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.

Contradictory Perspectives on Academic Support: Beyond Linear Logic

This paper employs a Square of Opposition as an interpretivist heuristic device in order to interrogate perceptions of academic support. The Square of Opposition is used to move beyond binary explanations of academic development subsumed within learner-/discipline-focussed practices or institutionally /epistemologically constrained systems; an exemplar data set is used to achieve this. The results of this analysis demonstrate that positions that might normally be understood as opposed in fact share common features, at least where some key concepts are concerned. In particular, two “contradictories” are explored: the first of these critiques the differences and similarities between contested meaning-making and knowledge dissemination and the second analyses the disjuncture between skills-focussed instruction and academic literacy as a social practice. This form of analysis offers new insights that directly speak to the ways in which we conceive of, and enact, teaching, personal tutoring and academic advising.

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.

Logics of (In)sane and (Un)reliable Beliefs

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given.

Negation and Opposition

The treatment of negation has long been linked to the treatment of opposition between propositions (or sentences) and between terms (or subsentential constituents). The primary types of opposition, usefully displayed on the post-Aristotelian Square of Opposition, are contradiction (two contradictories always differ in truth value) and contrariety (two contraries can both be false, but not both be true). The law of non-contradiction governs both oppositions, while the law of excluded middle applies only to contradictories. In principle, Aristotle’s semantic category of contradictory opposition lines up with the syntactic category of sentence (vs. constituent) negation, but in practice matters are more complicated, and while Klima’s diagnostics are helpful they are often not decisive. These complications are illustrated by the distribution of affixal negation, the phenomenon of logical double negation, the interaction of negation with quantifiers and modals, and the tendency for formal contradictory negation to be pragmatically strengthened to contrariety.

The Modal Square of Opposition and the Mental Models Theory

The mental models theory is a current approach trying to account for human thought and hence communication by highlighting the action of semantics and ignoring, to a large extent, syntax. However, it has been argued that the theory actually contains an underlying syntax related to any kind of modal logic. This paper delves into this last idea and is intended to show that the concepts of possibility and necessity as understood in it fulfill the basic requirement that, according to Fitting and Mendelsohn, every modal logic has to meet: to satisfy the relationships provided by the Aristotelian modal square of opposition.

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.