DOCSLIB.ORG

The Bounds of Logic Part 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

To Ik a Logical Term 37 Chapter 3 To Be a Logical Ternl logic; at the same time, the principles developed by Tarski in the 1930s are still the principles underlying logic in the early 1990s. My interest in Tarski is, Ileedless to say, not historical. I am interested in the modern conception of logic as it evolved out of Tarski's early work in semantics. The Task of Logic and the Origins of Semantics In "The Concept of Truth in Formalized Languages" (1933). "On the Concept of Logical Consequence" (1936a), and "The Establishment of Scientific Semantics" (1936b), Tarski describes the semantic project as comprising two tasks: Since the discovery ofgeneralized quantifiers by A. Mostowski (1957), the I. Definition of the gelleral concept of truth for formalized languages question "What is a logical term?" has taken on a significance it did not 2. Dclinition of the logical concepts of truth, consequence, consistency. have before. Are Mostowski's quantifiers Hlogical" quantifiers'! Do they etc. , differ in any significant way from the standard existential and universal The main purpose of (I) is to secure meta logic against semantic para­ I quantifiers? What logical operators, if any, has he left out? What. ill doxes. Tarski worried lest the ullcritical usc of semantic concepts prior to I all, are the first- and second-level predicates and rcla tions that can be his work concealed an inconsistency: a hidden fallacy would undermine construed as logical? the cntire venturc. Be therefore sought precise, materially, as well as One way in which I do not want to ask the question is, "What, ill Ihe formally, correct definitions of "truth" and related notions to serve as a I nature ofthings, makes a property or a relation logical'!" On this road lie hedge against paradox. This aspect of Tarski's work is well known. In t the controversies regarding necessity and apriority, and these, I bclieve, "Model Theory before 1945" R. Vaught (1974) puts Tarski's enterprise in should be left aside. Although some understanding of the modalities is a slightly different light: I essential for our enterprise, only their most general features come into [During the late I920s] Tarski had become dissatisfied with the notion of truth as play. A detailed study ofcomplex and intricate modal and epistemic issues it was heing lIsed. Since the notion "0 is true in '11" is highly intuitive (and perfectly would just divert our attention and is of little use here. But if "the nature clear for allY definite 0), it had heen possible to go even as far as the completeness of things" is not our measure, what is? What should our starting point be? theorem hy treating truth (consciously or unconsciously) essentially as an unde­ What strategy shall we decide upon? lined notion ---one with many ohvious properties .... But no one had made an analysis of truth, not even of exactly what is involved in treating it in the way just A promising approach is suggested by L. Tharp in "Which Logic Is the mcntioncd. At it time when it was quite well understood that 'all of mathematics' Right Logic'!" (1975). Tharp poses the question, What properties should a could he done, say, in ZF, with only the primitive notion E, this meant that the system oflogic have? In particular, is standard first-order logic the "right" theory of models (and hence much of meta logic) was indeed not part of mathe­ logic? To answer questions of this kind, he observes, it is crucial to have a matics. It sccms clear that this whole state of affairs was bound to cause a lack of dear idea about "the role logic is expected to play." 1 Tharp's point is sure-rootedness in metalogic.... [Tarski's] major contribution was to show that the notion "a is trlle in \11" can simply be defined inside of ordinary mathematics, worth taking, and it provides the clue we are searching for. If we identify for example, in ZF. 2 a central role of logic and, relative to that role, ask what expressions can function as logical terms, we will have found a perspective that makes our On both accounts the motivation for (I) has to do with the adequacy of question answerable, and significantly answerable at that. thc system designed to carry out the logical project, not with the logical The most suggestive discussion of the logical enterprise that I know of project itself. The goal of logic is notlhe mathematical definition of "true appears in A. Tarski's early papers on the foundations of semantics. scntcnce," and (I) is therefore a secondary, albeit crucially important, task Tarski's papers reveal the forces at work during the inception of modern ofTarski an logic. (2), on the other hand, does reflect Tarski's vision of the 39 Chapter 3 38 To Be a Logical Term role of logic. In paper after paper throughout the early 1930s Tarski In every deductive theory (apart from certain theories ofa particularly elementary 3 nature), however much we supplement the ordinary rules of inference by new described the logical project as follows: The goal is to develop and study purely structural rules, it is possible to construct sentences which follow, in the deductive systems. Given a formal system 2) with languagc L and a uSlIal sense, fwm the theorems of this theory, but which nevertheless cannot be 4 definition of "meaningful," i.e., "well-formed," sentcm:e for I"~ a (dosed) proved in this thcory on the basis of the accepted rules of inference. deductive system in !t:' is the set of all logical consequences of somc set X Tarski's conclusion was that proof theory provides only a partial ac­ of meaningful sentences of L "Logical consequcnce" was defined proof­ count of the logical concepts. A new method is called for that will permit theoretically in terms of logical axioms and rules of inferencc: if.w and .elf a more comprehensive systematization of the intuitive content of these are the sets of logical axioms and rules of inference of !.t', respectively, the set of logical cOllsequences of X ill !.t' is the smallcst set of well-formcd concepts. The intuitions underlying our informal notion of logical consequence sentences of L that includes X and and is closed undcr the rules in .w (and derivative concepts) are anchored, according to Tarski, in certain 91. In contemporary terminology, a deductive system is a jtJr/llul th('ory relationships between linguistic items and objects in (configurations of) within a logical framework !i:'. (Note that the logical framework itself can the world. The discipline that studies relationships of this kind is called be viewed as a deductive system, namely by taking X to be the set of logical axioms.) The task of logic, in this picture, is performcd in two steps: (a) semantics: We ... understand by semantics the totality of considerations concerning those the construction of a logical framework for formal (formalized) theories; concepts which, roughly speaking, ex.press certain connex.ions between the ex.pres­ (b) the investigation of the logical properties consistency, complctcncss, sions of a language and the objects and states of affairs referred to by these axiomatizability, etc.-~of formal theories relative to the logkal framc­ exprcssions. ~ work constructed in step (a). The concept of logical CO/l.\'('qll('llCl' (togcther The precise formulation of the intuitive content of the logical concepts is with that of a well-formed formula) is the key concept of Tarskil.111 logic. thcrcfore a job for semantics. (Although the relation between the set of Once the definition of "logical consequence" is given, we can easily scntences "/'11" and the universal quantification "('tx)Px," where x ranges obtain not only the notion of a deductive systcm but also those of a over the natural numbers and "n" stands for a name of a natural num­ logically true sentence; logically equivalent sets of sentences; an axiom ber, is not logical consequence, we will be able to characterize it ac­ system ofa set ofsentences; and axiomatizability, completeness, and con­ curately within the framework of Tarskian semantics, e.g., in terms of (JJ sistency of a set of sentences. The study of the conditions under which various formal theories possess these properties forms the subject matter complctellcss.) of metalogic. 2 The Semantic Definition of "Logical Consequence" and the Emergence Whence semantics? Prior to Tarski's "On the Concept of Logical Con­ sequence" the definitions of the logical concepts were proof-theoretical. of Models The need for semantic definitions of the same concepts arose when Tarski Tarski describes the intuitive content of the concept "logical consequence" realized that there was a serious gap between the proof-theoretic defini­ tions and the intuitive concepts they were intended to capture: many as follows: intuitive consequences of deductive systems could not be detected by the Certain considerations of an intuitive nature will form our starting-point. Consider any class K of sentences and a sentence X which follows from the sentences of this standard system of proof. Thus the sentence "For every natural number class. From an intuitive standpoint it can never happen that both the class K seems to follow, in some important sense, from the set of sentences n, Pn" consists only of true sentences and the sentence X is false. Moreover, ... we are "Pn," where Il is a natural number, but there is no way to express this f~\ct (.:oncerned here with thc concept of logical, Le.formal, consequence, and thus with by the proof method for standard first-order logic. This situation, Tarski a relation which is to be uniqucly dctermined by the form of the sentences between which it holds...
Recommended publications
  • Which Quantifiers Are Logical? a Combined Semantical and Inferential Criterion Solomon Feferman1

    Which Quantifiers Are Logical? a Combined Semantical and Inferential Criterion Solomon Feferman1

    Which Quantifiers are Logical? A combined semantical and inferential criterion Solomon Feferman1 Abstract. The aim of logic is to characterize the forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter; thus its concerns combine semantical and inferential notions in an essential way. Up to now most proposed characterizations of logicality of sentence generating operations have been given either in semantical or inferential terms. This paper offers a combined semantical and inferential criterion for logicality (improving one originally proposed by Jeffery Zucker) and shows that any quantifier that is to be counted as logical according to that criterion is definable in first order logic. The aim of logic is to characterize the forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter. The sentences involved are analyzed according to their logical (as opposed to grammatical) structure, i.e. how they are compounded from their parts by means of certain operations on propositions and predicates, of which the familiar ones are the connectives and quantifiers of first order logic. To spell this out in general, one must explain how the truth of compounds under given operations is determined by the truth of the parts, and characterize those forms of rules of inference for the given operations that insure preservation of truth. The so-called problem of “logical constants” (Gomez-Torrente 2002) is to determine all such operations.
  • The Nature of Logical Constants

    The Nature of Logical Constants

    Aporia vol. 27 no. 1—2017 The Nature of Logical Constants LAUREN RICHARDSON haracterizing the distinction between the “logical” and “non-logical” expressions of a language proves a challenging task, and one Cwith significant implications for the nature and scope of logic itself. It is often claimed that logical truths are statements that are “true by virtue of form,” and, likewise, that arguments are logically valid because of their respective “forms,” not because of their contents (Sider 2).1 Take, for example, a straightforward piece of reasoning: (P1) Maria is in Berlin or she is in Vienna. (P2) Maria is not in Berlin. (C) Maria is in Vienna. If this argument is valid—which, of course, in classical logic, it is—then it is typically held to be valid because of the structure of the argument and not because of certain material facts about the world. So it seems as if we should 1 Contrast the idea of truth preservation by virtue of form with the idea of an argument that is truth- preserving by virtue of the meaning of certain terms: (P1*):Nick is a bachelor. (C*): Nick is an unmarried man. We might think that the truth of (P1*) guarantees the truth of (C*), but it is not by virtue of the form of the argument; it is by virtue of the meaning of the expressions “bachelor” and “unmarried man.” Lauren Richardson is studying philosophy at the University of Chicago and will graduate in 2018. Her philosophical interests include philosophy of language, logic, metaethics, and feminism. After graduation, Lauren intends to pursue a graduate degree in philosophy.
  • Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2

    Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2

    Essential Logic Ronald C. Pine CHAPTER 10: SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY, PART 2 Introduction In the previous chapter there were many frustrating signs that something was wrong with our formal proof method that relied on only nine elementary rules of validity. Very simple, intuitive valid arguments could not be shown to be valid. For instance, the following intuitively valid arguments cannot be shown to be valid using only the nine rules. Somalia and Iran are both foreign policy risks. Therefore, Iran is a foreign policy risk. S I / I Either Obama or McCain was President of the United States in 2009.1 McCain was not President in 2010. So, Obama was President of the United States in 2010. (O v C) ~(O C) ~C / O If the computer networking system works, then Johnson and Kaneshiro will both be connected to the home office. Therefore, if the networking system works, Johnson will be connected to the home office. N (J K) / N J Either the Start II treaty is ratified or this landmark treaty will not be worth the paper it is written on. Therefore, if the Start II treaty is not ratified, this landmark treaty will not be worth the paper it is written on. R v ~W / ~R ~W 1 This or statement is obviously exclusive, so note the translation. 427 If the light is on, then the light switch must be on. So, if the light switch in not on, then the light is not on. L S / ~S ~L Thus, the nine elementary rules of validity covered in the previous chapter must be only part of a complete system for constructing formal proofs of validity.
  • Understanding the Logical Constants and Dispositions

    Understanding the Logical Constants and Dispositions

    Baltic International Yearbook of Cognition, Logic and Communication Volume 5 MEANING, UNDERSTANDING AND KNOWLEDGE Article 2 2009 Understanding the Logical Constants and Dispositions Corine Besson St Hugh's College, Oxford Follow this and additional works at: https://newprairiepress.org/biyclc This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Besson, Corine (2009) "Understanding the Logical Constants and Dispositions," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 5. https://doi.org/10.4148/biyclc.v5i0.279 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. Understanding the Logical Constants and Dispositions 2 The Baltic International Yearbook of not explore here how that explanatory connection might be accounted Cognition, Logic and Communication for. But however it is accounted for, it seems that the following min- imal constraint should be met by any account of competence with an October 2010 Volume 5: Meaning, Understanding and Knowledge expression: pages 1-24 DOI: 10.4148/biyclc.v5i0.279 (CT) An account of speakers’ understanding of an expres- sion should be consistent with their correct performances CORINE BESSON with that expression. St Hugh’s College, Oxford This is a weak constraint, since the connection between competence and performance is much tighter. But this will suffice for the purposes UNDERSTANDING THE LOGICAL CONSTANTS AND of this paper.
  • The Constituents of the Propositions of Logic Kevin C

    The Constituents of the Propositions of Logic Kevin C

    10 The Constituents of the Propositions of Logic Kevin C. Klement 1 Introduction Many founders of modern logic—Frege and Russell among them—bemoan- ed the tendency, still found in most textbook treatments, to define the subject matter of logic as “the laws of thought” or “the principles of inference”. Such descriptions fail to capture logic’s objective nature; they make it too dependent on human psychology or linguistic practices. It is one thing to identify what logic is not about. It is another to say what it is about. I tell my students that logic studies relationships between the truth-values of propositions that hold in virtue of their form. But even this characterization leaves me uneasy. I don’t really know what a “form” is, and even worse perhaps, I don’t really know what these “propositions” are that have these forms. If propositions are considered merely as sentences or linguistic assertions, the definition does not seem like much of an improvement over the psychological definitions. Language is a human invention, but logic is more than that, or so it seems. It is perhaps forgiveable then that at certain times Russell would not have been prepared to give a very good answer to the question “What is Logic?”, such as when he attempted, but failed, to compose a paper with that title in October 1912. Given that Russell had recently completed Principia Mathematica, a work alleging to establish the reducibility of mathematics to logic, one might think this overly generous. What does the claim that mathematics reduces to logic come to if we cannot independently speci- Published in Acquaintance, Knowledge and Logic: New Essays on Bertrand Russell’s Problems of Philosophy, edited by Donovan Wishon and Bernard Linsky.
  • A Absorption, 32 Addition, 33 Adequacy Theorem, 80 AEIO, 38 Agreement, 98 Algebra, 7 Argument, 4, 9 Artificial Intelligence

    A Absorption, 32 Addition, 33 Adequacy Theorem, 80 AEIO, 38 Agreement, 98 Algebra, 7 Argument, 4, 9 Artificial Intelligence

    Index A Constructive dilemma, 32 Absorption, 32 Contingent, 28 Addition, 33 Contradiction, 28, 46 Adequacy theorem, 80 Convince, 11 AEIO, 38 Corallary, 8 Agreement, 98 Cryptography, 9 Algebra, 7 Culture of proving, 4 Argument, 4, 9 Artificial Intelligence, 118 Assertions, 4, 19, 26 D Atomic, 60, 95 Data Analyst, 3 Axioms, 15, 93 Data Scientist, 3 Davis, M., 119 Decision Support Systems, 3 B Deduction, 4 Biconditional, 29 Deduction rules, 30, 93 Bill Nye, 11 Deduction theorem, 79, 106 Biology, 3 De Morgan, 29, 39 Bound variables, 98 Derivation, 30 Direct method, 43 Disjunction, 26 C Disjunctive Normal Form (DNF), 83 Calculus, 9 Distribution, 39 Cardinality, 44 Domain, 36, 99 Categorical forms, 38 Double Negation, 29 Categorical syllogism, 36 Dysjunctive syllogism, 32 Combinatorics, 22 Commutativity, 39 Complete, 77 E Completeness, 104 Econometrics, 3 Completeness theorem, 81, 82 Emojies, 16 Conjunction, 26, 33 Engineering, 9 Conjunctive Normal Form (CNF), 83 Entry, 59 Connectives, 26, 62 Euclid, 5 Consequences, 69 Euclidean Division Theorem, 48 Consistency, 77, 104 Euclid’s Theorem, 9 Constants, 36, 94 Ex falso quodlibet, 34 © Springer Nature Switzerland AG 2021 131 L. P. Cruz, Theoremus, https://doi.org/10.1007/978-3-030-68375-7 132 Index Existential generalization, 41 Law of Indentity (LOI), 30 Existential instantiation, 41 Law of Non-Contradiction (LNC), 30 Exists, 23 Leibnitz principle, 103 Exportation, 29 Lemma, 7 Logical equivalence, 98 Logic, equivalence, 29 F Logic, higher order, 93 Fallacy, 12, 42 Logic, non-classical, 30 Fallacy,
  • Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus

    Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus

    CORE Metadata, citation and similar papers at core.ac.uk Provided by Dagstuhl Research Online Publication Server Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus Ryuta Arisaka National Institute of Informatics, Tokyo, Japan [email protected] Abstract Development of a contraction-free BI sequent calculus, be the contraction-freeness implicit or explicit, has not been successful in the literature. We address this problem by presenting such a sequent system. Our calculus involves no structural rules. It should be an insight into non- formula contraction absorption in other non-classical logics. Contraction absorption in sequent calculus is associated to simpler cut elimination and to efficient proof searches. 1998 ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases cut-elimination, contraction-free, sequent calculus, proof theory, BI, logic combination Digital Object Identifier 10.4230/LIPIcs.FSCD.2016.8 1 Introduction Propositional BI [22] is a combined logic formed from propositional intuitionistic logic IL and propositional multiplicative fragment of intuitionistic linear logic MILL. Recall that IL, and respectively MILL, have the following logical connectives: {>0, ⊥0, ∧2, ∨2, ⊃2} (Cf. any standard text on the mathematical logic for intuitionistic logic; [16] for instance), and 1 respectively, {10, ⊗2, −◦2} (Cf. [11] for linear logic). A rough intuition about BI is that a BI expression is any expression that is constructable from (P, {>0, ⊥0, ∧2, ∨2, ⊃2, 10, ⊗2, −◦2}). P denotes some set of propositional letters. Following the popular convention in BI, we use the symbol ∗ in place of ⊗, and −∗ in place of −◦. In place of 1, we use ∗>, emphasising some link of its to >, as to be shortly stated.
  • What Does It Mean to Say That Logic Is Formal?

    What Does It Mean to Say That Logic Is Formal?

    WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? by John Gordon MacFarlane A.B., Philosophy, Harvard College, 1991 M.A., Philosophy, University of Pittsburgh, 1994 M.A., Classics, University of Pittsburgh, 1997 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2000 i Robert Brandom, Distinguished Service Professor of Philosophy (Director) Nuel Belnap, Alan Ross Anderson Distinguished Professor of Philosophy (Second Reader) Joseph Camp, Professor of Philosophy Danielle Macbeth, Associate Professor of Philosophy, Haverford College (Outside Reader) Kenneth Manders, Associate Professor of Philosophy Gerald Massey, Distinguished Service Professor of Philosophy ii WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL? John Gordon MacFarlane, PhD University of Pittsburgh, 2000 Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a pri- ori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topic-neutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought.
  • Transpositions Familiar to Artists, This Book Shows How Moves Can Be Made Between Established Positions and Completely New Ground

    Transpositions Familiar to Artists, This Book Shows How Moves Can Be Made Between Established Positions and Completely New Ground

    ORPHEUS New modes of epistemic relationships in artistic research ORPHEUS Research leads to new insights rupturing the existent fabric of knowledge. Situated in the still evolving field of artistic research, this book investigates a fundamental quality of this process. Building on the lessons of deconstruction, artistic research invents new modes of epistemic relationships that include aesthetic dimensions. Under the heading transposition, seventeen artists, musicians, and theorists explain how one thing may turn into another in a spatio- temporal play of identity and difference that has the power to expand into the unknown. By connecting materially concrete positions in a way Transpositions familiar to artists, this book shows how moves can be made between established positions and completely new ground. In doing so, research changes from a process that expands knowledge to one that creatively Aesthetico-Epistemic Operators reinvents it. in Artistic Research Michael Schwab is the founding editor-in-chief of the Journal for Artistic Research (JAR). He is senior researcher of MusicExperiment21 (Orpheus Institute, Ghent) and joint project leader of Transpositions: Artistic Data Exploration (University of Music and Performing Arts Graz; University of Applied Arts Vienna). Transpositions INSTITUTE Edited by Michael Schwab IN ISBN 9789462701410 S TIT U T 9 789462 701410 > E SERIES Transpositions cover final.indd 1 05/07/18 14:38 Transpositions: Aesthetico-Epistemic Operators in Artistic Research TRANSPOSITIONS: AESTHETICO- EPISTEMIC OPERATORS
  • MGF1121 Course Outline

    MGF1121 Course Outline

    MGF1121 Course Outline Introduction to Logic............................................................................................................. (3) (P) Description: This course is a study of both the formal and informal nature of human thought. It includes an examination of informal fallacies, sentential symbolic logic and deductive proofs, categorical propositions, syllogistic arguments and sorites. General Education Learning Outcome: The primary General Education Learning Outcome (GELO) for this course is Quantitative Reasoning, which is to understand and apply mathematical concepts and reasoning, and analyze and interpret various types of data. The GELO will be assessed through targeted questions on either the comprehensive final or an outside assignment. Prerequisite: MFG1100, MAT1033, or MAT1034 with a grade of “C” or better, OR the equivalent. Rationale: In order to function effectively and productively in an increasingly complex democratic society, students must be able to think for themselves to make the best possible decisions in the many and varied situations that will confront them. Knowledge of the basic concepts of logical reasoning, as offered in this course, will give students a firm foundation on which to base their decision-making processes. Students wishing to major in computer science, philosophy, mathematics, engineering and most natural sciences are required to have a working knowledge of symbolic logic and its applications. Impact Assessment: Introduction to Logic provides students with critical insight
  • HISTORY of LOGICAL CONSEQUENCE 1. Introduction

    HISTORY of LOGICAL CONSEQUENCE 1. Introduction

    HISTORY OF LOGICAL CONSEQUENCE CONRAD ASMUS AND GREG RESTALL 1. Introduction Consequence is a, if not the, core subject matter of logic. Aristotle's study of the syllogism instigated the task of categorising arguments into the logically good and the logically bad; the task remains an essential element of the study of logic. In a logically good argument, the conclusion follows validly from the premises; thus, the study of consequence and the study of validity are the same. In what follows, we will engage with a variety of approaches to consequence. The following neutral framework will enhance the discussion of this wide range of approaches. Consequences are conclusions of valid arguments. Arguments have two parts: a conclusion and a collection of premises. The conclusion and the premises are all entities of the same sort. We will call the conclusion and premises of an argument the argument's components and will refer to anything that can be an argument component as a proposition. The class of propositions is defined func- tionally (they are the entities which play the functional role of argument compo- nents); thus, the label should be interpreted as metaphysically neutral. Given the platonistic baggage often associated with the label \proposition", this may seem a strange choice but the label is already used for the argument components of many of the approaches below (discussions of Aristotlean and Medieval logic are two ex- amples). A consequence relation is a relation between collections of premises and conclusions; a collection of premises is related to a conclusion if and only if the latter is a consequence of the former.
  • Propositional Logic

    Propositional Logic

    Propositional Logic Fall 2014 Propositional Logic Fall 2014 1 / 30 Introduction Learning Outcomes for this Presentation Learning Outcomes . In this session we will Define the elements of propositional logic: statements and operations, including implication, and its converse, inverse, and negation. Translate English expressions into logical statements. Use both truth tables and derivations to demonstrate equivalence of logical statements. Define common tautologies, contradictions, and equivalences. Recognize and employ modus ponens and modus tollens and other forms of valid argumentation. Propositional Logic Fall 2014 2 / 30 A formal system for expressing truth and falsity. Provides a method of reasoning from given truths (called axioms) to new truths (called propositions or theorems). Logic is the skeleton that supports mathematical truth-making, i.e., proof-reading and proof-writing. Logic is the glue that holds computers and programs together. Introduction Learning Outcomes for this Presentation It's difficult, and possibly counterproductive, to provide a concise description of logic. What is logic? Why is it important? How will we use it in this class? Propositional Logic Fall 2014 3 / 30 Provides a method of reasoning from given truths (called axioms) to new truths (called propositions or theorems). Logic is the skeleton that supports mathematical truth-making, i.e., proof-reading and proof-writing. Logic is the glue that holds computers and programs together. Introduction Learning Outcomes for this Presentation It's difficult, and possibly counterproductive, to provide a concise description of logic. What is logic? A formal system for expressing truth and falsity. Why is it important? How will we use it in this class? Propositional Logic Fall 2014 3 / 30 Logic is the skeleton that supports mathematical truth-making, i.e., proof-reading and proof-writing.