Scuola Normale Superiore Classe di Lettere e Filosofia Corso di Perfezionamento in Filosofia The Principle of Analyticity of Logic A Philosophical and Formal Perspective Thesis Advisors: Candidate: Prof. Marcello D'Agostino Costanza Larese Prof. Massimo Mugnai July 2019 2 Contents Introduction 4 Summary 8 I Historical and philosophical reconstruction 12 1 Kant and the foundations of the analytic-synthetic distinction 14 1.1 The Kantian analytic-synthetic distinction . 14 1.1.1 Criteria for analyticity . 16 1.1.2 Containment . 19 1.1.3 Clarification, identity and contradiction . 26 1.1.4 Syntheticity and intuition . 33 1.2 Kant's pure general logic . 35 1.2.1 The topics of pure general logic . 37 1.2.2 The conception of pure general logic . 41 1.3 The relationship between analyticity and logic . 46 1.3.1 Logic as an instrument . 46 1.3.2 The status of logic . 47 2 Bolzano and the syntheticity of logic 54 2.1 Bolzano's analytic-synthetic distinction . 54 2.1.1 Preliminary notions: the method of substitution, validity, derivability . 54 2.1.2 Bolzano's conception of analyticity . 57 2.1.3 The notion of logical analyticity . 62 2.1.4 Language independence and synonymy . 65 2.1.5 Criticisms against Kant's analysis and analytic-synthetic dis- tinction . 66 2.2 The science of logic . 70 2.2.1 Grounding, deductive sciences and synthetic a priori .... 70 3 CONTENTS 4 2.2.2 Bolzano's thesis that logic is synthetic . 74 2.3 An evaluation . 76 2.3.1 A contradiction in Bolzano's system? The pragmatics of analyticity . 76 2.3.2 A deceptive terminological choice? Bolzano's place in the history of analyticity . 80 3 Frege and logical positivism: the golden age of the principle of analyticity of logic 90 3.1 Logic is analytic . 90 3.1.1 Logic in Frege's logicism . 90 3.1.2 Frege's analytic-synthetic distinction . 94 3.1.3 Frege's notion of analysis . 99 3.1.4 Frege on the analyticity of general logical laws . 104 3.1.5 Analyticity of Logic in Logical Empiricism . 107 3.2 Logic is tautologous . 110 3.2.1 The paradox of analysis . 110 3.2.2 Frege against the \legend of the sterility of pure logic" . 113 3.2.3 Frege after the Sinn-Bedeutung distinction . 118 3.2.4 Logical empiricism and the psychologistic solution . 122 3.2.5 Wittgenstein and the myth of the perfect language . 125 4 Hintikka's modern reconstruction of Kant's theory: against the traditional paradigm 130 4.1 Kantian premises of Hintikka's work . 131 4.1.1 Mathematical intuitions as singular representations . 132 4.1.2 From the mathematical method to the analytic-synthetic distinction . 137 4.1.3 Kant as \an heir to the constructional sense of analysis" . 140 4.2 Hintikka's main thesis: synthetical logical truths . 147 4.2.1 Hintikka's analytic-synthetic distinction . 147 4.2.2 A vindication of Kant's conception? . 152 4.3 Against logical positivism through modern means . 155 4.3.1 The theory of distributive normal forms . 156 4.3.2 Probability and semantic information . 161 4.3.3 An evaluation . 165 II Formal proposals 168 5 Depth Bounded First-Order Logics 170 CONTENTS 5 5.1 Depth Bounded Boolean Logics . 170 5.1.1 Is propositional logic really uninformative? . 170 5.1.2 Informational semantics . 172 5.1.3 Virtual information . 175 5.1.4 Tableaux rules for Depth Bounded Boolean Logics . 177 5.1.5 Observations on the relation with Hintikka's work and Kant's conceptions . 183 5.2 The idea of extending Hintikka's approach . 185 5.2.1 The project of Depth Bounded First-Order Logics . 185 5.2.2 First attempt: Propositionalization . 188 5.2.3 Second attempt: Skolem functions . 189 5.2.4 Final attempt: What is quantificational depth? . 192 5.3 Depth Bounded First-Order Logics . 198 5.3.1 Derivability relations `0;q with q ≥ 0 and intelim rules . 199 5.3.2 Derivability relations `k;q with k > 0 and q ≥ 0 and the PB rule . 204 5.3.3 Quantificational depth and parameter q ............ 208 5.3.4 Examples . 210 6 Depth Bounded Epistemic Logics 220 6.1 Analyticity of logic and logical omniscience . 220 6.1.1 Classical epistemic logics and logical omniscience . 220 6.1.2 The muddy children puzzle in S5 ............... 223 6.1.3 Ideal and realistic agents . 229 6.1.4 Degrees of logical omniscience . 231 6.2 Depth Bounded Epistemic Logics . 233 6.2.1 The structure . 233 6.2.2 Language and grammar . 235 6.2.3 Intermezzo I . 236 6.2.4 Models . 240 6.2.5 Intermezzo II . 242 6.2.6 The muddy children puzzle in Depth Bounded Epistemic Logics I . 245 6.2.7 Three notions of validity in a model . 247 6.2.8 Infinite notions of validity and the logics DBELxk ...... 252 6.2.9 Relationships between the logics DBELxk ........... 254 6.2.10 The muddy children puzzle in Depth Bounded Epistemic Logics II . 255 6.2.11 Different conclusions . 260 Conclusions 266 CONTENTS 6 Appendix 278 Bibliography 312 Introduction The subject of the present work is the principle of analyticity of logic. In order for the question `Is logic analytic?' to make sense and before trying to find an answer to this problem, it is obviously necessary to specify two preliminary issues, namely, the meaning of the term `analytic' and the meaning of the term `logic'. The former issue is somehow justified and expected: after all, analyticity represents one of the philosophical concepts par excellence and, as such, it has been at the core of a lively debate throughout the history of the discipline. But, despite possible appearances to the contrary, the second issue is probably more decisive than the former in determining the answer to the initial question: both the contents and the philosophical conceptions of logic play a fundamental role in the study of the epistemological status of this discipline. We could even say that the clarification of the concepts of analyticity and of logic constitutes in itself the decision on the analyticity of logic. This thesis studies the principle of analyticity of logic through two different, but related, methodologies, which individuate the two main parts of the work: the former offers a historical and philosophical reconstruction of the problem; the latter proposes two formal characterizations of the analytic-synthetic distinction. The reconstruction of the first part does not presume to be exhaustive and is restricted to the theories of the following philosophers: Kant, Bolzano, Frege and Hintikka. The material has been chosen according to the following criteria. First, this work aims at showing the `historical' nature of the principle of analyticity of logic, which has a certain genealogy and a precise starting point. Although after the Vienna Circle this tenet has been taken for granted, there are many and significant conceptions that criticize it. Theories holding that logic is either not analytic or synthetic are the main characters of our reconstruction. This explains, for example, why we have dedicated great attention to Bolzano, while leaving little margin to the logical empiricist movement, despite the fact that analyticity is probably more fundamental for the latter's thought than for the former's philosophical construction. As a result of this choice, theories of meaning and their connection to analyticity are completely overlooked, since they belong to the logical empiricists' interpretation of the analytic-synthetic distinction. In 7 INTRODUCTION 8 other words, the principle of analyticity of logic and the philosophers arguing for it are taken as a critical target, but the true focus is on the varieties of reactions against them. Second, each author that has been selected might be seen as a symbol for a peculiar answer to the question of the analyticity of logic: as we will see, each of them represent a definite approach to the problem. This observation clarifies why, for example, we have chosen only one author between Bolzano and Mill: al- though their philosophical presuppositions are completely different, the conclusion they reach on the status of logic goes surprisingly in the same direction. Last, but not least, we have chosen our protagonists among the most excellent figures of the modern and contemporary philosophical panorama for learning purposes. Despite the indisputable advantages, this choice has exposed our research to the risk of being too general and of privileging the horizontal rather than the vertical direction. We hope we have avoided this danger through an adequate methodological equipment. First of all, we have offered hopefully accurate textual readings of the relevant passages of our primary sources. This is the case for the excerpts of Kant's Critique of Pure Reason A 6-7/B 10-11 and A 151-152/B 190-191; Bolzano's The- ory of Science x148; Frege's Foundations of Arithmetic x3 and Hintikka's articles in Logic, Language-Games and Information. At the same time, we have tried to take into account a wide selection of secondary literature: we have considered both the most recent productions, such as, to make just one example, de Jong's use- ful article The Analytic-Synthetic Distinction and the Classical Model of Science: Kant, Bolzano and Frege, and the traditional interpretations, such as Coffa’s The Semantic Tradition from Kant to Carnap. To the Vienna Station and Proust's Questions of Form. Logic and the Analytic Proposition from Kant to Carnap. Consistently with the reflections expressed above, we have examined, for each author, first, his characterization of the analytic-synthetic distinction; second, his conception of logic; and, third, his standpoint on the status of logic with respect to analyticity.