Logical Operators
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1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Logic, Proofs
CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true. -
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. -
Lecture 1: Propositional Logic
Lecture 1: Propositional Logic Syntax Semantics Truth tables Implications and Equivalences Valid and Invalid arguments Normal forms Davis-Putnam Algorithm 1 Atomic propositions and logical connectives An atomic proposition is a statement or assertion that must be true or false. Examples of atomic propositions are: “5 is a prime” and “program terminates”. Propositional formulas are constructed from atomic propositions by using logical connectives. Connectives false true not and or conditional (implies) biconditional (equivalent) A typical propositional formula is The truth value of a propositional formula can be calculated from the truth values of the atomic propositions it contains. 2 Well-formed propositional formulas The well-formed formulas of propositional logic are obtained by using the construction rules below: An atomic proposition is a well-formed formula. If is a well-formed formula, then so is . If and are well-formed formulas, then so are , , , and . If is a well-formed formula, then so is . Alternatively, can use Backus-Naur Form (BNF) : formula ::= Atomic Proposition formula formula formula formula formula formula formula formula formula formula 3 Truth functions The truth of a propositional formula is a function of the truth values of the atomic propositions it contains. A truth assignment is a mapping that associates a truth value with each of the atomic propositions . Let be a truth assignment for . If we identify with false and with true, we can easily determine the truth value of under . The other logical connectives can be handled in a similar manner. Truth functions are sometimes called Boolean functions. 4 Truth tables for basic logical connectives A truth table shows whether a propositional formula is true or false for each possible truth assignment. -
The Family of Lodato Proximities Compatible with a Given Topological Space
Can. J. Math., Vol. XXVI, No. 2, 1974, pp. 388-404 THE FAMILY OF LODATO PROXIMITIES COMPATIBLE WITH A GIVEN TOPOLOGICAL SPACE W. J. THRON AND R. H. WARREN Compendium. Let (X, $~) be a topological space. By 99?i we denote the family of all Lodato proximities on X which induced". We show that 99? i is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of SDîi are developed. By means of one of these construc tions, all covers of any member of 99? i can be obtained. Several examples are given which relate 9D?i to the lattice 99? of all compatible proximities of Cech and the family 99?2 of all compatible proximities of Efremovic. The paper concludes with a chart which summarizes many of the structural properties of a», a»! and a»2. 1. Preliminaries and notation. M. W. Lodato in [51 and [6] has studied a symmetric generalized proximity structure (see Definition 1.2). Naimpally and Warrack [8] have called such a structure a Lodato proximity. We shall also use this name. The closure operator induced by a Lodato proximity satisfies the four Kuratowski closure conditions. This paper is primarily concerned with a study of the order structure of the family 99?i of all Lodato proximities which induce the same closure operator on a given set. Lodato characterized the least element in 99?i and those topo logical spaces for which 99?i ^ 0. Sharma and Naimpally [9] described the greatest member of 99? 1 and have given two methods for constructing members of 2»i. -
The Default OLP Configuration File Open-Logic-Config.Sty
The Default OLP Configuration File open-logic-config.sty Open Logic Project 2021-08-12 524d89a Description This file contains all commands and environments that are meant to be configured, changed, or adapted by a user generating their own text based on OLP text. Do not edit this file to customize your OLP-derived text! A file myversion.tex adapted from open-logic-complete.tex (or from any of the contributed example master files) will include myversion-config.sty if it exists. It will do so after it loads this file, so your myversion-config.sty will redefine the defaults. This means you won’t have to include everything, e.g., you can just change some tags and nothing else. You may copy and paste definitions you want to change into that file, or copy thi file, rename it myversion-config.sty and delete anything you’d like to leave as the default. Symbols Formula metavariabes Use the exclamation point symbol ! immediately in front of an uppercase letter in math mode for formula metavariables. By default, !A, !B, . are typeset as ϕ, ψ, χ, . if you use the command \olgreekformulas. If this is not desired, and you’d like A, B, C, . instead, use \ollatinformulas. If you issue \olalphagreekformulas, you’ll get α, β, γ,.... Greek symbols: prefer varphi and varepsilon Logical symbols The following commands are used in the OLP texts for logical symbols. Their definitions can be customized to produce different output. Truth Values • \True defaults to T and \False to F. 1 Other truth values Propositional Constants and Connectives • Falsity is \lfalse and defaults to ⊥. -
Mathematical Symbols
List of mathematical symbols This is a list of symbols used in all branches ofmathematics to express a formula or to represent aconstant . A mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations, a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. However, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning. Each symbol is shown both inHTML , whose display depends on the browser's access to an appropriate font installed on the particular device, and typeset as an image usingTeX . Contents Guide Basic symbols Symbols based on equality Symbols that point left or right Brackets Other non-letter symbols Letter-based symbols Letter modifiers Symbols based on Latin letters Symbols based on Hebrew or Greek letters Variations See also References External links Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol (note that this article doesn't -
Form and Content: an Introduction to Formal Logic
Connecticut College Digital Commons @ Connecticut College Open Educational Resources 2020 Form and Content: An Introduction to Formal Logic Derek D. Turner Connecticut College, [email protected] Follow this and additional works at: https://digitalcommons.conncoll.edu/oer Recommended Citation Turner, Derek D., "Form and Content: An Introduction to Formal Logic" (2020). Open Educational Resources. 1. https://digitalcommons.conncoll.edu/oer/1 This Book is brought to you for free and open access by Digital Commons @ Connecticut College. It has been accepted for inclusion in Open Educational Resources by an authorized administrator of Digital Commons @ Connecticut College. For more information, please contact [email protected]. The views expressed in this paper are solely those of the author. Form & Content Form and Content An Introduction to Formal Logic Derek Turner Connecticut College 2020 Susanne K. Langer. This bust is in Shain Library at Connecticut College, New London, CT. Photo by the author. 1 Form & Content ©2020 Derek D. Turner This work is published in 2020 under a Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 International License. You may share this text in any format or medium. You may not use it for commercial purposes. If you share it, you must give appropriate credit. If you remix, transform, add to, or modify the text in any way, you may not then redistribute the modified text. 2 Form & Content A Note to Students For many years, I had relied on one of the standard popular logic textbooks for my introductory formal logic course at Connecticut College. It is a perfectly good book, used in countless logic classes across the country. -
PART III – Sentential Logic
Logic: A Brief Introduction Ronald L. Hall, Stetson University PART III – Sentential Logic Chapter 7 – Truth Functional Sentences 7.1 Introduction What has been made abundantly clear in the previous discussion of categorical logic is that there are often trying challenges and sometimes even insurmountable difficulties in translating arguments in ordinary language into standard-form categorical syllogisms. Yet in order to apply the techniques of assessment available in categorical logic, such translations are absolutely necessary. These challenges and difficulties of translation must certainly be reckoned as limitations of this system of logical analysis. At the same time, the development of categorical logic made great advances in the study of argumentation insofar as it brought with it a realization that good (valid) reasoning is a matter of form and not content. This reduction of good (valid) reasoning to formal considerations made the assessment of arguments much more manageable than it was when each and every particular argument had to be assessed. Indeed, in categorical logic this reduction to form yielded, from all of the thousands of possible syllogisms, just fifteen valid syllogistic forms. Taking its cue from this advancement, modern symbolic logic tried to accomplish such a reduction without being hampered with the strict requirements of categorical form. To do this, modern logicians developed what is called sentential logic. As with categorical logic, modern sentential logic reduces the kinds of sentences to a small and thus manageable number. In fact, on the sentential reduction, there are only two basic sentential forms: simple sentences and compound sentences. (Wittgenstein called simple sentences “elementary sentences.”) But it would be too easy to think that this is all there is to it. -
Symbolic Logic Be Accessed At
This asset is intentionally omitted from this text. It may Symbolic Logic be accessed at www.mcescher.com. 1 Modern Logic and Its Symbolic Language (Belvedere by M.C. Escher) 2 The Symbols for Conjunction, Negation, and Disjunction 3 Conditional Statements and Material Implication 4 Argument Forms and Refutation by Logical Analogy 5 The Precise Meaning of “Invalid” and “Valid” 6 Testing Argument Validity Using Truth Tables 7 Some Common Argument Forms 8 Statement Forms and Material Equivalence 9 Logical Equivalence 10 The Three “Laws of Thought” 1 Modern Logic and Its Symbolic Language To have a full understanding of deductive reasoning we need a general theory of deduction. A general theory of deduction will have two objectives: (1) to explain the relations between premises and conclusions in deductive arguments, and (2) to provide techniques for discriminating between valid and invalid deduc- tions. Two great bodies of logical theory have sought to achieve these ends. The first is called classical (or Aristotelian) logic. The second, called modern, symbolic, or mathematical logic, is this chapter. Although these two great bodies of theory have similar aims, they proceed in very different ways. Modern logic does not build on the system of syllogisms. It does not begin with the analysis of categorical propositions. It does seek to dis- criminate valid from invalid arguments, although it does so using very different concepts and techniques. Therefore we must now begin afresh, developing a modern logical system that deals with some of the very same issues dealt with by traditional logic—and does so even more effectively. -
Appendix B-1
Appendix B-1 Alternative Notations and Connectives The logical language we have studied in this chapter is the best known classical system oflogic of statements. There are various alternative symbolic notations for the connectives we introduced, the most common of which are listed below. Alternative notation for connectives our symbol alternative( s) negation '" ---',-,p conjunction & 1\, • conditional -+ :::) (called 'horseshoe') Table B.I-l Furthermore, there are two connectives which we have not introduced, as they are not part of the system most commonly used. One, called "Quine's dagger", L is sufficiently powerful to be the only connective in a system equivalent to the one given. Its truth table is shown in (B.II.-2). 237 238 ApPENDIX B-1 P Q PlQ 1 1 0 1 0 0 0 1 0 0 0 1 Table B.I-2 Its nearest English correspondent is neither ... nor. As an interesting ex ercise one can show that negation can be defined in terms of this connec tive, and then that disjunction can be defined in terms of negation and this connective. We have already proven (Chapter, Exercise 7) that the five connective system can be reduced to one containing just V and "'. Therefore 1 suffices alone for the five connectives. Similarly, there is another connective, written as I, which is called the "Sheffer stroke", whose interpretation by the truth table is as follows: P Q PIQ 1 1 0 0 1 1 1 0 1 0 0 1 Table BJ-3 There is no simple English connective which has this meaning, but it can be expressed as 'not both p and q' or the neologism 'nand'. -
Baronett, Logic (4Th Ed.) Chapter Guide Chapter 7: Propositional Logic the Basic Components of Propositional Logic Are Statement
Baronett, Logic (4th ed.) Chapter Guide Chapter 7: Propositional Logic The basic components of propositional logic are statements. A. Logical Operators and Translations There are two types of statement: simple and complex, or compound. A simple statement is one that does not contain another statement as a component. These statements are represented by capital letters A to Z. A compound statement contains at least one simple statement as a component, along with a logical operator, or connective. There are five compound types represented by logical operators: Compound Type Negation Conjunction Disjunction Conditional Biconditional Operator ~ • v Operator Name Tilde Dot Wedge Horseshoe Triple Bar B. Compound Statements Here are four rules to help organize your thinking about translating complex statements: 1. Remember that every operator except the negation is always placed in between statements. 2. Tilde is always placed to the left of whatever is to be negated. 3. Tilde never goes by itself in between two statements. 4. Parentheses, brackets, and braces are required to eliminate ambiguity in a complex statement. Following these rules assures that a compound statement is a well-formed formula. There is always only one main operator in any statement, and that operator is either one of the four that appear in between statements or the tilde that appears in front of the statement that is negated. C. Truth Functions Every simple statement has a truth value: it is either true or false. The truth value of a compound statement is uniquely determined by the meaning of the five operators and the truth values of its component propositions.