Here I attempt to clarify the general sense of the question that forms the background of Hegel's section on contradiction: What is the essence of contradiction? To what extent does this question pose a philosophical problem for Hegel? By considering this problem can we come to understand contradiction as a relation pertaining to "objective logic"? Translated by Erin Flynn & Kenneth R. Westphal. Originally published as "Über Hegels Lehre vom Widerspruch," in: Dieter Henrich, ed., Hegels Wissenschaft der Logik: Formation und (...) Rekonstruktion (Stuttgart: Klett-Cotta, 1986), 107-28. (shrink)

This chapter tells how, within a century, the notions of judgment and inference were driven out of logical theory and replaced by propositions and consequence. Systematic considerations guide the treatment. The history is unashamedly Whiggish: the current position is shown as the outcome, or even culmination, of a historical development.

The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The (...) second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. -/- Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently. (shrink)

Supporters of G.W.F. Hegel's philosophy have largely shied away from relating his logic to modern symbolic or mathematical approaches. While it has predominantly been the non-Greek discipline of algebra that has informed modern mathematical logic, philosopher Paul Redding argues that the approaches of Plato and Aristotle to logic were deeply shaped by the arithmetic and geometry of classical Greek culture. And by ignoring the fact that Hegel's logic also has this deep mathematical dimension, conventional Hegelians have missed some of Hegel's (...) greatest insights. In Conceptual Harmonies, Redding develops an account of Hegel's logic against a classical and modern historical background that is rarely considered. He stresses Hegel's attention to the Platonic background of Aristotle's original syllogistic and beyond. He then links these Platonic elements to Leibniz's modern revitalization of the logical tradition and then to new forms of algebraic geometry emerging in Hegel's lifetime. Redding thereby reestablishes aspects of Hegel's philosophy that are essential if Hegel is to be taken as a thinker relevant not only to contemporary philosophy, but also to current philosophical conceptions of logic. (shrink)

This paper examines Hegel’s claim that philosophy “has no other object than God‘ as a claim about the essentiality of the idea of God to philosophy. On this idealist interpretation, even atheistic philosophies would presuppose rationally evaluable ideas of God, despite denials of the existence of anything corresponding to those ideas. This interpretation is then applied to Hegel’s version of idealism in relation to those of two predecessors, Leibniz and Kant. Hegel criticizes the idea of the Christian God present within (...) his predecessors in terms of his own heterodox reading of the Trinity in order to resolve a paradox affecting them -- the “paradox of perspectivism‘. (shrink)

Hermann Grassmann's ideas on the nature and foundations of mathematics were published as an integral part of his mathematical treatise, the Ausdehnungslehre, in 1844. In spite of its notoriously obscure style we can better understand the work if we view it as an expression of the dialectical philosophy of his mentor, the theologian F. Schleiermacher. The relation to Schleiermacher is presented here through an analysis of the principal ideas of the Ausdehnungslehre.

This two-volume work brings together a comprehensive selection of mathematical works from the period 1707-1930. During this time the foundations of modern mathematics were laid, and From Kant to Hilbert provides an overview of the foundational work in each of the main branches of mathmeatics with narratives showing how they were linked. Now available as a separate volume.

Self-taught mathematician and father of Boolean algebra, George Boole (1815-1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as a (...) system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind. (shrink)

This chapter discusses the complex conditions for the emergence of 19th-century symbolic logic. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole and above all of his German follower Ernst Schröder.

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners (...) of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

This paper considers the relation between the American pragmatist Charles Sanders Peirce and the German idealist G. W. F. Hegel . While Peirce engaged with Hegel’s thought quite extensively, his often critical comments on the latter have made it hard to see any genuine common ground between the two; recent ways of reading Hegel, however, suggest how this might be possible, where the connections between their respective metaphysical positions and views of the categories are explored here. Issues relating to their (...) epistemology and philosophies of languages are considered in a companion article. (shrink)

Ever since I abandoned the philosophy of Kant and Hegel, I have sought solutions of philosophical problems by means of analysis, and I remain firmly persuaded, in spite of some modern tendencies to the contrary, that only by analysing is progress possible. (Russell, My Philosophical Development, ch. 1).

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Contemporary hybrid logic is based on the idea of using formulas as terms, an idea invented and explored by Arthur Prior in the mid-1960s. But Prior’s own work on hybrid logic remains largely undiscussed. This is unfortunate, since hybridisation played a role that was both central to and problematic for his philosophical views on tense. In this paper I introduce hybrid logic from a contemporary perspective, and then examine the role it played in Prior’s work.