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Logical Truth, Contradictions, Inconsistency, and Logical Equivalence
Logical Truth, Contradictions, Inconsistency, and Logical Equivalence Logical Truth • The semantical concepts of logical truth, contradiction, inconsistency, and logi- cal equivalence in Predicate Logic are straightforward adaptations of the corre- sponding concepts in Sentence Logic. • A closed sentence X of Predicate Logic is logically true, X, if and only if X is true in all interpretations. • The logical truth of a sentence is proved directly using general reasoning in se- mantics. • Given soundness, one can also prove the logical truth of a sentence X by provid- ing a derivation with no premises. • The result of the derivation is that X is theorem, ` X. An Example • (∀x)Fx ⊃ (∃x)Fx. – Suppose d satisfies ‘(∀x)Fx’. – Then all x-variants of d satisfy ‘Fx’. – Since the domain D is non-empty, some x-variant of d satisfies ‘Fx’. – So d satisfies ‘(∃x)Fx’ – Therefore d satisfies ‘(∀x)Fx ⊃ (∃x)Fx’, QED. • ` (∀x)Fx ⊃ (∃x)Fx. 1 (∀x)Fx P 2 Fa 1 ∀ E 3 (∃x)Fx 1 ∃ I Contradictions • A closed sentence X of Predicate Logic is a contradiction if and only if X is false in all interpretations. • A sentence X is false in all interpretations if and only if its negation ∼X is true on all interpretations. • Therefore, one may directly demonstrate that a sentence is a contradiction by proving that its negation is a logical truth. • If the ∼X of a sentence is a logical truth, then given completeness, it is a theorem, and hence ∼X can be derived from no premises. 1 • If a sentence X is such that if it is true in any interpretation, both Y and ∼Y are true in that interpretation, then X cannot be true on any interpretation. -
The Real Estate Marketplace Glossary: How to Talk the Talk
Federal Trade Commission ftc.gov The Real Estate Marketplace Glossary: How to Talk the Talk Buying a home can be exciting. It also can be somewhat daunting, even if you’ve done it before. You will deal with mortgage options, credit reports, loan applications, contracts, points, appraisals, change orders, inspections, warranties, walk-throughs, settlement sheets, escrow accounts, recording fees, insurance, taxes...the list goes on. No doubt you will hear and see words and terms you’ve never heard before. Just what do they all mean? The Federal Trade Commission, the agency that promotes competition and protects consumers, has prepared this glossary to help you better understand the terms commonly used in the real estate and mortgage marketplace. A Annual Percentage Rate (APR): The cost of Appraisal: A professional analysis used a loan or other financing as an annual rate. to estimate the value of the property. This The APR includes the interest rate, points, includes examples of sales of similar prop- broker fees and certain other credit charges erties. a borrower is required to pay. Appraiser: A professional who conducts an Annuity: An amount paid yearly or at other analysis of the property, including examples regular intervals, often at a guaranteed of sales of similar properties in order to de- minimum amount. Also, a type of insurance velop an estimate of the value of the prop- policy in which the policy holder makes erty. The analysis is called an “appraisal.” payments for a fixed period or until a stated age, and then receives annuity payments Appreciation: An increase in the market from the insurance company. -
Philosophical Theory-Construction and the Self-Image of Philosophy
Open Journal of Philosophy, 2014, 4, 231-243 Published Online August 2014 in SciRes. http://www.scirp.org/journal/ojpp http://dx.doi.org/10.4236/ojpp.2014.43031 Philosophical Theory-Construction and the Self-Image of Philosophy Niels Skovgaard Olsen Department of Philosophy, University of Konstanz, Konstanz, Germany Email: [email protected] Received 25 May 2014; revised 28 June 2014; accepted 10 July 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract This article takes its point of departure in a criticism of the views on meta-philosophy of P.M.S. Hacker for being too dismissive of the possibility of philosophical theory-construction. But its real aim is to put forward an explanatory hypothesis for the lack of a body of established truths and universal research programs in philosophy along with the outline of a positive account of what philosophical theories are and of how to assess them. A corollary of the present account is that it allows us to account for the objective dimension of philosophical discourse without taking re- course to the problematic idea of there being worldly facts that function as truth-makers for phi- losophical claims. Keywords Meta-Philosophy, Hacker, Williamson, Philosophical Theories 1. Introduction The aim of this article is to use a critical discussion of the self-image of philosophy presented by P. M. S. Hacker as a platform for presenting an alternative, which offers an account of how to think about the purpose and cha- racter of philosophical theories. -
Denying a Dualism: Goodman's Repudiation of the Analytic/Synthetic Distinction
Midwest Studies in Philosophy, 28, 2004, 226-238. Denying a Dualism: Goodman’s Repudiation of the Analytic/Synthetic Distinction Catherine Z. Elgin The analytic synthetic/distinction forms the backbone of much modern Western philosophy. It underwrites a conception of the relation of representations to reality which affords an understanding of cognition. Its repudiation thus requires a fundamental reconception and perhaps a radical revision of philosophy. Many philosophers believe that the repudiation of the analytic/synthetic distinction and kindred dualisms constitutes a major loss, possibly even an irrecoverable loss, for philosophy. Nelson Goodman thinks otherwise. He believes that it liberates philosophy from unwarranted restrictions, creating opportunities for the development of powerful new approaches to and reconceptions of seemingly intractable problems. In this article I want to sketch some of the consequences of Goodman’s reconception. My focus is not on Goodman’s reasons for denying the dualism, but on some of the ways its absence affects his position. I do not contend that the Goodman obsessed over the issue. I have no reason to think that the repudiation of the distinction was a central factor in his intellectual life. But by considering the function that the analytic/synthetic distinction has performed in traditional philosophy, and appreciating what is lost and gained in repudiating it, we gain insight into Goodman’s contributions. I begin then by reviewing the distinction and the conception of philosophy it supports. The analytic/synthetic distinction is a distinction between truths that depend entirely on meaning and truths that depend on both meaning and fact. In the early modern period, it was cast as a distinction between relations of ideas and matters of fact. -
God As Both Ideal and Real Being in the Aristotelian Metaphysics
God As Both Ideal and Real Being In the Aristotelian Metaphysics Martin J. Henn St. Mary College Aristotle asserts in Metaphysics r, 1003a21ff. that "there exists a science which theorizes on Being insofar as Being, and on those attributes which belong to it in virtue of its own nature."' In order that we may discover the nature of Being Aristotle tells us that we must first recognize that the term "Being" is spoken in many ways, but always in relation to a certain unitary nature, and not homonymously (cf. Met. r, 1003a33-4). Beings share the same name "eovta," yet they are not homonyms, for their Being is one and the same, not manifold and diverse. Nor are beings synonyms, for synonymy is sameness of name among things belonging to the same genus (as, say, a man and an ox are both called "animal"), and Being is no genus. Furthermore, synonyms are things sharing a common intrinsic nature. But things are called "beings" precisely because they share a common relation to some one extrinsic nature. Thus, beings are neither homonyms nor synonyms, yet their core essence, i.e. their Being as such, is one and the same. Thus, the unitary Being of beings must rest in some unifying nature extrinsic to their respective specific essences. Aristotle's dialectical investigations into Being eventually lead us to this extrinsic nature in Book A, i.e. to God, the primary Essence beyond all specific essences. In the pre-lambda books of the Metaphysics, however, this extrinsic nature remains very much up for grabs. -
Pluralisms About Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected]
University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 8-9-2019 Pluralisms about Truth and Logic Nathan Kellen University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Kellen, Nathan, "Pluralisms about Truth and Logic" (2019). Doctoral Dissertations. 2263. https://opencommons.uconn.edu/dissertations/2263 Pluralisms about Truth and Logic Nathan Kellen, PhD University of Connecticut, 2019 Abstract: In this dissertation I analyze two theories, truth pluralism and logical pluralism, as well as the theoretical connections between them, including whether they can be combined into a single, coherent framework. I begin by arguing that truth pluralism is a combination of realist and anti-realist intuitions, and that we should recognize these motivations when categorizing and formulating truth pluralist views. I then introduce logical functionalism, which analyzes logical consequence as a functional concept. I show how one can both build theories from the ground up and analyze existing views within the functionalist framework. One upshot of logical functionalism is a unified account of logical monism, pluralism and nihilism. I conclude with two negative arguments. First, I argue that the most prominent form of logical pluralism faces a serious dilemma: it either must give up on one of the core principles of logical consequence, and thus fail to be a theory of logic at all, or it must give up on pluralism itself. I call this \The Normative Problem for Logical Pluralism", and argue that it is unsolvable for the most prominent form of logical pluralism. Second, I examine an argument given by multiple truth pluralists that purports to show that truth pluralists must also be logical pluralists. -
Philosophical Logic 2018 Course Description (Tentative) January 15, 2018
Philosophical Logic 2018 Course description (tentative) January 15, 2018 Valentin Goranko Introduction This 7.5 hp course is mainly intended for students in philosophy and is generally accessible to a broad audience with basic background on formal classical logic and general appreciation of philosophical aspects of logic. Practical information The course will be given in English. It will comprise 18 two-hour long sessions combining lectures and exercises, grouped in 2 sessions per week over 9 teaching weeks. The weekly pairs of 2-hour time slots (incl. short breaks) allocated for these sessions, will be on Mondays during 10.00-12.00 and 13.00-15.00, except for the first lecture. The course will begin on Wednesday, April 4, 2018 (Week 14) at 10.00 am in D220, S¨odra huset, hus D. Lecturer's info: Name: Valentin Goranko Email: [email protected] Homepage: http://www2.philosophy.su.se/goranko Course webpage: http://www2.philosophy.su.se/goranko/Courses2018/PhilLogic-2018.html Prerequisites The course will be accessible to a broad audience with introductory background on classical formal logic. Some basic knowledge of modal logics would be an advantage but not a prerequisite. Brief description Philosophical logic studies a variety of non-classical logical systems intended to formalise and reason about various philosophical concepts and ideas. They include a broad family of modal logics, as well as many- valued, intuitionistic, relevant, conditional, non-monotonic, para-consistent, etc. logics. Modal logics extend classical logic with additional intensional logical operators, reflecting different modes of truth, including alethic, epistemic, doxastic, temporal, deontic, agentive, etc. -
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. -
Verifying the Unification Algorithm In
Verifying the Uni¯cation Algorithm in LCF Lawrence C. Paulson Computer Laboratory Corn Exchange Street Cambridge CB2 3QG England August 1984 Abstract. Manna and Waldinger's theory of substitutions and uni¯ca- tion has been veri¯ed using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an exam- ple of interaction with LCF. Translating the theory into LCF's domain- theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Cor- rectness of uni¯cation is expressed using predicates for such properties as idempotence and most-generality. The veri¯cation is presented as a series of lemmas. The LCF proofs are compared with the original ones, and with other approaches. It appears di±cult to ¯nd a logic that is both simple and exible, especially for proving termination. Contents 1 Introduction 2 2 Overview of uni¯cation 2 3 Overview of LCF 4 3.1 The logic PPLAMBDA :::::::::::::::::::::::::: 4 3.2 The meta-language ML :::::::::::::::::::::::::: 5 3.3 Goal-directed proof :::::::::::::::::::::::::::: 6 3.4 Recursive data structures ::::::::::::::::::::::::: 7 4 Di®erences between the formal and informal theories 7 4.1 Logical framework :::::::::::::::::::::::::::: 8 4.2 Data structure for expressions :::::::::::::::::::::: 8 4.3 Sets and substitutions :::::::::::::::::::::::::: 9 4.4 The induction principle :::::::::::::::::::::::::: 10 5 Constructing theories in LCF 10 5.1 Expressions :::::::::::::::::::::::::::::::: -
BY PLATO• ARISTOTLE • .AND AQUINAS I
i / REF1,l!;CTit.>NS ON ECONOMIC PROBLEMS / BY PLATO• ARISTOTLE • .AND AQUINAS ii ~FLECTIONS ON ECONO:MIC PROBLEMS 1 BY PLA'I'O, ARISTOTLE, JJJD AQUINAS, By EUGENE LAIDIBEL ,,SWEARINGEN Bachelor of Science Oklahoma Agricultural and Mechanical Collage Stillwater, Oklahoma 1941 Submitted to the Depertmeut of Economics Oklahoma Agricultural and Mechanical College In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE 1948 iii f.. 'I. I ···· i·: ,\ H.: :. :· ··: ! • • ~ ' , ~ • • !·:.· : i_ ·, 1r 1i1. cr~~rJ3t L l: i{ ,\ I~ Y , '•T •)() 1 0 ,1 8 API-'HOV~D BY: .J ,.· 1.., J l.;"t .. ---- -··- - ·- ______.,.. I 7 -.. JI J ~ L / \ l v·~~ u ' ~) (;_,LA { 7 {- ' r ~ (\.7 __\ _. ...A'_ ..;f_ ../-_" ...._!)_.... ..." ___ ......._ ·;...;;; ··-----/ 1--.,i-----' ~-.._.._ :_..(__,,---- ....... Member of the Report Committee 1..j lj:,;7 (\ - . "'·- -· _ .,. ·--'--C. r, .~-}, .~- Q_ · -~ Q.- 1Head of the Department . · ~ Dean of the Graduate School 502 04 0 .~ -,. iv . r l Preface The purpose and plan of this report are set out in the Introduction. Here, I only wish to express my gratitude to Professor Russell H. Baugh who has helped me greatly in the preparation of this report by discussing the various subjects as they were in the process of being prepared. I am very much indebted to Dr. Harold D. Hantz for his commentaries on the report and for the inspiration which his classes in Philosophy have furnished me as I attempted to correlate some of the material found in these two fields, Ecor!.Omics and Philosophy. I should like also to acknowledge that I owe my first introduction into the relationships of Economics and Philosophy to Dean Raymond Thomas, and his com~ents on this report have been of great value. -
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.