Language, Logic, and Mathematics in Schopenhauer

Contents
An Introduction to Language, Logic and Mathematicsin Schopenhauer
	1 Interpretations and Contradictions
	2 System and World
	3 Language, Logic and Mathematics
	4 Intuition and Diagrams
	5 Overview of the Volume
		5.1 Part I: Language
		5.2 Part II: Logic
		5.3 Part III: Mathematics
	References
Part I Language
	Language as an “Indispensable Tool and Organ” of Reason: Intuition, Concept and Word in Schopenhauer
		1 Introduction
		2 Real Empirical Objects
		3 Concepts
		4 Words
		5 Conclusive Remarks
		References
	Problems in Reconstructing Schopenhauer's Theory of Meaning: With Reference to His Influence on Wittgenstein
		1 Introduction
		2 Schopenhauer's Theory of Language and its Links to the Representational and Use Theories
		3 The Ambivalent Use of the Terms ``Concept'' and ``Word''
		4 Consequences of Ambivalence for Understanding Schopenhauer's Theory of Meaning
		5 Meaning as Sense and Reference
		6 Use Theory of Meaning as Unnatural Way of Language Acquisition
		7 Conclusions and Prospects
		References
	Concept Diagrams and the Context Principle
		1 Introduction
		2 Euler-Type Diagrams and Concept Diagrams
			2.1 Euler and Euler Diagrams
			2.2 Concept Diagrams
		3 The Primacy Question in Logic
			3.1 The Kant–Frege Thesis
			3.2 Criticism of the Kant–Frege Thesis
		4 Schopenhauer's Concept Diagrams and the Context Principle
			4.1 Representationalism and Explanation
				4.1.1 Concepts and Spheres
				4.1.2 Origin, Emergence, and Development of Concepts
				4.1.3 Interpretation
			4.2 Rationalism and Understanding
				4.2.1 Translating and Understanding Languages
				4.2.2 Schopenhauer's Use Theory of Meaning and His Context Principle
				4.2.3 Interpretation
		5 Rational Representationalism
		References
	A Comment on Lemanski's “Concept Diagrams and the Context Principle”
		References
	The World as Will and I-Language: Schopenhauer's Philosophy as Precursor of Cognitive Sciences
		1 Chomsky's Conception of E- and I-Language
		2 Schopenhauer's E- and I-Language
		3 Language Processing According to Schopenhauer and the Primacy of I-Language
		4 Results
		References
	Schopenhauer's Perceptive Invective
		1 Introduction
		2 Schopenhauer's Philosophy of Language
		3 Learning from the Tirades
		4 Conclusion
		References
Part II Logic
	Schopenhauer's Eulerian Diagrams
		1 Introduction
		2 Eulerian Diagrams
		3 Spheres and Concepts
		4 Relations of Concepts
		5 Complex Diagrams
		6 Conclusion
		References
	Schopenhauer's Logic in Its Historical Context
		1 Introduction
		2 Logic as an Already-Perfected Science and Intuitive Way of Thinking
		3 The History of Logic After Aristotle
			3.1 The Laws of Thought at the Beginning of Logic
			3.2 The Scholastic Mnemonics
			3.3 Hypothetical and Disjunctive Inferences
			3.4 The Separation of Concepts and Representations of Perception
			3.5 The Fourth Figure
			3.6 Criticisms of Aristotelian Logic
		4 Left Off Schopenhauer's List: The Port-Royal Logic and Kant's Logic
		References
	Arthur Schopenhauer on Naturalness in Logic
		1 Introduction
		2 Unnatural and Natural Deduction in Syllogistics
			2.1 Unnaturalness of FIV
			2.2 Naturalness of FI
			2.3 Naturalness of FII
			2.4 Naturalness of FIII
		3 Natural Completeness and Unnatural Redundancy
			3.1 Mental-Linguistic Completeness and Redundancy
			3.2 Diagrammatic Completeness and Redundancy
		4 Conclusion and Outlook
			4.1 Summary
			4.2 Schopenhauer's Logic in the Context of Current Debates
		References
	Schopenhauer and the Equational Form of Predication
		1 Introduction
		2 Schopenhauer's Use of Equality Signs in Logic
		3 Schopenhauer's Account of Quantity and Quality of Judgments
		4 Possible Sources for Schopenhauer's Equational Symbolism
			4.1 Reimarus, Kant, and Fichte
			4.2 Herbart
			4.3 Maimon
			4.4 Schulze
		5 A Tentative Assessment
		References
	From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry
		1 Introduction
		2 Some Background on Logical Geometry
			2.1 Defining the Aristotelian Relations in a Boolean Algebra
			2.2 The α-Structures and Their Properties
		3 Two Basic Examples
			3.1 From an Euler Diagram to a Classical Square of Opposition
			3.2 From an Euler Diagram to a Degenerate Square of Opposition
		4 Partitions, Euler Diagrams and Aristotelian Diagrams
		5 Boolean Subtypes of Aristotelian Diagrams
		6 Conclusion
		References
	Metalogic, Schopenhauer and Universal Logic
		1 Indiosyncralogical Schopenhauer
		2 Metamathematics, Metalogic, and Universal Logic
			2.1 Origin and Nature of Metamathematics
			2.2 From Metamathematics to Metalogic
			2.3 From Metalogic to Universal Logic
		3 Schopenhauer's Theory of the Metalogical
			3.1 The Tricky and Crutchy Euclid
			3.2 Metalogical Truths: Where They Are and What They Are
			3.3 The Femininity and Triviality of Metalogical Truths
		4 Formulation, Axiomatization, Interaction, Reflection
			4.1 Reformulations, Semiotical Changes, and Mathematical Interaction
			4.2 The Modern Axiomatic Methodology
			4.3 Multi-Level Analysis and Productive Self-Reflection
		Acknowledgements and Memories
		References
Part III Mathematics
	Schopenhauer and the Mathematical Intuition as the Foundation of Geometry
		1 Introduction
		2 Intuition in Mathematics
		3 Role and Purpose of Mathematics
		4 A British Debate
			4.1 Whewell on the Study of Mathematics
			4.2 Hamilton's Review
		5 Intuition and the Foundations of Geometry
		References
	Schopenhauer on Intuition and Proof in Mathematics
		1 Preliminary Remarks
		2 Schopenhauer's Theory of Cognition
		3 The Principle of Sufficient Reason
		4 Geometrical Proof Procedures
		5 Some Problems
		6 On Disinclination for Mathematics
		References
	Schopenhauer on Diagrammatic Proof
		1 Two Challenges to The Legitimacy of Picture Proofs
		2 Schopenhauer's Advocacy
		3 Applying Schopenhauer's Remarks
			3.1 Particularity and Generality
			3.2 Misleading Pictures
		4 Conclusion
		References
	From Necessary Truths to Feelings: The Foundations of Mathematics in Leibniz and Schopenhauer
		1 A Denied Source
		2 `Necessary Truths' and Reminiscence
		3 Necessary Truths in Euclid's System
		4 Confused Ideas Between Images and Feeling
		References