This paper investigates some classical oppositional categories, like synthetic vs. analytic, posterior vs. prior, imagination vs. grammar, metaphor vs. hermeneutics, metaphysics vs. observation, innovation vs. routine, and image vs. sound, and the role they play in epistemology and philosophy of science. The epistemological framework of objective cognitive constructivism is of special interest in these investigations. Oppositional relations are formally represented using algebraic lattice structures like the cube and the hexagon of opposition, with applications in the contexts of modern color theory, (...) Kantian philosophy, Jungian psychology, and linguistics. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to reasoning fail, so (...) that ignorance of their nature is profound. (shrink)

The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations of mathematicians expended on refining the methods of (...) reasoning with formulae and predicate calculus. Next, two main groups of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained, methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several other problems constitute new, specifically diagrammatic ways of reliable reasoning. (shrink)

It is suggested that Charles Sanders Peirce's triadic semiotics provides a framework for a diagrammatic representation of a sign's proper structure. The action of signs is described at the logical and psychological levels. The role of (unconscious) abductive inference is analyzed, and a diagram of reasoning is offered. A series of interpretants transform brute facts into interpretable signs thereby providing human experience with value or meaning. The triadic structure helps in de-mystifying the relations between Penrose's three worlds when the latter (...) are considered as constituting a semiotic triangle. (shrink)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This paper proposes that (...) transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

This paper is informed by Charles Sanders Peirce’s philosophy as semiotics or the doctrine of signs. The paper’s purpose is to explore Peirce’s category of abduction as not being limited to the inference to the best explanation. In the context of the logic of discovery, abduction is posited as a necessary although not sufficient condition for the production of meanings. The structure of a genuine sign is triadic and represents a synthesis between precognitive ideas and conceptual representations. The novel model (...) of reasoning is offered, based on the mathematical formalism borrowed from Gauss’ interpretation of the complex number. It is suggested that this model in a form of a diagram not only represents a semiotic process-structure but also overcomes the long-standing paradox of new knowledge. For Peirce, it is a diagram as a visual representation that may yield solutions to the otherwise unsolvable logical problems. What appears to us as a paradox is the very presence of abductive, or hypothetical, inference, as Peircean generic category of Firstness within the Thirdness of the total thought-process. Firstness , Secondness , and Thirdness together constitute a dynamic structure of experience. (shrink)

Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a (...) natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs. (shrink)