From Mathematics in Logic to Logic in Mathematics: Boole and Frege Aliou Tall Submitted for the degree of PhD University of York Department of Philosophy October 2002·· Abstract This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme. I Contents Page Part One: Setting The Scene 1. Introduction 2 1.1 Remarks on Gillies' 'The Fregean Revolution in Logic' 2 1.1.1 Gillies' General Thesis 3 1.1.2 Gillies on Peano's Research Programme 7 1.2 Outline of The project 11 Part Two: Boole: Mathematics in Logic 2. Background to Boole's Logical Calculus 15 2.1 A Resume of Aristotelian Logic 16 2.1.1 Categorical Propositions 16 2.1.2 Conversion and Syllogism 20 2.2 Logical Ambience in Britain Before Boole 28 2.2.1 Hamilton: The Quantification of The Predicate 29 2.2.2 Hamilton versus de Morgan 32 2.2.3 de Morgan: The Universe of Discourse 33 2.3 The Move towards Abstractness in Algebra 35 2.3.1 The Emergence of Symbolical Algebra in Britain 36 2.3.2 Boole's Contributions to Symbolical Algebra 40 3. Boole's Computational Procedure in Logic 47 3.1 Boole's Algebraic Notation 50 3.1.1 The System of Signs 51 3.1.2 The Laws of Boo le's Algebra of Logic 52 II 3.2 Boole's Procedures of Computation 56 3.2.1 Development of Logical Function 56 3.2.2 The Logical Interpretation 62 3.2.3 Elimination of Logical Symbols 66 3.3 Boole's 'Primary Propositions' 71 3.3.1 The Expression of 'Primary Propositions' 71 3.3.2 Reduction of System of Propositions 74 3.4 Systematisation and Generalisation of Aristotelian Logic 77 3.4.1 Boole's Laws of Conversion 78 3.4.2 Boole's Solution of Syllogism 83 3.5 Boole's 'Secondary Propositions' 90 3.5.1 Calculus of Hypothetical Propositions and 'Cases' 90 3.5.2 Propositional Calculus and 'Time' 99 3.6 Some Criticisms of Boole's Logical Calculus 105 3.6.1 Boole's Mathematicism 105 3.6.2 The Inconsistent Treatment of The Symbol 'v' 115 4. Some Further Developments of Boole's Logical Calculus 121 4.1 Venn: Diagrannnatic Representation of Boole's Logical Calculus 121 4.1.1 A Compartmental View 122 4.1.2 A Geometrical Illustration of Boole' s logical calculus 127 4.2 Modem Algebra of Logic 136 4.3 Axiomatic Methods for 'Boolean Algebras' 145 4.3.1 Huntington: Sets of Independent Postulates for Boolean Algebra 145 4.3.2 Sheffer: A Set of Five Independent Postulates for Boolean Algebras152 Part Three: Frege: Logic in Mathematics 5. An Account of the First Two Parts of Frege's Begriffsschrift 157 5.1 What is Logic about? 158 5.1.1 Logic as universal 158 III 5.1.2 Psychologism versus Logic 159 5.1.3 Boole and Frege on Logic 161 5.2 The Project of Begriffsschrift 164 5.2.1 An Ideal of Rigour in The Proof 165 5.2.2 Leibniz and the Concept of Formal Proof 166 5.3 A Formula-Language For Pure Thought 171 5.3.1 Frege's Symbolic Notation 171 5.3.2 Frege's New Theory of Judgement: Function I Argument 179 5.3.3 Frege's Theory of Quantification 190 5.4 A Formal System of Logic 203 5.4.1 A Euclidean Procedure 203 5.4.2 The Axiomatisation of Propositional Calculus 206 5.5 The Place of Begriffsschrift in the History of Logic 212 6. Boole-as-Frege-Discusses-Him 218 6.1 The Standard Account of The Discussion 220 6.2 Boole and Frege: The Same Subject-Matter 224 6.2.1 The Perspicuous Representation of Logical Relations 225 6.2.2 The Deductive Calculus of Reasoning 227 6.3 The Relation of Logic to Mathematics 231 6.3.1 Boole: Logic as an Auxiliary Part of Mathematics 231 6.3.2 Frege: Mathematics as a Branch of Logic 233 6.4 The Nodal Point ofFrege's Discussion of Boo le 237 6.4.1 Judgements as Prior to Concepts 238 6.4.2 Functional Analysis of Mathematical Judgments 244 6.4.3 Representation of Generality in Mathematics 246 6.5 A Picture ofBoole-as-Frege-Discusses-Him 250 Part Four: Conclusion 7. Metamathematics: A Return to Boole's Research Programme 253 IV 7.1 The Importance of Boole in The History of Logic 255 7.2 Boole's Anticipation of Metamathematics 264 7.2.1 Boole's Logic as Formalism 265 7.2.2 Boole's Logic as Semantics 270 7.3 Lowenheim's Revival of Boole's Research Programme 272 7.4 Back to Gillies' 'The Fregean Revolution in Logic' 277 7.5 An Overall Portrait of Boole and Frege 280 List of References 283 v Acknowledgement I would like to thank all the people in the Philosophy Department at the University of York for all the help and support that I have received during my happy and fruitful stay as a postgraduate student there. In particular, I would like to thank Dr. Marie McGinn, Dr. Joseph Melia, and Dr. Samir Okasha for their guidance and helpful comments. Most of all, I would like to thank warmly my supervisor, Professor Thomas Baldwin, whose wisdom, munificence, diligence and support guided and inspired my pleasant voyage of philosophical discovery, and without whom this project would not, and could not, have been possible at all. I would like to thank Anna Baldwin, Donald and Rarnzieh Munro for the emotional support and for the food which make my sojourn more bearable, and John S Robinson, Evgeni Dimitrov Vachkov, Olusegun Mayegun, Debbie Brooke and Marysia Koc for their help. I would like to thank Professor Ro bert Rasmussen for editing the project, and Dr Raymond Swaray, Joseph Akpokodje, Jorge Villacampa, Abass Bundu for their friendship. I would also like to extend my gratitude to those undergraduates who have been sources of fruitful discussion and practical guidance. Finally, I would like to thank Mr. Alan Furness, former British ambassador in Senegal, for the respect that he has shown for philosophy as a meritorious and genuine area of study; and to acknowledge the support of the British Foreign Connnonwealth Office and the British Council of Senegal and Manchester for the Chevening Scholarship which made my first two years of study possible. VI Dedication To my wife Sara, and my children Gabriel and Abraham: For unparalleled love and devotion and for bearing the greatest emotional burden of my absence throughout the course of my studies. To my mother and the memory of my father. VII Part One: Setting The Scene 1. Introduction o my Lord! Advance me in knowledge. Surah 20, Taha, verse 114 History, if viewed as a repository for more than anecdote or chronology, could produce a decisive transformation in the image of science by which we are now possessed. Thomas Kuhn 1.1 Remarks on Gillies' 'The Fregean Revolution in Logic' In the historiography of science concerned with the historical development of knowledge, there has been a fixation on empirical sciences for illustrations. For instance, although Kuhn himself acted as a commentator in a session in which the growth of mathematics was discussed in the light of his work at the fiftieth-anniversary meeting of the 'History of Science Society' held inl974 at Norwalk, he never wrote anything about non-empirical sciences. Therefore, the book, Revolutions in Mathematics edited by Donald Gillies is more than welcome. In particular, in the paper on 'The Fregean Revolution in Logic', Gillies espouses Kuhn's general analysis of revolutions in science in order to approach the Fregean revolution. Gillies holds that Frege (1848-1925) is a revolutionary in logic and that his revolution can be compared to 'the Copernican revolution in astronomy and physics' in which the Aristotelian-Ptolemaic paradigm was 'overthrown and irrevocably discarded' and replaced by the Newtonian paradigm. Thus Gillies appears to be reliant upon Kuhn's concept of 'paradigm', which has in his view just the right degree of precision for the analysis of revolutions in science and mathematics (Gillies 1992, p.