DOCSLIB.ORG

Alfred Tarski Updated by Karl Frinkle © Draft Date April 24, 2019

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Alfred Tarski Updated by Karl Frinkle © Draft Date April 24, 2019 Logic Alfred Tarski updated by Karl Frinkle © Draft date April 24, 2019 Contents Preface for New Edition ix Preface xi From the Preface to the Original Edition xvii Compilation of Rules and Laws xix I Elements of Logic1 1 On The Use of Variables3 1.1 Constants and Variables......................3 1.2 Expressions Containing Variables.................4 1.3 Formation of Sentences by Means of Variables..........6 1.4 Universal and Existential Quantifiers, Free and Bound Variables.....................7 1.5 The Importance of Variables in Mathematics.......... 10 Chapter 1 Exercises........................... 11 2 On The Sentential Calculus 15 2.1 The Old Logic and the New Logic................ 15 2.2 Negations, Conjunctions, and Disjunctions............ 16 2.3 Conditional Sentences and Implications in Material Meaning. 19 2.4 The Use of Implication in Mathematics............. 22 2.5 Equivalence of Sentences...................... 25 2.6 Formulations of Definitions and Its Rules............ 26 2.7 The Laws of Sentential Calculus................. 29 2.8 Symbolism and Truth Tables................... 30 2.9 Application of Laws of Sentential Calculus in Inference..... 36 2.10 Rules of Inference, Complete Proofs............... 38 Chapter 2 Exercises........................... 41 v vi Contents 3 On The Theory of Identity 47 3.1 Logical Concepts Outside Sentential Calculus.......... 47 3.2 Fundamental Laws of the Theory of Identity........... 48 3.3 Identity of Objects, Their Designations, and Quotation Marks. 53 3.4 Equality in Arithmetic and Geometry.............. 55 3.5 Numerical Quantifiers....................... 57 Chapter 3 Exercises........................... 59 4 On The Theory of Classes 63 4.1 Classes and Their Elements.................... 63 4.2 Classes and Sentential Functions with One Free Variable.... 64 4.3 Universal and Null Classes..................... 67 4.4 Fundamental Relations Among Classes.............. 68 4.5 Operations on Classes....................... 71 4.6 Cardinal Number of a Class.................... 75 Chapter 4 Exercises........................... 77 5 On The Theory of Relations 81 5.1 Relations and Sentential Functions with Two Free Variables.. 81 5.2 Calculus of Relations........................ 83 5.3 Some Properties of Relations................... 86 5.4 Relations which are Reflexive, Symmetric, and Transitive... 88 5.5 Ordering Relations......................... 90 5.6 One-many Relations or Functions................. 91 5.7 Bijective Functions......................... 94 5.8 Many-Termed Relations...................... 95 5.9 The Importance of Logic for Other Sciences........... 97 Chapter 5 Exercises........................... 97 II The Deductive Method and Formal Systems 103 6 On The Deductive Method 105 6.1 Primitive and Defined Terms, Axioms and Theorems...... 105 6.2 Model and Interpretation of a Deductive Theory........ 107 6.3 Law of Deduction.......................... 113 6.4 Selection of Axioms and Primitive Terms............ 116 6.5 Formalization of Definitions and Proofs............. 117 6.6 Consistency and Completeness of a Deductive Theory..... 119 6.7 Widened Conception of the Methodology of Deductive Sciences........................ 121 Chapter 6 Exercises........................... 122 Contents vii 7 Propositional Languages and the Deductive Method 133 7.1 Formal Languages......................... 133 7.2 Metalogical Theorems About j=P ................. 135 7.3 Formal Systems........................... 136 7.4 Metalogical Theorems About `PS ................ 140 7.5 The Deduction Theorem for PS .................. 141 7.6 Concepts of Semantic Completeness............... 145 Chapter 7 Exercises........................... 155 III Applications of Logic and Methodology: Constructing Mathematical Theories 157 8 Construction of a Mathematical Theory: Laws of Order for Numbers 159 8.1 Primitive Terms and Axioms on Fundamental Relations Among Numbers.................... 159 8.2 Laws of Irreflexivity and Indirect Proofs............. 161 8.3 Further Theorems on the Fundamental Relations........ 162 8.4 Other Relations Among Numbers................. 166 Chapter 8 Exercises........................... 171 9 Construction of a Mathematical Theory: Laws of Addition and Subtraction 173 9.1 Axioms Concerning Addition, Properties of Groups....... 173 9.2 Further Commutative and Associative Laws........... 175 9.3 Laws of Monotony For Addition and Their Converses...... 176 9.4 Closed Systems of Sentences.................... 180 9.5 Consequences of the Laws of Monotony............. 182 9.6 Definition of Subtraction; Inverse Operations.......... 189 9.7 Definitions Containing Identity.................. 191 9.8 Theorems on Subtraction..................... 192 Chapter 9 Exercises........................... 195 10 Methodological Considerations on the Constructed Theory 199 10.1 Elimination of Superfluous Axioms in the Original Axiomatic System.................... 199 10.2 Independence of the Axioms of the Simplified System..... 203 10.3 Elimination of Superfluous Primitive Terms........... 205 10.4 Further Simplifications of the Axiomatic System........ 207 10.5 Problem of the Consistency of the Constructed Theory..... 213 10.6 Problem of the Completeness of the Constructed Theory.... 214 Chapter 10 Exercises........................... 215 viii Contents 11 Foundations of Arithmetic for Real Numbers 219 11.1 First Axiomatic System for the Arithmetic of Real Numbers.......................... 219 11.2 Characterization of the First Axiomatic System......... 220 11.3 Second Axiomatic System for the Arithmetic of Real Numbers.......................... 222 11.4 Characterization of the Second Axiomatic System........ 223 11.5 Equipollence of the Two Systems................. 225 Chapter 11 Exercises........................... 226 Suggested Reading 229 Index 233 Preface for New Edition I should start off by saying that this text is, in this mathematician's opinion, the best, basic introduction to logic textbook available. Many current logical textbooks fail to focus on the most important aspect of logic to the sciences, which is, the deductive method. The rigorous approach to building a deduc- tive theory based on an axiomatic system, and the consequences of selecting axioms and primitive terms is thoroughly, and carefully explained, yet readily accessible to the reader. Many texts spend too much time dealing with logical symbolism, truth ta- bles, and clever word puzzles, while others tend to be only accessible to those studying the fundamental semantics and intricacies of logic. This text is the perfect balance between sufficiently deep logical theory and the application of this theory to algebraic axiomatic systems. One might thus ask: Why would you wish to edit or alter such a text if you praise it so highly? The text itself has been `updated' several times, the last being in 1994. In the last two updates, very little has been done to make the language of the writing or the symbolism more current. Furthermore, af- ter listening to the comments of the students in my Honor's Logic Courses, I realized that the students tolerated the message of the book, just not the style in which that message was presented. Most textbooks today focus heavily on highlighting key items such as definitions, tables, laws, theorems, examples, etc... I have attempted to make the text more easily parsed upon reading. One of the more important aspects of the upgrade to this text is in the rigorous, formal proof system-style approach to proving many of the theorems presented in this text. All previous versions of this text were very quick to wave hands at many of the more subtle steps required to rigorously prove many of the theorems given. In doing so, many logical laws and clever insights were missed, and I have attempted to include them in the details added. I have also taken the liberty of either incorporating optional sections of the text (cordoned off by *'s) or removing them. After using this text numerous ix x Preface for New Edition times, it became clear which material really had an impact on the overall theme of the course, and what material was either not necessary or added confusion to the learning process. In a similar fashion, some of the homework problems have been reworked. Less emphasis has been put on geometrically oriented problems, and most of the questions starting off with \Give examples from..... of such and such a property..." have simply been removed or replaced. I have found Dr. Tarski's writing style to be rather refreshing and con- versational in style, and have therefore attempted to leave most of his words intact. Any additions or further explanations hopefully blend in as seamlessly as I intended. At this point, I need to give a special thanks to two very dedicated stu- dents: Laiken McMurl and Michaela Metts, who were students in my Fall 2015 Logic course. These two students are the definition of ideal students, they read the text multiple times, had intelligent discourse over all of the material, and critiqued every aspect of the book. Many of the modifications done in the text were the direct result of input from these two students. For instance, the com- pilation of rules, laws, and logical identities found after the set of prefaces to this text was one important suggestion they made which I feel will benefit all se- rious students of this text. Included in this list are some clever logical identities that may help solve some of the later exercises and are not discussed in the text. Added in 2019 : After a few years of teaching
Load more

Recommended publications
  • Rules of Replacement II, §7.4
    Philosophy 109, Modern Logic, Queens College Russell Marcus, Instructor email: [email protected] website: http://philosophy.thatmarcusfamily.org Office phone: (718) 997-5287 Rules of Replacement II, §7.4 I. The Last Five Rules of Replacement See the appendix at the end of the lesson for truth tables proving equivalence for each. Transposition (Trans) P e Q :: -Q e -P You may switch the antecedent and consequent of a conditional statement, as long as you negate (or un-negate) both. Often used with (HS). Also, traditionally, called the ‘contrapositive’. Sample Derivation: 1. A e B 2. D e -B / A e -D 3. --B e -D 2, Trans 4. A e -D 1, 3, DN, HS QED Transposition can be tricky when only one term is negated: A e -B becomes, by Trans: --B e -A which becomes, by DN B e -A Equivalently, but doing the double negation first: A e -B becomes, by DN: --A e -B becomes, by Trans: B e -A Either way, you can include the DN on the line with Trans. Material Implication (Impl) P e Q :: -P w Q Implication allows you to change a statement from a disjunction to a conditional, or vice versa. It’s often easier to work with disjunctions. You can use (DM) to get conjunctions. You may be able to use distribution, which doesn’t apply to conditionals. On the other hand, sometimes, you just want to work with conditionals. You can use (HS) and (MP). Proofs are overdetermined by our system - there are many ways to do them.
    [Show full text]
  • 7.1 Rules of Implication I
    Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.
    [Show full text]
  • The Incorrect Usage of Propositional Logic in Game Theory
    The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself Holger I. MEINHARDT ∗ August 13, 2018 Recently, we had to realize that more and more game theoretical articles have been pub- lished in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that “if A then B” is valid, they suppose that the assumptions “A and not B” are valid to derive a contradiction in order to deduce “if A then B”. Hence, they want to establish the equivalent proposition “A∧ not B implies A ∧ notA” to conclude that “if A then B”is valid. In fact, they prove that a truth implies a falsehood, which is a wrong statement. As a consequence, “if A then B” is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles. Keywords: Transferable Utility Game, Solution Concepts, Axiomatization, Propositional Logic, Material Implication, Circular Reasoning (circulus in probando), Indirect Proof, Proof by Contradiction, Proof by Contraposition, Cooperative Oligopoly Games 2010 Mathematics Subject Classifications: 03B05, 91A12, 91B24 JEL Classifications: C71 arXiv:1509.05883v1 [cs.GT] 19 Sep 2015 ∗Holger I. Meinhardt, Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Englerstr. 11, Building: 11.40, D-76128 Karlsruhe. E-mail: [email protected] The Incorrect Usage of Propositional Logic in Game Theory 1 INTRODUCTION During the last decades, game theory has encountered a great success while becoming the major analysis tool for studying conflicts and cooperation among rational decision makers.
    [Show full text]
  • Simple Laws About Nonprominent Properties of Binary Relations
    Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality .
    [Show full text]
  • Tables of Implications and Tautologies from Symbolic Logic
    Tables of Implications and Tautologies from Symbolic Logic Dr. Robert B. Heckendorn Computer Science Department, University of Idaho March 17, 2021 Here are some tables of logical equivalents and implications that I have found useful over the years. Where there are classical names for things I have included them. \Isolation by Parts" is my own invention. By tautology I mean equivalent left and right hand side and by implication I mean the left hand expression implies the right hand. I use the tilde and overbar interchangeably to represent negation e.g. ∼x is the same as x. Enjoy! Table 1: Properties of All Two-bit Operators. The Comm. is short for commutative and Assoc. is short for associative. Iff is short for \if and only if". Truth Name Comm./ Binary And/Or/Not Nands Only Table Assoc. Op 0000 False CA 0 0 (a " (a " a)) " (a " (a " a)) 0001 And CA a ^ b a ^ b (a " b) " (a " b) 0010 Minus b − a a ^ b (b " (a " a)) " (a " (a " a)) 0011 B A b b b 0100 Minus a − b a ^ b (a " (a " a)) " (a " (a " b)) 0101 A A a a a 0110 Xor/NotEqual CA a ⊕ b (a ^ b) _ (a ^ b)(b " (a " a)) " (a " (a " b)) (a _ b) ^ (a _ b) 0111 Or CA a _ b a _ b (a " a) " (b " b) 1000 Nor C a # b a ^ b ((a " a) " (b " b)) " ((a " a) " a) 1001 Iff/Equal CA a $ b (a _ b) ^ (a _ b) ((a " a) " (b " b)) " (a " b) (a ^ b) _ (a ^ b) 1010 Not A a a a " a 1011 Imply a ! b a _ b (a " (a " b)) 1100 Not B b b b " b 1101 Imply b ! a a _ b (b " (a " a)) 1110 Nand C a " b a _ b a " b 1111 True CA 1 1 (a " a) " a 1 Table 2: Tautologies (Logical Identities) Commutative Property: p ^ q $ q
    [Show full text]
  • 46. on the Completeness O F the Leibnizian Modal System with a Restriction by Setsuo SAITO Shibaura Institute of Technology,Tokyo (Comm
    198 [Vol. 42, 46. On the Completeness o f the Leibnizian Modal System with a Restriction By Setsuo SAITO Shibaura Institute of Technology,Tokyo (Comm. by Zyoiti SUETUNA,M.J.A., March 12, 1966) § 1. Introduction. The purpose of this paper is to show the completeness of a modal system which will be called Lo in the following. In my previous paper [1], in order to show an example of defence of _.circular definition, the following definition was given: A statement is analytic if and only if it is consistent with every statement that expresses what is possible. This definition, roughly speaking, is materially equivalent to Carnap's definition of L-truth which is suggested by Leibniz' conception that a necessary truth must hold in all possible worlds (cf. Carnap [2], p.10). If "analytic" is replaced by "necessary" in the above definition, this definition becomes as follows: A statement is necessary if and only if it is consistent with every statement that expresses what is possible. This reformed definition is symbolized by modal signs as follows: Opm(q)[Qq~O(p•q)1, where p and q are propositional variables. Let us replace it by the following axiom-schema and rule: Axiom-schema. D a [Q,9 Q(a •,S)], where a, ,9 are arbitrary formulas. Rule of inference. If H Q p Q(a •p), then i- a, where a is an arbitrary formula and p is a propositional variable not contained in a. We call L (the Leibnizian modal system) the system obtained from the usual propositional calculus by adding the above axiom schema and rule and the rule of replacement of material equivalents.
    [Show full text]
  • A General Account of Coinduction Up-To Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot
    A General Account of Coinduction Up-To Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot To cite this version: Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot. A General Account of Coinduction Up-To. Acta Informatica, Springer Verlag, 2016, 10.1007/s00236-016-0271-4. hal-01442724 HAL Id: hal-01442724 https://hal.archives-ouvertes.fr/hal-01442724 Submitted on 20 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A General Account of Coinduction Up-To ∗ Filippo Bonchi Daniela Petrişan Damien Pous Jurriaan Rot May 2016 Abstract Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras.
    [Show full text]
  • 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.
    [Show full text]
  • Argument Forms and Fallacies
    6.6 Common Argument Forms and Fallacies 1. Common Valid Argument Forms: In the previous section (6.4), we learned how to determine whether or not an argument is valid using truth tables. There are certain forms of valid and invalid argument that are extremely common. If we memorize some of these common argument forms, it will save us time because we will be able to immediately recognize whether or not certain arguments are valid or invalid without having to draw out a truth table. Let’s begin: 1. Disjunctive Syllogism: The following argument is valid: “The coin is either in my right hand or my left hand. It’s not in my right hand. So, it must be in my left hand.” Let “R”=”The coin is in my right hand” and let “L”=”The coin is in my left hand”. The argument in symbolic form is this: R ˅ L ~R __________________________________________________ L Any argument with the form just stated is valid. This form of argument is called a disjunctive syllogism. Basically, the argument gives you two options and says that, since one option is FALSE, the other option must be TRUE. 2. Pure Hypothetical Syllogism: The following argument is valid: “If you hit the ball in on this turn, you’ll get a hole in one; and if you get a hole in one you’ll win the game. So, if you hit the ball in on this turn, you’ll win the game.” Let “B”=”You hit the ball in on this turn”, “H”=”You get a hole in one”, and “W”=”you win the game”.
    [Show full text]
  • Philosophy 109, Modern Logic Russell Marcus
    Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus Reference Sheeet for What Follows Names of Languages PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic Basic Truth Tables - á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 Rules of Inference Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2 Rules of Equivalence DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á Six Derived Rules for the Biconditional Rules of Inference Rules of Equivalence Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof.
    [Show full text]
  • Introduction to Relations
    CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition:A binary relation from a set A to a set B is a subset R ⊆ A × B: If (a; b) 2 R we say a is related to b by R. A is the domain of R, and B is the codomain of R. If A = B, R is called a binary relation on the set A. Notation: • If (a; b) 2 R, then we write aRb. • If (a; b) 62 R, then we write aR6 b. Discussion Notice that a relation is simply a subset of A × B. If (a; b) 2 R, where R is some relation from A to B, we think of a as being assigned to b. In these senses students often associate relations with functions. In fact, a function is a special case of a relation as you will see in Example 1.2.4. Be warned, however, that a relation may differ from a function in two possible ways. If R is an arbitrary relation from A to B, then • it is possible to have both (a; b) 2 R and (a; b0) 2 R, where b0 6= b; that is, an element in A could be related to any number of elements of B; or • it is possible to have some element a in A that is not related to any element in B at all. 204 1. RELATIONS AND THEIR PROPERTIES 205 Often the relations in our examples do have special properties, but be careful not to assume that a given relation must have any of these properties.
    [Show full text]
  • From Peirce's Algebra of Relations to Tarski's Calculus of Relations
    The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations Author(s): Roger D. Maddux Source: Studia Logica: An International Journal for Symbolic Logic, Vol. 50, No. 3/4, Algebraic Logic (1991), pp. 421-455 Published by: Springer Stable URL: http://www.jstor.org/stable/20015596 Accessed: 02/03/2009 14:54 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Springer is collaborating with JSTOR to digitize, preserve and extend access to Studia Logica: An International Journal for Symbolic Logic.
    [Show full text]