The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has recently (...) become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)

An eleventh-century Greek text, in which a fourth-century patristic text is discussed, gives an outline of a solution to the Liar Paradox. The eleventh-century text is probably the first medieval treatment of the Liar. Long passages from both texts are translated in this article. The solution to the Liar Paradox, which they entail, is analysed and compared with the results of modern scholarship on several Latin solutions to this paradox. It is found to be a solution, which bears some analogies (...) to contemporary game semantics. Further, an overview of other Byzantine scholia on the Liar Paradox is provided. The findings and the originality of the discussed solution to the Liar Paradox suggest a change in the way in which Byzantine Logic is traditionally regarded in contemporary scholarship. (shrink)

In this article, I offer two different formalizations for prescriptions which correspond to two different forms of biblical prohibitions. I discuss the known fact that the prohibitive commandments of the Decalogue according to the Septuagint and the Vulgate, Exodus 20 and Deuteronomy 5, are formulated with normative future tense indicatives. However, the Greek and Latin sources provide in Mark 10:19 variants of five biblical prohibitive commandments which are formulated with prohibitive subjunctives. I argue that there are semantic differences between normative (...) future tense indicatives and prohibitive subjunctives. These semantic differences are of importance for the understanding of the Decalogue. (shrink)

This article discusses rationality gaps triggered by self-referential/cyclic choice, the latter being understood as choosing according to a norm that refers to the choosing itself. The Crocodile Paradox is reformulated and analyzed as a game—named CP—whose Nash equilibrium is shown to trigger a cyclic choice and to invite a rationality gap. It is shown that choosing the Nash equilibrium of CP conforms to the principles Wolfgang Spohn and Haim Gaifman introduced to, allegedly, guarantee acyclicity but, in fact, does not prevent (...) self-referential/cyclic choice and rationality gaps. It is shown that CP is a counter-example to Gaifman's solution of the rationality gaps problem. (shrink)

In this paper several assumptions concerning omniscience and future contingents on the one side, and omniscience and self-reference on the other, areexamined with respect to a classical and a three-valued semantic setting.Interesting features of both settings are highlighted and their basic assumptions concerning omniscience are explored. To generate a context in which the notion of omniscience does not deviate from some basic intuitions, two special futurity operators are introduced in this article: one for what will definitely take place and another (...) one for what is indeterminate as to whether it will take place. Once these operators are introduced, some puzzles about omniscience in combination with future contingents are removed. An analogous solution to some puzzles concerning omniscience and selfreferentiality is also provided. (shrink)

I try to explain the difference between three kinds of negation: external negation, negation of the predicate and privation. Further I use polygons of opposition as heuristic devices to show that a logic which contains all three mentioned kinds of negation must be a fragment of a Łukasiewicz-four-valued predicate logic. I show, further, that, this analysis can be elaborated so as to comprise additional kinds of privation. This would increase the truth-values in question and bring fragments of (more generally speaking) (...) Łukasiewicz-n-valued predicate logics into the scene. (shrink)

After a short presentation of Aristotle’s views on morally acceptable pleasures vis-á-vis the hedonist and the Platonic views, the Byzantine commentaries published in CAG 19.2 and 20 on knowledge as pleasure are discussed. It is shown that the Byzantine commentators are eventually keen in discovering problems in the Aristotelian account, in a way reminiscent of their Christian premises and akin to Platonism.

Aristotle produced several arguments to vindicate the futura contingentia and to refute the conception of modalities which do not allow incidental facts. This conception was coined mainly by Diodorus Cronus and implied the view that whatever may happen, is to happen necessarily. Although Aristotle condemned this view and refuted the theology which it implies, Diodorean modalities were employed by the scholastics to support their theology. Abaelard's Diodorean formula reads: God wishes no more and no less than what He is able (...) to do - i. e. God's ability to do something implies necessity. In the Summa theologiae, Thomas Aquinas employed Diodorean modalities along with this result of Abaelard's. Leibniz himself confessed his debt to Diodorean modalities as well as to the work of Abaelard in formulating his own ontological proof. Kurt Gödel was under the influence of Leibniz when he wrote his »Ontological Proof«, which employs Diodorean modalities. — For the Greek-speaking scholars of the Middle Ages, however, Aristotelian influences were stronger than Diodorean as regards theory building on modalities. Philosophers from the East from the 2nd to the 11th century A.D., such as Alexander of Aphrodisias, John Philoponus and Michael Psellos, condemned Diodorean modalities as fallacious. In the same period, Greek Church Fathers such as Cyril of Alexandria, Maximus Confessor and John of Damascus gave an orthodox account of God and the modalities, according to which God is able to do whatever He wishes. The absence of Leibniz-like modal ontological proofs in the Greek tradition seems more plausible under these circumstances. (shrink)

Aristotle produced several arguments to vindicate the futura contingentia and to refute the conception of modalities which do not allow incidental facts. This conception was coined mainly by Diodorus Cronus and implied the view that whatever may happen, is to happen necessarily. Although Aristotle condemned this view and refuted the theology which it implies, Diodorean modalities were employed by the scholastics to support their theology. Abaelard's Diodorean formula reads: God wishes no more and no less than what He is able (...) to do - i. e. God's ability to do something implies necessity. In the Summa theologiae, Thomas Aquinas employed Diodorean modalities along with this result of Abaelard's. Leibniz himself confessed his debt to Diodorean modalities as well as to the work of Abaelard in formulating his own ontological proof. Kurt Gödel was under the influence of Leibniz when he wrote his »Ontological Proof«, which employs Diodorean modalities. — For the Greek-speaking scholars of the Middle Ages, however, Aristotelian influences were stronger than Diodorean as regards theory building on modalities. Philosophers from the East from the 2nd to the 11th century A.D., such as Alexander of Aphrodisias, John Philoponus and Michael Psellos, condemned Diodorean modalities as fallacious. In the same period, Greek Church Fathers such as Cyril of Alexandria, Maximus Confessor and John of Damascus gave an orthodox account of God and the modalities, according to which God is able to do whatever He wishes. The absence of Leibniz-like modal ontological proofs in the Greek tradition seems more plausible under these circumstances. (shrink)

The present volume is the first comprehensive reference work for research on part-whole relations. The Handbook of Mereology offers a wide scope, inclusive presentation of contemporary research on part-whole relations that draws out systematic, historical, and interdisciplinary trajectories, shows the subject’s fertility, and inspires future explorations. In particular, we want to impress that mereology is much more than the study of axiomatised systems. The relationship between part and whole is a basic schema of cognitive organisation that operates not only at (...) the level of language and propositional thought, but also at the level of sensory input processing, especially visual and auditory. In the natural, social, and human sciences, as well as in the Humanities, part-whole relations organize all three: data domains, methods, and theories. In short, part-whole relations play a fundamental role in how we perceive and interact with nature, how we speak and think about the world and ourselves, as societies and as individuals. (shrink)