Notes on Mathematical Logic
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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. -
Notes on Mathematical Logic David W. Kueker
Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. -
Chapter 3 – Describing Syntax and Semantics CS-4337 Organization of Programming Languages
!" # Chapter 3 – Describing Syntax and Semantics CS-4337 Organization of Programming Languages Dr. Chris Irwin Davis Email: [email protected] Phone: (972) 883-3574 Office: ECSS 4.705 Chapter 3 Topics • Introduction • The General Problem of Describing Syntax • Formal Methods of Describing Syntax • Attribute Grammars • Describing the Meanings of Programs: Dynamic Semantics 1-2 Introduction •Syntax: the form or structure of the expressions, statements, and program units •Semantics: the meaning of the expressions, statements, and program units •Syntax and semantics provide a language’s definition – Users of a language definition •Other language designers •Implementers •Programmers (the users of the language) 1-3 The General Problem of Describing Syntax: Terminology •A sentence is a string of characters over some alphabet •A language is a set of sentences •A lexeme is the lowest level syntactic unit of a language (e.g., *, sum, begin) •A token is a category of lexemes (e.g., identifier) 1-4 Example: Lexemes and Tokens index = 2 * count + 17 Lexemes Tokens index identifier = equal_sign 2 int_literal * mult_op count identifier + plus_op 17 int_literal ; semicolon Formal Definition of Languages • Recognizers – A recognition device reads input strings over the alphabet of the language and decides whether the input strings belong to the language – Example: syntax analysis part of a compiler - Detailed discussion of syntax analysis appears in Chapter 4 • Generators – A device that generates sentences of a language – One can determine if the syntax of a particular sentence is syntactically correct by comparing it to the structure of the generator 1-5 Formal Methods of Describing Syntax •Formal language-generation mechanisms, usually called grammars, are commonly used to describe the syntax of programming languages. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
A Simple Definition of General Semantics Ben Hauck *
A SIMPLE DEFINITION OF GENERAL SEMANTICS BEN HAUCK * [A] number of isolated facts does not produce a science any more than a heap of bricks produces a house. The isolated facts must be put in order and brought into mutual structural relations in the form of some theory. Then, only, do we have a science, something to start from, to analyze, ponder on, criticize, and improve. – Alfred Korzybski Science & Sanity: An Introduction to Non-Aristotelian Systems and General Semantics1 The term, ‘semantic reaction’ will be used as covering both semantic reflexes and states. In the present work, we are interested in [semantic reactions], from a psychophysiological, theoretical and experimental point of view, which include the corresponding states. – Alfred Korzybski Science & Sanity: An Introduction to Non-Aristotelian Systems and General Semantics2 OR A NUMBER OF DECADES and perhaps for all of its life, general semantics has Fsuffered from an identity crisis. People have long had difficulty defining the term general semantics for others. Of those people who have settled on definitions, many of their definitions are too vague, too general, or just plain awkward. The bulk of these definitions is of the awkward sort, more like descriptions than definitions3, leading to a hazy image of general semantics and a difficulty in categorizing it in the grand scheme of fields. Because of awkward definitions, people learning of general semantics for the first time can’t relate to it, so they don’t become interested in it. *Ben Hauck webmasters the websites for the Institute of General Semantics and the New York Soci- ety for General Semantics. -
Scholasticism Old and New : an Introduction to Scholastic
f^frrninnamvfuv^ii^ 3 1924 102 136 409 DATE DUE 1 i The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924102136409 SCHOLASTICISM OLD AND NEW Vetera Novis Augere. SCHOLASTICISM OLD AND NEW AN INTRODUCTION TO SCHOLASTIC PHILOSOPHY MEDIEVAL AND MODEKN BY M. DE WULF DOCTOR OF LAWS, l.ciCTOR OF PHILOSOPHy AND LETTERS, PROFESSOR AT THE UNIVERSITY OF LOUVAIN TRANSLATED BY P. COFFEY, D.Ph. PROFESSOR UF PllILOSOPilV, MAISOOTH COLLtOB, IRELAND Jublin M H. GILL & SON, Lm LONGMANS, GREEN & 00. 39 PATBKNOSTEK EOW BOMBAV AND CALCUTTA. 1910 Printed and Bound in Ireland, ^3f ; PEEFATOEY NOTE. My object in translating Professor De Wulf's Introduction a la Philosophie Neo-scolastique has been fourfold : firstly, to give tlie advocates and supporters of " modern " systems of philosopby, as opposed " to scholasticism" —whether in its medieval or in its modern form—an opportunity of obtaining better and more authentic information about the latter system than books in English are usually found to contain ; secondly, to help students of scholastic philosophy to take in the main principles of schol- asticism in one connected view, and to equip them with a more accurate historical and critical appre- ciation of the system than they are ever hkely to derive from an unaided study of stereotyped manuals thirdly, to give aU Enghsh readers interested in philosophy of whatsoever kind an insight into the meaning, the spirit and the progress of the move- ment which has been developing during the last quarter of a century for the revival of scholastic philosophy ; fourthly, to prepare the way for trans- lations or adaptations of the Louvain Cours de philosophie, and to draw attention to the vahie of the work already done and hkely to be done in the well-known Belgian centre of the new scholasticism. -
Scholasticism, the Medieval Synthesis CVSP 202 General Lecture Monday, March 23, 2015 Encountering Aristotle in the Middle-Ages Hani Hassan [email protected]
Scholasticism, The Medieval Synthesis CVSP 202 General Lecture Monday, March 23, 2015 Encountering Aristotle in the middle-ages Hani Hassan [email protected] PART I – SCHOLASTIC PHILOSOPHY SCHOLASTIC: “from Middle French scholastique, from Latin scholasticus "learned," from Greek skholastikos "studious, learned"” 1 Came to be associated with the ‘teachers’ and churchmen in European Universities whose work was generally rooted in Aristotle and the Church Fathers. A. Re-discovering Aristotle & Co – c. 12th Century The Toledo school: translating Arabic & Hebrew philosophy and science as well as Arabic & Hebrew translations of Aristotle and commentaries on Aristotle (notably works of Ibn Rushd and Maimonides) Faith and Reason united From Toledo, through Provence, to… th by German painter Ludwig Seitz Palermo, Sicily – 13 Century – Under the rule of Roger of Sicily (Norman (1844–1908) ruler) th William of Moerbeke (Dutch cleric): greatest translator of the 13 century, and (possibly) ‘colleague’ of Thomas Aquinas. B. Greek Philosophy in the Christian monotheistic world: Boethius to scholasticism Boethius: (c. 475 – c. 526) philosopher, poet, politician – best known for his work Consolation of Philosophy, written as a dialogue between Boethius and 'Lady Philosophy', addressing issues ranging from the nature and essence of God, to evil, and ethics; translated Aristotle’s works on logic into Latin, and wrote commentaries on said works. John Scotus Eriugena (c. 815 – c. 877) Irish philosopher of the early monastic period; translated into Latin the works of pseudo-Dionysius, developed a Christian Neoplatonic world view. Anselm (c. 1033 – 1109) Archbishop of Canterbury from 1093 to 1109; best known for his “ontological argument” for the existence of God in chapter two of the Proslogion (translated into English: Discourse on the Existence of God); referred to by many as one of the founders of scholasticism. -
What Is Mathematical Logic? a Survey
What is mathematical logic? A survey John N. Crossley1 School of Computer Science and Software Engineering, Monash University, Clayton, Victoria, Australia 3800 [email protected] 1 Introduction What is mathematical logic? Mathematical logic is the application of mathemat- ical techniques to logic. What is logic? I believe I am following the ancient Greek philosopher Aristotle when I say that logic is the (correct) rearranging of facts to find the information that we want. Logic has two aspects: formal and informal. In a sense logic belongs to ev- eryone although we often accuse others of being illogical. Informal logic exists whenever we have a language. In particular Indian Logic has been known for a very long time. Formal (often called, ‘mathematical’) logic has its origins in ancient Greece in the West with Aristotle. Mathematical logic has two sides: syntax and semantics. Syntax is how we say things; semantics is what we mean. By looking at the way that we behave and the way the world behaves, Aris- totle was able to elicit some basic laws. His style of categorizing logic led to the notion of the syllogism. The most famous example of a syllogism is All men are mortal Socrates is a man [Therefore] Socrates is mortal Nowadays we mathematicians would write this as ∀x(Man(x) → Mortal (x)) Man(S) (1) Mortal (S) One very general form of the above rule is A (A → B) (2) B otherwise know as modus ponens or detachment: the A is ‘detached’ from the formula (A → B) leaving B. This is just one example of a logical rule. -
In Defense of Radical Empiricism
In Defense of Radical Empiricism A thesis presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Master of Arts Ryan D. Ross May 2015 © 2015 Ryan D. Ross. All Rights Reserved. 2 This thesis titled In Defense of Radical Empiricism by RYAN D. ROSS has been approved for the Department of Philosophy and the College of Arts and Sciences by John W. Bender Professor of Philosophy Robert Frank Dean, College of Arts and Sciences 3 Abstract ROSS, RYAN D., M. A., May 2015, Philosophy In Defense of Radical Empiricism Director of Thesis: John W. Bender Laurence BonJour defends a moderate version of rationalism against rivaling empiricist epistemologies. His moderate rationalism maintains that some beliefs are justified a priori in a way that does not reduce to mere analyticity, but he tempers this strong claim by saying that such justification is both fallible and empirically defeasible. With the aim of ruling out radical empiricism (the form of empiricism that repudiates the a priori), BonJour puts forth what he calls the “master argument.” According to this argument, the resources available to radical empiricists are too slender to allow for justified empirical beliefs that go beyond what is immediately available to sense- perception, e.g., what we see, hear, and taste. If so, then radical empiricists are committed to a severe form of skepticism, one in which it is impossible to have justified beliefs about the distant past, the future, unobserved aspects of the present, etc. Worse, radical empiricists, who pride themselves on their scientific worldview, would be unable to account for justified beliefs about the abstract, theoretical claims of science itself! Clearly, the master argument is intended to hit the radical empiricist where it hurts. -
Logical Positivism, Operationalism, and Behaviorism
This dissertation has been microfilmed exactly as received 70-6878 SHANAB, Robert E lias Abu, 1939- LOGICAL POSITIVISM, OPERATIONALISM, AND BEHAVIORISM. The Ohio State University, Ph.D., 1969 Philosophy University Microfilms, Inc., Ann Arbor, Michigan LOGICAL POSITIVISM, OPERATIONALISM, AND BEHAVIORISM DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Robert Elias Abu Shanab, B.A., A.M. ******** The Ohio State University 1969 Approved by / Adviser Department of Philosophy Dedicated to Professor Virgil Hinshaw, ACKNOWLEDGMENTS I am especially indebted to my adviser, Professor Virgil Hinshaw, Jr. Several of his suggestions have been incorporated in the final manuscript. I wish also to express my thanks to Professor Charles F. Kielkopf. Finally I wish to extend affection and gratitude to my wife for encouragement, patience and for the hours spent typing and retyping manuscripts. ii VITA September 29, 1939 B o m - Jerusalem, Palestine 1962 ........ ........... B.A. , San Jose State College, San Jose, California 1964 ................... M.A., San Jose State College, San Jose, California 1965-1966 ............. Instructor, College of San Mateo, San Mateo, California 1967-1968 ....... Teaching Assistant, Department of Philosophy, The Ohio State University, Columbus, Ohio 1969 ................... Lecturer, The Ohio State University, Newark, Ohio iii CONTENTS. Page ACKNOWLEDGMENTS ..... ....................... ii V I T A ..............................................iii -
Protestant Scholasticism: Some Methodological Considerations in the Study of Its Development
PROTESTANT SCHOLASTICISM: SOME METHODOLOGICAL CONSIDERATIONS IN THE STUDY OF ITS DEVELOPMENT WILLEM J.VAN ASSELT Utrecht This present issue of the Dutch Review of Church History brings together papers by European and American church historians presented during a colloquium at Utrecht University on June 6, 2001. All of them have in common that, in one way or another, they have contributed to the development of new directions in the research of scholasticism, in its medieval, Reformation and post-Reformation forms. One of the reasons for organising this colloquium was to present the book Reformation and Scholasticism edited by Eef Dekker and the present writer.' A second, more important, reason was to introduce the results of the new research on - scholasticism - which some call 'the new school approach' to a broader public. Let me explain this. For the adherents of this approach, it has always been a curious phenomenon that the post-Reformation period of Protestant theology is one of the least known in the history of Christian thought and, at the same time a period in the interpretation of which there are many hidden agendas. Post-Reformation theology is usually presented as a highly obscure period characterised by the return of medieval dialectic and Aristotelian logic to the Protestant classroom and, therefore a distortion or perversion of Reformation theology. For too long, this theology has been the victim of the attempt of modern theo- logians to claim the Reformers as forerunners of modernity. Time and again the theology of the post-Reformation period is read exclusively in the light of modern issues and not on its own terms and in the light of its own concern and context.