Truth-Bearers and Truth Value*
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Rhetorical Analysis
RHETORICAL ANALYSIS PURPOSE Almost every text makes an argument. Rhetorical analysis is the process of evaluating elements of a text and determining how those elements impact the success or failure of that argument. Often rhetorical analyses address written arguments, but visual, oral, or other kinds of “texts” can also be analyzed. RHETORICAL FEATURES – WHAT TO ANALYZE Asking the right questions about how a text is constructed will help you determine the focus of your rhetorical analysis. A good rhetorical analysis does not try to address every element of a text; discuss just those aspects with the greatest [positive or negative] impact on the text’s effectiveness. THE RHETORICAL SITUATION Remember that no text exists in a vacuum. The rhetorical situation of a text refers to the context in which it is written and read, the audience to whom it is directed, and the purpose of the writer. THE RHETORICAL APPEALS A writer makes many strategic decisions when attempting to persuade an audience. Considering the following rhetorical appeals will help you understand some of these strategies and their effect on an argument. Generally, writers should incorporate a variety of different rhetorical appeals rather than relying on only one kind. Ethos (appeal to the writer’s credibility) What is the writer’s purpose (to argue, explain, teach, defend, call to action, etc.)? Do you trust the writer? Why? Is the writer an authority on the subject? What credentials does the writer have? Does the writer address other viewpoints? How does the writer’s word choice or tone affect how you view the writer? Pathos (a ppeal to emotion or to an audience’s values or beliefs) Who is the target audience for the argument? How is the writer trying to make the audience feel (i.e., sad, happy, angry, guilty)? Is the writer making any assumptions about the background, knowledge, values, etc. -
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. -
Logic, Sets, and Proofs David A
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical statements with capital letters A; B. Logical statements be combined to form new logical statements as follows: Name Notation Conjunction A and B Disjunction A or B Negation not A :A Implication A implies B if A, then B A ) B Equivalence A if and only if B A , B Here are some examples of conjunction, disjunction and negation: x > 1 and x < 3: This is true when x is in the open interval (1; 3). x > 1 or x < 3: This is true for all real numbers x. :(x > 1): This is the same as x ≤ 1. Here are two logical statements that are true: x > 4 ) x > 2. x2 = 1 , (x = 1 or x = −1). Note that \x = 1 or x = −1" is usually written x = ±1. Converses, Contrapositives, and Tautologies. We begin with converses and contrapositives: • The converse of \A implies B" is \B implies A". • The contrapositive of \A implies B" is \:B implies :A" Thus the statement \x > 4 ) x > 2" has: • Converse: x > 2 ) x > 4. • Contrapositive: x ≤ 2 ) x ≤ 4. 1 Some logical statements are guaranteed to always be true. These are tautologies. Here are two tautologies that involve converses and contrapositives: • (A if and only if B) , ((A implies B) and (B implies A)). In other words, A and B are equivalent exactly when both A ) B and its converse are true. -
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. -
Topics in Philosophical Logic
Topics in Philosophical Logic The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Litland, Jon. 2012. Topics in Philosophical Logic. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9527318 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA © Jon Litland All rights reserved. Warren Goldfarb Jon Litland Topics in Philosophical Logic Abstract In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the log- ical constants to be given by their elimination rules. I then consider the question: given a set of introduction (elimination) rules , what are the R strongest elimination (introduction) rules that are validated by an assertabil- ity (consequence) conditional meaning-theory based on ? I prove that the R intuitionistic introduction (elimination) rules are the strongest rules that are validated by the intuitionistic elimination (introduction) rules. I then prove that intuitionistic logic is the strongest logic that can be given either an assertability-conditional or consequence-conditional meaning-theory. In “Grounding Grounding” I discuss the notion of grounding. My discus- sion revolves around the problem of iterated grounding-claims. -
Gottfried Wilhelm Leibniz (1646-1716)
Gottfried Wilhelm Leibniz (1646-1716) • His father, a professor of Philosophy, died when he was small, and he was brought up by his mother. • He learnt Latin at school in Leipzig, but taught himself much more and also taught himself some Greek, possibly because he wanted to read his father’s books. • He studied law and logic at Leipzig University from the age of fourteen – which was not exceptionally young for that time. • His Ph D thesis “De Arte Combinatoria” was completed in 1666 at the University of Altdorf. He was offered a chair there but turned it down. • He then met, and worked for, Baron von Boineburg (at one stage prime minister in the government of Mainz), as a secretary, librarian and lawyer – and was also a personal friend. • Over the years he earned his living mainly as a lawyer and diplomat, working at different times for the states of Mainz, Hanover and Brandenburg. • But he is famous as a mathematician and philosopher. • By his own account, his interest in mathematics developed quite late. • An early interest was mechanics. – He was interested in the works of Huygens and Wren on collisions. – He published Hypothesis Physica Nova in 1671. The hypothesis was that motion depends on the action of a spirit ( a hypothesis shared by Kepler– but not Newton). – At this stage he was already communicating with scientists in London and in Paris. (Over his life he had around 600 scientific correspondents, all over the world.) – He met Huygens in Paris in 1672, while on a political mission, and started working with him. -
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. -
Philosophy of Language in the Twentieth Century Jason Stanley Rutgers University
Philosophy of Language in the Twentieth Century Jason Stanley Rutgers University In the Twentieth Century, Logic and Philosophy of Language are two of the few areas of philosophy in which philosophers made indisputable progress. For example, even now many of the foremost living ethicists present their theories as somewhat more explicit versions of the ideas of Kant, Mill, or Aristotle. In contrast, it would be patently absurd for a contemporary philosopher of language or logician to think of herself as working in the shadow of any figure who died before the Twentieth Century began. Advances in these disciplines make even the most unaccomplished of its practitioners vastly more sophisticated than Kant. There were previous periods in which the problems of language and logic were studied extensively (e.g. the medieval period). But from the perspective of the progress made in the last 120 years, previous work is at most a source of interesting data or occasional insight. All systematic theorizing about content that meets contemporary standards of rigor has been done subsequently. The advances Philosophy of Language has made in the Twentieth Century are of course the result of the remarkable progress made in logic. Few other philosophical disciplines gained as much from the developments in logic as the Philosophy of Language. In the course of presenting the first formal system in the Begriffsscrift , Gottlob Frege developed a formal language. Subsequently, logicians provided rigorous semantics for formal languages, in order to define truth in a model, and thereby characterize logical consequence. Such rigor was required in order to enable logicians to carry out semantic proofs about formal systems in a formal system, thereby providing semantics with the same benefits as increased formalization had provided for other branches of mathematics. -
Chapter 4 Statement Logic
Chapter 4 Statement Logic In this chapter, we treat sentential or statement logic, logical systems built on sentential connectives like and, or, not, if, and if and only if. Our goals here are threefold: 1. To lay the foundation for a later treatment of full predicate logic. 2. To begin to understand how we might formalize linguistic theory in logical terms. 3. To begin to think about how logic relates to natural language syntax and semantics (if at all!). 4.1 The intuition The basic idea is that we will define a formal language with a syntax and a semantics. That is, we have a set of rules for how statements can be constructed and then a separate set of rules for how those statements can be interpreted. We then develop—in precise terms—how we might prove various things about sets of those statements. 4.2 Basic Syntax We start with an finite set of letters: {p,q,r,s,...}. These become the infinite set of atomic statements when we add in primes, e.g. p′,p′′,p′′′,.... This infinite set can be defined recursively. 46 CHAPTER 4. STATEMENT LOGIC 47 Definition 4 (Atomic statements) Recursive definition: 1. Any letter {p,q,r,s,...} is an atomic statement. 2. Any atomic statement followed by a prime, e.g. p′, q′′,... is an atomic statement. 3. There are no other atomic statements. The first clause provides for a finite number of atomic statements: 26. The second clause is the recursive one, allowing for an infinite number of atomic statements built on the finite set provided by the first clause. -
Epictetus, Stoicism, and Slavery
Epictetus, Stoicism, and Slavery Defense Date: March 29, 2011 By: Angela Marie Funk Classics Department Advisor: Dr. Peter Hunt (Classics) Committee: Dr. Jacqueline Elliott (Classics) and Dr. Claudia Mills (Philosophy) Funk 1 Abstract: Epictetus was an ex-slave and a leading Stoic philosopher in the Roman Empire during the second-century. His devoted student, Arrian, recorded Epictetus’ lectures and conversations in eight books titled Discourses, of which only four are extant. As an ex- slave and teacher, one expects to see him deal with the topic of slavery and freedom in great detail. However, few scholars have researched the relationship of Epictetus’ personal life and his views on slavery. In order to understand Epictetus’ perspective, it is essential to understand the political culture of his day and the social views on slavery. During his early years, Epictetus lived in Rome and was Epaphroditus’ slave. Epaphroditus was an abusive master, who served Nero as an administrative secretary. Around the same period, Seneca was a tutor and advisor to Nero. He was a Stoic philosopher, who counseled Nero on political issues and advocated the practice of clemency. In the mid to late first-century, Seneca spoke for a fair and kind treatment of slaves. He held a powerful position not only as an advisor to Nero, but also as a senator. While he promoted the humane treatment of slaves, he did not actively work to abolish slavery. Epaphroditus and Seneca both had profound influences in the way Epictetus viewed slaves and ex-slaves, relationships of former slaves and masters, and the meaning of freedom.