This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students’ transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction (...) between contradictories and contraries, and the square of opposition. This strategy helps to improve students’ understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics. (shrink)

The traditional Christian view that God foreknows the future exclusively in terms of what will and will not come to pass is partially rooted in two ancient Hellenistic philosophical assumptions. Hellenistic philosophers universally assumed that propositions asserting ‘ x will occur’ contradict propositions asserting ‘ x will not occur’ and generally assumed that the gods lose significant providential advantage if they know the future partly as a domain of possibilities rather than exclusively in terms of what will and will not (...) occur. Both assumptions continue to influence people in the direction of the traditional understanding of God's knowledge of the future. In this essay I argue that the first assumption is unnecessary and the second largely misguided. (shrink)

This essay explores the validity of Gregory Boyd’s open theistic account of the nature of the future. In particular, it is an investigation into whether Boyd’s logical square of opposition for future contingents provides a model of reality for free will theists that can preserve both bivalence and a classical conception of omniscience. In what follows, I argue that it can.

At the beginning of Book X of his last work The Laws Plato turns his attention from violent and outrageous actions in general to the particular case of undisciplined and presumptuous behaviour in matters of religion: “We have already stated summarily what the punishment should be for temple-robbing, whether by open force or secretly. But the punishments for the various sorts of insolence in speech or action with regard to the gods, which a man can show in word or deed, (...) have to be proclaimed after we have provided an exordium. Let this be it: ‘No one believing, as the laws prescribe, in the existence of the gods has ever yet performed an impious action willingly, or uttered a lawless word. Anyone acting in such a way is in one of three conditions: either, first, he does not believe the proposition aforesaid; or, second, he believes that though the gods exist they have no concern about men; or, third, he believes that they can easily be won over by the bribery of prayer and sacrifice”. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

This book offers a unique synthesis of past and current work on the structure, meaning, and use of negation and negative expressions, a topic that has engaged thinkers from Aristotle and the Buddha to Freud and Chomsky. Horn's masterful study melds a review of scholarship in philosophy, psychology, and linguistics with original research, providing a full picture of negation in natural language and thought; this new edition adds a comprehensive preface and bibliography, surveying research since the book's original publication.

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles ("contraposition" (...) and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

Venn diagrams, which are widely used in introductory logic courses, provide a convenient and illuminating way of presenting the various theories concerning the nature of law. When combined with the Aristotelian square of opposition, these diagrams show not only how the theories are related to one another, logically, which is essential to understanding them, but also which theories are compossible. One surprising result of this approach is that it shows the substantive compatibility of the theories of law set forth by (...) H. L. A. Hart and Ronald Dworkin, who are usually pitted against one another. I show, through quotation and visual representation, that there is no essential disagreement between these jurisprudential stalwarts concerning the relation of law and morality. (shrink)

In the recent debate on future contingents and the nature of the future, authors such as G. A. Boyd, W. L. Craig, and E. Hess have made use of various logical notions, such as the Aristotelian relations of contradiction and contrariety, and the ‘open future square of opposition.’ My aim in this paper is not to enter into this philosophical debate itself, but rather to highlight, at a more abstract methodological level, the important role that Aristotelian diagrams can play in (...) organizing and clarifying the debate. After providing a brief survey of the specific ways in which Boyd and Hess make use of Aristotelian relations and diagrams in the debate on the nature of the future, I argue that the position of open theism is best represented by means of a hexagon of opposition. Next, I show that on the classical theist account, this hexagon of opposition ‘collapses’ into a single pair of contradictory statements. This collapse from a hexagon into a pair has several aspects, which can all be seen as different manifestations of a single underlying change. (shrink)

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. (shrink)