Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. (...) Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: chords inferentially related to subchords, pairs of chords in the space of major–minor tonality and arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments. (shrink)

This book marks the inauguration of the Historia Logicae book series, which seeks to publish high-quality monographs, dissertations, textbooks, proceedings, and anthologies on the history of logic in either German or English. Serving as the inaugural volume in this series, the book explores the contemporary interpretation of logic across many centuries and cultures. The first section of the volume comprises a compilation of papers dedicated to ancient and medieval logic, examining prominent thinkers such as Plato, Aristotle, Seneca, Porphyry, Proclus, Boethius, (...) Buridan, and Kilwardby. The second section shifts focus towards early-modern and contemporary logics, including the works of Caramuel, Kant, Drobisch, Husserl, and Jaskowski. Topics of discussion encompass the relationship between mathematics and logic, natural deduction, intuitionism, graph theory, categorical judgments, theories of truth, modal logic, opposition relations, logic diagrams, definitions of formal logic, many-sorted logic, model theory, and paraconsistent logic. (shrink)

Abstract“Let us think of the intention to play chess”. On the Role of Chess Analogies in Wittgenstein’s Philosophy starting from 1929. Chess analogies represent a neglected topic in the studies on Wittgenstein. However, already a closer look at the Philosophical Investigations shows the great variety of contexts in which there are analogies to very different aspects of chess. An examination of the entire Nachlass illustrates Wittgenstein’s ongoing interest in chess which began in 1929 and lasted until his death in 1951. (...) The integration of a thorough analysis of the references to chess sheds new light on an adequate understanding of the transition from his early philosophy of the Tractatus logico-philosophicus to his later philosophy, especially in the years 1929 to 1931. In our language of analysis, chess is understood as a codex-based strategy game. In this context and in contrast to it, we consider selected instructive uses of chess analogies in Wittgenstein's Nachlass. By means of such analogies similarities as well as filigree or significant differences in the meanings of linguistic expressions can be very convincingly illustrated, misunderstandings can be removed and complete clarity can be attained. Starting from Wittgenstein's considerations, the method of finding and inventing chess analogies can be developed in perspective into a creative form of practice of linguistic-analytic philosophizing. (shrink)

Ontological commitments and other problems concerning existence arise in connection with various aspects of logical theories. The semantics of quantification theory is usually formulated in such a manner that theorems are all and only those formulae which come out true under all interpretations in all non-empty domains. There are several approaches to include the empty domain. Paradoxically this apparent semantic extension means surrendering several formulae which are valid and intuitively plausible.