteenth century, a moderate shift towards subjectivism and psychologism was observed in both of these disciplines. This was influenced by modern epistemologists (George Berkeley, John Locke and others). Logicians and metaphysicians of Vilnius University developed an interest in the topics of human understanding and cognition. However, this shift did not take a radical form – while the majority of eighteenth-century authors sought a combination of scholastic and modern ideas, a critical and even hostile stance towards modern philosophy and science is still noticeable in many of the analysed sources. An overview of the research supported by statistical data is followed by some preliminary remarks on how and where investigation of this topic might be continued in the future.

Concession and modal calculus

Contrary to modal logic, the linguistic approach to modality, without in any way minimizing modal relationships as such (for example by way of the “square of opposition”), tends towards a relative calculation of the knowledge and the beliefs of the speaker. In this article the (universal) modal of the universe of beliefs or of possible worlds will be updated, focusing on how the signs of concession can be interpreted, then defining the “modal calculus” (conceived as a system suitable for determining the modal content of any proposition) in order to apply it to concessive relationships.

Visual Thinking in Logic Notebooks and Alba amicorum

This chapter turns to the notebooks of seventeenth- and eighteenth-century students of logic in Paris and Leuven to demonstrate further the ways in which visual representations, including diagrams of the Porphyrian tree and the square of opposition, along with allegorical and figurative illustrations, were critical in the teaching and development of Aristotelian scholastic philosophy. It shows that lecture notebooks, as well as contemporaneous alba amicorum, incorporate visual materials as a mode of philosophical thought in itself. The visual representations of lecture notebooks and friendship albums are discussed in the same chapter as both these bound manuscript sources functioned as a locus in which students could manipulate visual materials to reflect on philosophical questions in their own voices and with a certain amount of freedom. Furthermore, it shows the iconographic overlaps among the prints and drawings found in these sources.