Space and Time As Relations: the Theoretical Approach of Leibniz
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Chapter 5: Methods of Proof for Boolean Logic
Chapter 5: Methods of Proof for Boolean Logic § 5.1 Valid inference steps Conjunction elimination Sometimes called simplification. From a conjunction, infer any of the conjuncts. • From P ∧ Q, infer P (or infer Q). Conjunction introduction Sometimes called conjunction. From a pair of sentences, infer their conjunction. • From P and Q, infer P ∧ Q. § 5.2 Proof by cases This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy. In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction. • From P ∨ Q, and from the fact that S follows from P and S also follows from Q, infer S. The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction. In practice, this method of proof requires the use of “subproofs”—we will take these up in the next chapter when we look at formal proofs. -
David Lewis on Convention
David Lewis on Convention Ernie Lepore and Matthew Stone Center for Cognitive Science Rutgers University David Lewis’s landmark Convention starts its exploration of the notion of a convention with a brilliant insight: we need a distinctive social competence to solve coordination problems. Convention, for Lewis, is the canonical form that this social competence takes when it is grounded in agents’ knowledge and experience of one another’s self-consciously flexible behavior. Lewis meant for his theory to describe a wide range of cultural devices we use to act together effectively; but he was particularly concerned in applying this notion to make sense of our knowledge of meaning. In this chapter, we give an overview of Lewis’s theory of convention, and explore its implications for linguistic theory, and especially for problems at the interface of the semantics and pragmatics of natural language. In §1, we discuss Lewis’s understanding of coordination problems, emphasizing how coordination allows for a uniform characterization of practical activity and of signaling in communication. In §2, we introduce Lewis’s account of convention and show how he uses it to make sense of the idea that a linguistic expression can come to be associated with its meaning by a convention. Lewis’s account has come in for a lot of criticism, and we close in §3 by addressing some of the key difficulties in thinking of meaning as conventional in Lewis’s sense. The critical literature on Lewis’s account of convention is much wider than we can fully survey in this chapter, and so we recommend for a discussion of convention as a more general phenomenon Rescorla (2011). -
Arxiv:2103.15570V2 [Physics.Hist-Ph] 22 Jun 2021 to This Article in Its Purpose and Content
An Analysis of the Concept of Inertial Frame Boris Čulina Department of Mathematics University of Applied Sciences Velika Gorica Zagrebačka cesta 5, 10410 Velika Gorica, Croatia e-mail: [email protected] Abstract. The concept of inertial frame of reference is analysed. It has been shown that this fundamental concept of physics is not clear enough. A definition of inertial frame of reference is proposed which expresses its key inherent property. The definition is operational and powerful. Many other properties of inertial frames follow from the definition or it makes them plausible. In particular, the definition shows why physical laws obey space and time symmetries and the principle of relativity, it resolves the problem of clock synchronization and the role of light in it, as well as the problem of the geometry of inertial frames. keywords: inertial frame of reference, space and time symmetries, the principle of relativity, clock synchronization, physical geometry The concept of inertial frame is a fundamental concept of physics. The opinion of the author is that not enough attention has been paid to such a significant concept, not only in textbooks, but also in the scientific literature. In the scientific literature, many particular issues related to the concept of inertial frame have been addressed, but, as far as the author is aware, a sys- tematic analysis of this concept has not been made. DiSalle’s article [DiS20] in the Stanford Encyclopedia of Philosophy gives an overview of the histori- cal development of the concept of an inertial frame as an essential part of the historical development of physics. -
This Electronic Thesis Or Dissertation Has Been Downloaded from Explore Bristol Research
This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Jenkins, Clare Helen Elizabeth Title: Jansenism as literature : a study into the influence of Augustinian theology on seventeenth-century French literature General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. Jansenism as Literature: A Study into the Influence of Augustinian Theology on Seventeenth-Century French Literature Clare Helen Elizabeth Jenkins A Dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy in the Faculty of Arts. -
2. Aristotle's Concept of the State
Page No.13 2. Aristotle's concept of the state Olivera Z. Mijuskovic Full Member International Association of Greek Philosophy University of Athens, Greece ORCID iD: http://orcid.org/0000-0001-5950-7146 URL: http://worldphilosophynetwork.weebly.com E-Mail: [email protected] Abstract: In contrast to a little bit utopian standpoint offered by Plato in his teachings about the state or politeia where rulers aren`t “in love with power but in virtue”, Aristotle's teaching on the same subject seems very realistic and pragmatic. In his most important writing in this field called "Politics", Aristotle classified authority in the form of two main parts: the correct authority and moose authority. In this sense, correct forms of government are 1.basileus, 2.aristocracy and 3.politeia. These forms of government are based on the common good. Bad or moose forms of government are those that are based on the property of an individual or small governmental structures and they are: 1.tiranny, 2.oligarchy and 3.democracy. Also, Aristotle's political thinking is not separate from the ethical principles so he states that the government should be reflected in the true virtue that is "law" or the "golden mean". Keywords: Government; stat; , virtue; democracy; authority; politeia; golden mean. Vol. 4 No. 4 (2016) Issue- December ISSN 2347-6869 (E) & ISSN 2347-2146 (P) Aristotle's concept of the state by Olivera Z. Mijuskovic Page no. 13-20 Page No.14 Aristotle's concept of the state 1.1. Aristotle`s “Politics” Politics in its defined form becomes affirmed by the ancient Greek world. -
Aristotle and the Productive Class: Did Aristotle Argue
ARISTOTLE AND THE PRODUCTIVE CLASS: DID ARISTOTLE ARGUE IN FAVOR OF EDUCATION FOR ALL? by DEANNA M. HACKWORTH, B.A. A THESIS IN PHILOSOPHY Submitted to the Graduate Faculty of Texas Tech University in Pardal Fulfillment of the Requirements for the Degree of MASTER OF ARTS Approved August, 2003 ACKNOWLEDGMENTS I am grateful to many people who offered criticism, support, and comments during the writing of this paper, without whom, it is doubtful this thesis would have ever been completed. First and foremost, I must thank the chair of my committee, Howard Curzer, whose support, encouragement, and vast knowledge of Aristotle were both enlightening and infinitely helpfijl. For his continuous help in my attempt to understand the capabilities approach first developed by Sen, and his unwavering commitment to keep my interpretation of Aristotle truthful and correct, I am indebted to Walter Schaller. An earUer and much different version of this paper sparked conversations between myself and Eric Carter, which helped to shape my thinking on Nussbaum, and to whom I would like to offer sincere thanks. LaKrisha Mauldin and Stiv Fleishman edited and provided comments on several drafts, allowing me to see errors that I would not have been able to see otherwise, and proving invaluable in the final hours before the defense of this paper. Several unpublished articles written and sent to me by Dr. Fred Miller proved fertile ground to shape my thinking of objections to my project, and I am forever gratefial. Last but certainly not least, I would like to offer thanks to my family, for giving me large amounts of uninterrupted time which allowed me to work on this project, and supported me to the extent they were able as I endeavored to become a philosopher. -
Radin Centre for the Study of Religion University of Toronto [email protected]
Long Commentary on the De Anima of Aristotle. Written by Averroes (Ibn Rushd) of Cordoba. Translated and with introduction and notes by Richard C. Taylor, with Therese-Anne Druart, subeditor. New Haven CT: Yale University Press, 2009. 498 pages. ISBN: 978-0-0300-11668-7. $85.00 US, $97.95 CDN. Born in Andalusia in 1126, Averroes (or Ibn Rushd) is generally considered one of the most influential Arabic classical philosophers, and his impact on Western thought, through the interpretations of Thomas Aquinas, Duns Scotus, and others, can hardly be overestimated. Although Averroes wrote extensively in the fields of religious law (Bidayat al-Mujtahid wa Nihayat al-Muqtasad), philosophy (The Incoherence of the Incoherence) and their relationship (Decisive Treatise), his nickname, the Great Commentator, is a result of Averroes’ prolific commentaries on Greek philosophy, and particularly the works of Aristotle. The date of completion of Averroes’ Long Commentary is, although debatable, usually given at around 1186 C.E., and while its chronological relationship to the Short and Middle Commentaries (and therefore status as Averroes’ ‘final opinion’ on the De Anima) is debatable, the Long Commentary certainly presents Averroes’ most detailed examination of the De Anima. In form, it is basically a line-by-line exegesis, with Averroes reproducing Aristotle’s work and then proceeding to provide explanation, extension, and interpretation. Averroes’ Long Commentary is key to tracing how Averroes interpreted Aristotle’s work on the soul and how his own concept of the soul developed from his reading of Aristotle. The current translation of Averroes’ Long Commentary on the De Anima of Aristotle by Richard C. -
Against Logical Form
Against logical form Zolta´n Gendler Szabo´ Conceptions of logical form are stranded between extremes. On one side are those who think the logical form of a sentence has little to do with logic; on the other, those who think it has little to do with the sentence. Most of us would prefer a conception that strikes a balance: logical form that is an objective feature of a sentence and captures its logical character. I will argue that we cannot get what we want. What are these extreme conceptions? In linguistics, logical form is typically con- ceived of as a level of representation where ambiguities have been resolved. According to one highly developed view—Chomsky’s minimalism—logical form is one of the outputs of the derivation of a sentence. The derivation begins with a set of lexical items and after initial mergers it splits into two: on one branch phonological operations are applied without semantic effect; on the other are semantic operations without phono- logical realization. At the end of the first branch is phonological form, the input to the articulatory–perceptual system; and at the end of the second is logical form, the input to the conceptual–intentional system.1 Thus conceived, logical form encompasses all and only information required for interpretation. But semantic and logical information do not fully overlap. The connectives “and” and “but” are surely not synonyms, but the difference in meaning probably does not concern logic. On the other hand, it is of utmost logical importance whether “finitely many” or “equinumerous” are logical constants even though it is hard to see how this information could be essential for their interpretation. -
Truth-Bearers and Truth Value*
Truth-Bearers and Truth Value* I. Introduction The purpose of this document is to explain the following concepts and the relationships between them: statements, propositions, and truth value. In what follows each of these will be discussed in turn. II. Language and Truth-Bearers A. Statements 1. Introduction For present purposes, we will define the term “statement” as follows. Statement: A meaningful declarative sentence.1 It is useful to make sure that the definition of “statement” is clearly understood. 2. Sentences in General To begin with, a statement is a kind of sentence. Obviously, not every string of words is a sentence. Consider: “John store.” Here we have two nouns with a period after them—there is no verb. Grammatically, this is not a sentence—it is just a collection of words with a dot after them. Consider: “If I went to the store.” This isn’t a sentence either. “I went to the store.” is a sentence. However, using the word “if” transforms this string of words into a mere clause that requires another clause to complete it. For example, the following is a sentence: “If I went to the store, I would buy milk.” This issue is not merely one of conforming to arbitrary rules. Remember, a grammatically correct sentence expresses a complete thought.2 The construction “If I went to the store.” does not do this. One wants to By Dr. Robert Tierney. This document is being used by Dr. Tierney for teaching purposes and is not intended for use or publication in any other manner. 1 More precisely, a statement is a meaningful declarative sentence-type. -
HISTORICAL ANTECEDENTS of ST. PIUS X's DECREE on FREQUENT COMMUNION JOHN A
HISTORICAL ANTECEDENTS OF ST. PIUS X's DECREE ON FREQUENT COMMUNION JOHN A. HARDON, SJ. West Baden College HPHE highest tribute to the apostolic genius of St. Pius X was paid by * his successor on the day he raised him to the honors of the altar: "in the profound vision which he had of the Church as a society, Pius X recognized that it was the Blessed Sacrament which had the power to nourish its intimate life substantially, and to elevate it high above all other human societies." To this end "he overcame the prejudices springing from an erroneous practice and resolutely promoted frequent, even daily, Communion among the faithful," thereby leading "the spouse of Christ into a new era of Euchari^tic life."1 In order to appreciate the benefits which Pius X conferred on the Church by his decree on frequent Communion, we might profitably examine the past half-century to see how the practice which he advo cated has revitalized the spiritual life of millions of the faithful. Another way is to go back in history over the centuries preceding St. Pius and show that the discipline which he promulgated in 1905 is at once a vindication of the Church's fidelity to her ancient traditions and a proof of her vitality to be rid of whatever threatens to destroy her divine mission as the sanctifier of souls. The present study will follow the latter method, with an effort to cover all the principal factors in this Eucharistic development which had its roots in the apostolic age but was not destined to bear full fruit until the present time. -
Ibn Rushd and the Enlightenment Project in the Islamic World
Religions 2015, 6, 1082–1106; doi:10.3390/rel6031082 OPEN ACCESS religions ISSN 2077-1444 www.mdpi.com/journal/religions Article Qur’anic Interpretation and the Problem of Literalism: Ibn Rushd and the Enlightenment Project in the Islamic World Chryssi Sidiropoulou Department of Philosophy, Faculty of Arts and Sciences, Boğaziçi University, TB 350, Bebek 343 42, Istanbul, Turkey; E-Mail: [email protected]; Tel.: +90-212-3595400 (ext. 7210); Fax: +90-212-2872469 Academic Editor: Jonathan Hill Received: 4 May 2015 / Accepted: 26 August 2015 / Published: 11 September 2015 Abstract: This article examines the claim that Ibn Rushd of Cordoba (“Averroës,” 12th century B.C.) is a precursor of the Enlightenment and a source of inspiration for the emancipation of contemporary Islamic societies. The paper critically discusses the fascination that Ibn Rushd has exercised on several thinkers, from Ernest Renan to Salman Rushdie, and highlights the problem of literalism in Qur’anic interpretation. Based on Ibn Rushd’s Decisive Treatise (Fasl al-maqāl), the paper investigates Ibn Rushd’s proposed division of (Muslim) society into three distinct classes. The main question here is whether there is a substantial link between the people of the Muslim community, given the three distinct kinds of assent (tasdīq) introduced by Ibn Rushd. I argue that if such a link cannot be supplied, then it is hard to see in Ibn Rushd an enlightened social model for today’s Muslim societies. Furthermore, that the great majority of people are prevented from having any contact with non-literal interpretation of the Scripture and non-revealed ways of thinking. -
Leibniz' De Arte Combinatoria
Leibniz’ De arte combinatoria John N. Martin Department of Philosophy University of Cincinnati Cincinnati, OH 45221 [email protected] John N. Martin, 2003 Page 1 Leibniz’ De arte combinatoria I. INTRODUCTION Logicians, philosophers and to judge from the Internet even the general public are vaguely aware that Leibniz held views about logic that anticipate modern ideas of proof system and algorithm. Though there are many places in Leibniz’ works that might be cited as evidence for such claims, popular works cite virtually only two of Leibniz’ shorter papers, Characteristica universalis and De arte combinatoria. Curiously, though there are hundreds, maybe thousands, of references to these papers, nothing serious has been written in recent decades about the papers themselves that could be called a professional exegesis or discussion of their logical content. The purpose of this short paper is to remedy that lack by offering a “reconstruction” of the system Leibniz sketches in De arte combinatoria, which of the two essays is the one more focused on the notions of proof and algorithm. A point of caution about method should be made at the outset. Any modern “reconstruction” of views in the history of logic is by its nature a compromise. It is an attempt to preserve as much of the original content, including its terminology and formulas, as is possible while simultaneously meeting the standards of modern metatheory. For example, if it is possible to do justice to the original by observing standard formats, then they should be Page 2 followed. For example, if it is fair to the text, it is desirable to define the syntax inductively, state definitions set theoretically, develop notions of proof within an axiom or natural deduction system, and define semantic ideas in a recursive manner parallel to syntax.