DOCSLIB.ORG

Essence and Necessity, and the Aristotelian Modal Syllogistic: a Historical and Analytical Study Daniel James Vecchio Marquette University

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Marquette University e-Publications@Marquette Dissertations (2009 -) Dissertations, Theses, and Professional Projects Essence and Necessity, and the Aristotelian Modal Syllogistic: A Historical and Analytical Study Daniel James Vecchio Marquette University Recommended Citation Vecchio, Daniel James, "Essence and Necessity, and the Aristotelian Modal Syllogistic: A Historical and Analytical Study" (2016). Dissertations (2009 -). 686. http://epublications.marquette.edu/dissertations_mu/686 ESSENCE AND NECESSITY, AND THE ARISTOTELIAN MODAL SYLLOGISTIC: A HISTORICAL AND ANALYTICAL STUDY by Daniel James Vecchio, B.A., M.A. A Dissertation submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Milwaukee, Wisconsin December 2016 ABSTRACT ESSENCE AND NECESSITY, AND THE ARISTOTELIAN MODAL SYLLOGISTIC: A HISTORICAL AND ANALYTICAL STUDY Daniel James Vecchio, B.A., M.A. Marquette University, 2016 The following is a critical and historical account of Aristotelian Essentialism informed by recent work on Aristotle’s modal syllogistic. The semantics of the modal syllogistic are interpreted in a way that is motivated by Aristotle, and also make his validity claims in the Prior Analytics consistent to a higher degree than previously developed interpretative models. In Chapter One, ancient and contemporary objections to the Aristotelian modal syllogistic are discussed. A resolution to apparent inconsistencies in Aristotle’s modal syllogistic is proposed and developed out of recent work by Patterson, Rini, and Malink. In particular, I argue that the semantics of negation is distinct in modal context from those of assertoric negative claims. Given my interpretive model of Aristotle’s semantics, in Chapter Two, I provide proofs for each of the mixed apodictic syllogisms, and propose a method of using Venn Diagrams to visualize the validity claims Aristotle makes in the Prior Analytics. Chapter Three explores how Aristotle’s syllogistic fits within Aristotle’s philosophy of science and demonstration, particularly within the context of the Posterior Analytics. Consideration is given to the Aristotelian understanding of the relationship among necessity, explanation, definition, and essence. Chapter Four applies Aristotelian modal logic in contemporary contexts. I contrast Aristotelian modality and essentialism with contemporary modalism based upon the semantics of possible worlds, e.g. Kripke and Putnam. I also develop an account of how Aristotelian modal logic can ground a sortal dependent theory of identity, as discussed by Wiggins. i ACKNOWLEDGEMENTS Daniel James Vecchio, B.A., M.A. I am grateful Dr. Owen Goldin, who first proposed at a Philosophy Department social that I take a look at Aristotle’s modal syllogistic to see if I could make any sense of it. As director of my dissertation project, Dr. Goldin has provided an enormous amount of support— meeting with me remotely, at all hours, while I wrote the bulk of this dissertation in Madrid, Spain. I am also thankful for the support and helpful comments of my other committee members: Dr. Michael Wreen, Dr. Noel Adams, and Dr. Marko Malink. Dr. Wreen has guided me in becoming a better writer, and a more rigorous and logical thinker. Without the tools he taught me in the Philosophy of Logic, this dissertation would never have gotten off the ground. Dr. Adams has also been an extraordinary mentor and friend, who has provided insightful comments at the outset of my project. Dr. Malink has been of central importance to me for the past few years, both in terms of his brilliant work on Aristotle’s modal syllogistic, and in terms of meeting with me to guide and teach me so much about this topic. Finally, I would like to acknowledge the loving support of my family. In particular, I should mention the tireless efforts of my father, James, who often served as a second set of eyes, in reviewing my writing. I would like to thanks my mother, Kathleen who did not live to see this project through to its completion, but whose love gave me the foundation to believe that I could complete my dissertation. I would like to thank my mother-in-law, Graciela, who is always praying for me and our family, and has provided so much to me and my family as I complete this dissertation. Also, I would like to thank my beautiful daughter, Vivian, whose recent birth gave me a special impetus to complete the dissertation, though the demands of the dissertation often called me away from her. I ii dedicate this work to my darling wife, Gabriela, who has believed in me and supported me through this challenging period. Without you, Gabriela, I could not have done this. I love you. Finally, I am grateful to God, who is the source of all good things. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS………………………………………………………………………………………… i LIST OF FIGURES………………………………………………………………………………………………….. v ABBREVIATIONS OF THE WORKS OF ARISTOTLE…………………………………………………. vi CHAPTER ONE…………………………………………………………………………………………………………… 1 1.0 Introduction………………………………………………………………………….. 1 1.1 Symbolization……………………………………………………………………….. 6 1.2 Basic Modal Syllogisms………………………………………………………... 6 1.3 The Two Barbaras……………………………………………………………….. 13 1.4 The Modal Copula………………………………………………………………….. 22 1.5 Malink on the Metaphysics of Aristotelian Modality………………... 28 1.6 Malink’s Heterodox Interpretation of Aristotelian Semantics….. 29 1.7 An Adjustment to Malink’s Heterodox Interpretation……………... 40 1.8 Conclusion…………………………………………………………………………… 62 TWO…………………………………………………………………………………………………………. 65 2.0 Explication of Argument Forms along with Venn Diagrams……. 65 2.1 The First Figure……………………………………………………………………. 68 2.2 The Second Figure………………………………………………………………… 76 2.3 The Third Figure…………………………………………………………………… 89 2.4 Conclusion………………………………………………………………………….... 101 THREE……………………………………………………………………………………………………….. 105 3.0 Introduction………………………………………………………………………… 105 iv 3.1 Transcending the De Re/De Dicto Distinction……………………….. 106 3.2 The Modality of Definition, Explanation, and Essence…………….. 109 3.3 Towards an Aristotelian Demonstrative Science……………………. 133 3.4 Knowledge of Essences………………………………………………………… 151 3.5 Aristotelian Essentialism and Aristotelian Science………………… 166 3.6 Conclusion…………………………………………………………………………… 170 FOUR………………………………………………………………………………………………………… 173 4.0 Introduction…………………………………………………………………………. 173 4.1 The Circularity Problem………………………………………………………... 184 4.2 Actualism…………………………………………………………………………….. 186 4.3 Possible Worlds and Essences………………………………………………. 192 4.4 Transworld Individuals and Counterpart Theory………………….. 195 4.5 Identity and Essence……………………………………………………………… 204 4.5.1 Leibniz’s Laws and Regulating Identity Relations…………. 205 4.5.2 Relative Theory of Identity…………………………………………. 209 4.5.3 Sortal Dependency Theory of Identity………………………… 213 4.5.4 Identity Grounded on Aristotelian Essentialism and Modality……………………………………………………………… 222 4.6 Conclusion, or Counteracting Indispensability………………………… 232 BIBLIOGRAPHY…………………………………………………………………………………………………….. 236 APPENDIX……………………………………………………………………………………………………………. 245 v LIST OF FIGURES Fig. 1: Assertoric Categorical Propositions…………………………………………………………… 66 Fig. 2: Apodictic Categorical Propositions…………………………………………………………….. 67 Fig. 3: Comparison of Barbara-LXL to Barbara-XLL……………………………………………… 69 Fig. 4: Comparison of Celarent-LXL to Celarent-XLL……………………………………………… 71 Fig. 5: Comparison of Darii-LXL to Darii-XLL………………………………………………………… 73 Fig. 6: Comparison of Ferio-LXL to Ferio-XLL………………………………………………………… 75 Fig. 7: Comparison of Cesare-LXL to Cesare-XLL……………………………………………………. 77 Fig 8: Comparison of Camestres-XLL to Camestres-LXL…………………………………………. 78 Fig. 9: Comparison of Festino-LXL to Festino-XLL………………………………………………….. 80 Fig. 10: Comparison of Baroco-LLL to Baroco-XLL………………………………………………... 83 Fig. 11: Comparison of Baroco-LLL to Baroco-LXL………………………………………………... 87 Fig. 12: Comparison of Baroco-LXX to Baroco-XLX………………………………………………… 88 Fig. 13: Comparison of Darapti-LXL to Darapti-XLL……………………………………………….. 90 Fig. 14: Comparison of Felapton-LXL to Felapton-XLL…………………………………………… 92 Fig. 15: Comparison of Disamis-XLL to Disamis-LXL……………………………………………… 94 Fig. 16: Comparison of Datisi-LXL to Datisi-XLL…………………………………………………….. 95 Fig. 17: Comparison of Bocardo-LLL to Bocardo-LXL…………………………………………… 97 Fig. 18: Comparison of Bocardo-LLL to Bocardo-XLL…………………………………………… 98 Fig. 19: Comparison of Ferison-LXL to Ferison-XLL……………………………………………… 100 vi ABBREVIATIONS OF THE WORKS OF ARISTOTLE APo. Posterior Analytics APr. Prior Analytics Cat. Categories De An. de Anima De In. de Interpretatione Meta. Metaphysics PA de Partibus Animalium Phys. Physics Top. Topics 1 CHAPTER ONE 1.0 Introduction Aristotle’s modal logic is considered by many commentators to be a murky mess. The oft repeated phrase on this subject, coined by Günther Patzig, is that the modal syllogistic is “a realm of darkness” (1968, 86 fn.21). Patzig was specifically referencing the attempt by Albrect Becker to interpret Aristotle’s use of “modal operators” in the “traditional way”. This is, by no means, a recent condemnation. Alexander of Aphrodisias reports that Aristotle’s earliest commentators disagreed with many of his supposed valid forms of argument. Theophrastus and Eudemus thought that no valid mixed apodictic syllogism, a syllogism with one apodictic premise and one non-apodictic premise, could have an apodictic conclusion. Rather, they insisted that the weaker modality of the premises should rule over the conclusion.
Recommended publications
  • Aristotle's Assertoric Syllogistic and Modern Relevance Logic*

    Aristotle's Assertoric Syllogistic and Modern Relevance Logic*

    Aristotle’s assertoric syllogistic and modern relevance logic* Abstract: This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. Keywords: Aristotle, Syllogism, Relevance Logic, History of Logic, Logic, Relevance 1.
  • The Modal Logic of Potential Infinity, with an Application to Free Choice

    The Modal Logic of Potential Infinity, with an Application to Free Choice

    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
  • DRAFT: Final Version in Journal of Philosophical Logic Paul Hovda

    DRAFT: Final Version in Journal of Philosophical Logic Paul Hovda

    Tensed mereology DRAFT: final version in Journal of Philosophical Logic Paul Hovda There are at least three main approaches to thinking about the way parthood logically interacts with time. Roughly: the eternalist perdurantist approach, on which the primary parthood relation is eternal and two-placed, and objects that persist persist by having temporal parts; the parameterist approach, on which the primary parthood relation is eternal and logically three-placed (x is part of y at t) and objects may or may not have temporal parts; and the tensed approach, on which the primary parthood relation is two-placed but (in many cases) temporary, in the sense that it may be that x is part of y, though x was not part of y. (These characterizations are brief; too brief, in fact, as our discussion will eventually show.) Orthogonally, there are different approaches to questions like Peter van Inwa- gen's \Special Composition Question" (SCQ): under what conditions do some objects compose something?1 (Let us, for the moment, work with an undefined notion of \compose;" we will get more precise later.) One central divide is be- tween those who answer the SCQ with \Under any conditions!" and those who disagree. In general, we can distinguish \plenitudinous" conceptions of compo- sition, that accept this answer, from \sparse" conceptions, that do not. (van Inwagen uses the term \universalist" where we use \plenitudinous.") A question closely related to the SCQ is: under what conditions do some objects compose more than one thing? We may distinguish “flat” conceptions of composition, on which the answer is \Under no conditions!" from others.
  • Marko Malink (NYU)

    Marko Malink (NYU)

    Demonstration by reductio ad impossibile in Posterior Analytics 1.26 Marko Malink (New York University) Contents 1. Aristotle’s thesis in Posterior Analytics 1. 26 ................................................................................. 1 2. Direct negative demonstration vs. demonstration by reductio ................................................. 14 3. Aristotle’s argument from priority in nature .............................................................................. 25 4. Priority in nature for a-propositions ............................................................................................ 38 5. Priority in nature for e-, i-, and o-propositions .......................................................................... 55 6. Accounting for Aristotle’s thesis in Posterior Analytics 1. 26 ................................................... 65 7. Parts and wholes ............................................................................................................................. 77 References ............................................................................................................................................ 89 1. Aristotle’s thesis in Posterior Analytics 1. 26 At the beginning of the Analytics, Aristotle states that the subject of the treatise is demonstration (ἀπόδειξις). A demonstration, for Aristotle, is a kind of deductive argument. It is a deduction through which, when we possess it, we have scientific knowledge (ἐπιστήμη). While the Prior Analytics deals with deduction in
  • C.S. Peirce's Abduction from the Prior Analytics

    C.S. Peirce's Abduction from the Prior Analytics

    Providence College DigitalCommons@Providence Community Scholar Publications Providence College Community Scholars 7-1996 C.S. Peirce's Abduction from the Prior Analytics William Paul Haas Follow this and additional works at: https://digitalcommons.providence.edu/comm_scholar_pubs Haas, William Paul, "C.S. Peirce's Abduction from the Prior Analytics" (1996). Community Scholar Publications. 2. https://digitalcommons.providence.edu/comm_scholar_pubs/2 This Article is brought to you for free and open access by the Providence College Community Scholars at DigitalCommons@Providence. It has been accepted for inclusion in Community Scholar Publications by an authorized administrator of DigitalCommons@Providence. For more information, please contact [email protected]. WILLIAM PAUL HAAS July 1996 C.S. PEIRCE'S ABDUCTION FROM THE PRIOR ANALYTICS In his Ancient Formal Logic. Professor Joseph Bochenski finds Aristotle's description of syllogisms based upon hypotheses to be "difficult to understand." Noting that we do not have the treatise which Aristotle promised to write, Bochenski laments the fact that the Prior Analytics, where it is treated most explicitly, "is either corrupted or (which is more probable) was hastily written and contains logical errors." (1) Charles Sanders Peirce wrestled with the same difficult text when he attempted to establish the Aristotelian roots of his theory of abductive or hypothetical reasoning. However, Peirce opted for the explanation that the fault was with the corrupted text, not with Aristotle's exposition. Peirce interpreted the text of Book II, Chapter 25 thus: Accordingly, when he opens the next chapter with the word ' Ajray (¿y-q a word evidently chosen to form a pendant to 'Errctyuyrj, we feel sure that this is what he is coming to.
  • Indiscernible Logic: Using the Logical Fallacies of the Illicit Major Term and the Illicit Minor Term As Litigation Tools

    Indiscernible Logic: Using the Logical Fallacies of the Illicit Major Term and the Illicit Minor Term As Litigation Tools

    WLR_47-1_RICE (FINAL FORMAT) 10/28/2010 3:35:39 PM INDISCERNIBLE LOGIC: USING THE LOGICAL FALLACIES OF THE ILLICIT MAJOR TERM AND THE ILLICIT MINOR TERM AS LITIGATION TOOLS ∗ STEPHEN M. RICE I. INTRODUCTION Baseball, like litigation, is at once elegant in its simplicity and infinite in its complexities and variations. As a result of its complexities, baseball, like litigation, is subject to an infinite number of potential outcomes. Both baseball and litigation are complex systems, managed by specialized sets of rules. However, the results of baseball games, like the results of litigation, turn on a series of indiscernible, seemingly invisible, rules. These indiscernible rules are essential to success in baseball, in the same way the rules of philosophic logic are essential to success in litigation. This article will evaluate one of the philosophical rules of logic;1 demonstrate how it is easily violated without notice, resulting in a logical fallacy known as the Fallacy of the Illicit Major or Minor Term,2 chronicle how courts have identified this logical fallacy and used it to evaluate legal arguments;3 and describe how essential this rule and the fallacy that follows its breach is to essential effective advocacy. However, because many lawyers are unfamiliar with philosophical logic, or why it is important, this article begins with a story about a familiar subject that is, in many ways, like the rules of philosophic logic: the game of baseball. ∗ Stephen M. Rice is an Assistant Professor of Law, Liberty University School of Law. I appreciate the efforts of my research assistant, Ms.
  • Immediate Inference

    Immediate Inference

    Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P.
  • Catalogue of Titles of Works Attributed to Aristotle

    Catalogue of Titles of Works Attributed to Aristotle

    Catalogue of Titles of works attributed by Aristotle 1 To enhance readability of the translations and usability of the catalogues, I have inserted the following bold headings into the lists. These have no authority in any manuscript, but are based on a theory about the composition of the lists described in chapter 3. The text and numbering follows that of O. Gigon, Librorum deperditorum fragmenta. PART ONE: Titles in Diogenes Laertius (D) I. Universal works (ta kathalou) A. The treatises (ta syntagmatika) 1. The dialogues or exoterica (ta dialogika ex terika) 2. The works in propria persona or lectures (ta autopros pa akroamatika) a. Instrumental works (ta organika) b. Practical works (ta praktika) c. Productive Works (ta poi tika) d. Theoretical works (ta the r tika) . Natural philosophy (ta physiologia) . Mathematics (ta math matika) B. Notebooks (ta hypomn matika) II. Intermediate works (ta metaxu) III. Particular works (ta merika) PART TWO: Titles in the Vita Hesychii (H) This list is organized in the same way as D, with two exceptions. First, IA2c “productive works” has dropped out. Second, there is an appendix, organized as follows: IV. Appendix A. Intermediate or Particular works B. Treatises C. Notebooks D. Falsely ascribed works PART THREE: Titles in Ptolemy al-Garib (A) This list is organized in the same way as D, except it contains none of the Intermediate or Particular works. It was written in Arabic, and later translated into Latin, and then reconstructed into Greek, which I here translate. PART FOUR: Titles in the order of Bekker (B) The modern edition contains works only in IA2 (“the works in propria persona”), and replaces the theoretical works before the practical and productive, as follows.
  • Probabilistic Semantics for Modal Logic

    Probabilistic Semantics for Modal Logic

    Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators).
  • Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

    Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

    axioms Article Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics Lorenz Demey 1,2 1 Center for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, Belgium; [email protected] 2 KU Leuven Institute for Artificial Intelligence, KU Leuven, 3000 Leuven, Belgium Abstract: Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distin- guish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic. Keywords: Aristotelian diagram; non-normal modal logic; square of opposition; logical geometry; neighborhood semantics; bitstring semantics MSC: 03B45; 03A05 Citation: Demey, L. Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. Axioms 2021, 10, 128. https://doi.org/ 1. Introduction 10.3390/axioms10030128 Aristotelian diagrams, such as the so-called square of opposition, visualize a number Academic Editor: Radko Mesiar of formulas from some logical system, as well as certain logical relations holding between them.
  • The Beginnings of Formal Logic: Deduction in Aristotle's Topics Vs

    The Beginnings of Formal Logic: Deduction in Aristotle's Topics Vs

    Phronesis 60 (�0�5) �67-309 brill.com/phro The Beginnings of Formal Logic: Deduction in Aristotle’s Topics vs. Prior Analytics Marko Malink Department of Philosophy, New York University, 5 Washington Place, New York, NY 10003. USA [email protected] Abstract It is widely agreed that Aristotle’s Prior Analytics, but not the Topics, marks the begin- ning of formal logic. There is less agreement as to why this is so. What are the distinctive features in virtue of which Aristotle’s discussion of deductions (syllogismoi) qualifies as formal logic in the one treatise but not in the other? To answer this question, I argue that in the Prior Analytics—unlike in the Topics—Aristotle is concerned to make fully explicit all the premisses that are necessary to derive the conclusion in a given deduction. Keywords formal logic – deduction – syllogismos – premiss – Prior Analytics – Topics 1 Introduction It is widely agreed that Aristotle’s Prior Analytics marks the beginning of formal logic.1 Aristotle’s main concern in this treatise is with deductions (syllogismoi). Deductions also play an important role in the Topics, which was written before the Prior Analytics.2 The two treatises start from the same definition of what 1 See e.g. Cornford 1935, 264; Russell 1946, 219; Ross 1949, 29; Bocheński 1956, 74; Allen 2001, 13; Ebert and Nortmann 2007, 106-7; Striker 2009, p. xi. 2 While the chronological order of Aristotle’s works cannot be determined with any certainty, scholars agree that the Topics was written before the Prior Analytics; see Brandis 1835, 252-9; Ross 1939, 251-2; Bocheński 1956, 49-51; Kneale and Kneale 1962, 23-4; Brunschwig 1967, © koninklijke brill nv, leiden, ���5 | doi �0.��63/�5685�84-��34��86 268 Malink a deduction is (stated in the first chapter of each).
  • Boxes and Diamonds: an Open Introduction to Modal Logic

    Boxes and Diamonds: an Open Introduction to Modal Logic

    Boxes and Diamonds An Open Introduction to Modal Logic F19 Boxes and Diamonds The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université de Lorraine Jeremy Avigad, Carnegie Mellon University Tim Button, University College London Walter Dean, University of Warwick Gillian Russell, Dianoia Institute of Philosophy Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, Columbia University Dana Hägg, University of Calgary Zesen Qian, Carnegie Mellon University Boxes and Diamonds An Open Introduction to Modal Logic Remixed by Richard Zach Fall 2019 The Open Logic Project would like to acknowledge the gener- ous support of the Taylor Institute of Teaching and Learning of the University of Calgary, and the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Cover illustrations by Matthew Leadbeater, used under a Cre- ative Commons Attribution-NonCommercial 4.0 International Li- cense. Typeset in Baskervald X and Nimbus Sans by LATEX. This version of Boxes and Diamonds is revision ed40131 (2021-07- 11), with content generated from Open Logic Text revision a36bf42 (2021-09-21). Free download at: https://bd.openlogicproject.org/ Boxes and Diamonds by Richard Zach is licensed under a Creative Commons At- tribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Cre- ative Commons Attribution 4.0 Interna- tional License. Contents Preface xi Introduction xii I Normal Modal Logics1 1 Syntax and Semantics2 1.1 Introduction....................