Mathematical Logic
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Propositional Logic, Modal Logic, Propo- Sitional Dynamic Logic and first-Order Logic
A I L Madhavan Mukund Chennai Mathematical Institute E-mail: [email protected] Abstract ese are lecture notes for an introductory course on logic aimed at graduate students in Com- puter Science. e notes cover techniques and results from propositional logic, modal logic, propo- sitional dynamic logic and first-order logic. e notes are based on a course taught to first year PhD students at SPIC Mathematical Institute, Madras, during August–December, . Contents Propositional Logic . Syntax . . Semantics . . Axiomatisations . . Maximal Consistent Sets and Completeness . . Compactness and Strong Completeness . Modal Logic . Syntax . . Semantics . . Correspondence eory . . Axiomatising valid formulas . . Bisimulations and expressiveness . . Decidability: Filtrations and the finite model property . . Labelled transition systems and multi-modal logic . Dynamic Logic . Syntax . . Semantics . . Axiomatising valid formulas . First-Order Logic . Syntax . . Semantics . . Formalisations in first-order logic . . Satisfiability: Henkin’s reduction to propositional logic . . Compactness and the L¨owenheim-Skolem eorem . . A Complete Axiomatisation . . Variants of the L¨owenheim-Skolem eorem . . Elementary Classes . . Elementarily Equivalent Structures . . An Algebraic Characterisation of Elementary Equivalence . . Decidability . Propositional Logic . Syntax P = f g We begin with a countably infinite set of atomic propositions p0, p1,... and two logical con- nectives : (read as not) and _ (read as or). e set Φ of formulas of propositional logic is the smallest set satisfying the following conditions: • Every atomic proposition p is a member of Φ. • If α is a member of Φ, so is (:α). • If α and β are members of Φ, so is (α _ β). We shall normally omit parentheses unless we need to explicitly clarify the structure of a formula. -
On Basic Probability Logic Inequalities †
mathematics Article On Basic Probability Logic Inequalities † Marija Boriˇci´cJoksimovi´c Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca154, 11000 Belgrade, Serbia; [email protected] † The conclusions given in this paper were partially presented at the European Summer Meetings of the Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on 12–18 July 2012. Abstract: We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A ! B, and B ! C have probabilities a, b, c, r, and s, respectively, then for probability p of A ! C, we have f (a, b, c, r, s) ≤ p ≤ g(a, b, c, r, s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner. Keywords: inequality; probability logic; inference rule MSC: 03B48; 03B05; 60E15; 26D20; 60A05 1. Introduction The main part of probabilization of logical inference rules is defining the correspond- Citation: Boriˇci´cJoksimovi´c,M. On ing best possible bounds for probabilities of propositions. Some of them, connected with Basic Probability Logic Inequalities. conjunction and disjunction, can be obtained immediately from the well-known Boole’s Mathematics 2021, 9, 1409. -
Propositional Logic - Basics
Outline Propositional Logic - Basics K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 14 January and 16 January, 2013 Subramani Propositonal Logic Outline Outline 1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Subramani Propositonal Logic (i) The Law! (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? Subramani Propositonal Logic (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! (ii) Mathematics. Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. -
Logic in Computer Science
P. Madhusudan Logic in Computer Science Rough course notes November 5, 2020 Draft. Copyright P. Madhusudan Contents 1 Logic over Structures: A Single Known Structure, Classes of Structures, and All Structures ................................... 1 1.1 Logic on a Fixed Known Structure . .1 1.2 Logic on a Fixed Class of Structures . .3 1.3 Logic on All Structures . .5 1.4 Logics over structures: Theories and Questions . .7 2 Propositional Logic ............................................. 11 2.1 Propositional Logic . 11 2.1.1 Syntax . 11 2.1.2 What does the above mean with ::=, etc? . 11 2.2 Some definitions and theorems . 14 2.3 Compactness Theorem . 15 2.4 Resolution . 18 3 Quantifier Elimination and Decidability .......................... 23 3.1 Quantifiier Elimination . 23 3.2 Dense Linear Orders without Endpoints . 24 3.3 Quantifier Elimination for rationals with addition: ¹Q, 0, 1, ¸, −, <, =º ......................................... 27 3.4 The Theory of Reals with Addition . 30 3.4.1 Aside: Axiomatizations . 30 3.4.2 Other theories that admit quantifier elimination . 31 4 Validity of FOL is undecidable and is r.e.-hard .................... 33 4.1 Validity of FOL is r.e.-hard . 34 4.2 Trakhtenbrot’s theorem: Validity of FOL over finite models is undecidable, and co-r.e. hard . 39 5 Quantifier-free theory of equality ................................ 43 5.1 Decidability using Bounded Models . 43 v vi Contents 5.2 An Algorithm for Conjunctive Formulas . 44 5.2.1 Computing ¹º ................................... 47 5.3 Axioms for The Theory of Equality . 49 6 Completeness Theorem: FO Validity is r.e. ........................ 53 6.1 Prenex Normal Form . -
Tarski's Threat to the T-Schema
Tarski’s Threat to the T-Schema MSc Thesis (Afstudeerscriptie) written by Cian Chartier (born November 10th, 1986 in Montr´eal, Canada) under the supervision of Prof. dr. Frank Veltman, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30, 2011 Prof. Dr. Frank Veltman Prof. Dr. Michiel van Lambalgen Prof. Dr. Benedikt L¨owe Dr. Robert van Rooij Abstract This thesis examines truth theories. First, four relevant programs in philos- ophy are considered. Second, four truth theories are compared according to a range of criteria. The truth theories are categorised according to Leitgeb’s eight criteria for a truth theory. The four truth theories are then compared with each-other based on three new criteria. In this way their relative use- fulness in pursuing some of the aims of the four programs is evaluated. This presents a springboard for future work on truth: proposing ideas for differ- ent truth theories that advance unambiguously different programs on the basis of the four truth theories proposed. 1 Acknowledgements I first thank my supervisor Prof. Dr. Frank Veltman for regularly read- ing and providing constructive feedback, recommending some papers along the way. This thesis has been much improved by his corrections and input, including extensive discussions and suggestions of the ideas proposed here. His patience in working with me on the thesis has been much appreciated. Among the other professors in the ILLC, Dr. Robert van Rooij also provided helpful discussion. -
1 Sets and Set Notation. Definition 1 (Naive Definition of a Set)
LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set. -
CS245 Logic and Computation
CS245 Logic and Computation Alice Gao December 9, 2019 Contents 1 Propositional Logic 3 1.1 Translations .................................... 3 1.2 Structural Induction ............................... 8 1.2.1 A template for structural induction on well-formed propositional for- mulas ................................... 8 1.3 The Semantics of an Implication ........................ 12 1.4 Tautology, Contradiction, and Satisfiable but Not a Tautology ........ 13 1.5 Logical Equivalence ................................ 14 1.6 Analyzing Conditional Code ........................... 16 1.7 Circuit Design ................................... 17 1.8 Tautological Consequence ............................ 18 1.9 Formal Deduction ................................. 21 1.9.1 Rules of Formal Deduction ........................ 21 1.9.2 Format of a Formal Deduction Proof .................. 23 1.9.3 Strategies for writing a formal deduction proof ............ 23 1.9.4 And elimination and introduction .................... 25 1.9.5 Implication introduction and elimination ................ 26 1.9.6 Or introduction and elimination ..................... 28 1.9.7 Negation introduction and elimination ................. 30 1.9.8 Putting them together! .......................... 33 1.9.9 Putting them together: Additional exercises .............. 37 1.9.10 Other problems .............................. 38 1.10 Soundness and Completeness of Formal Deduction ............... 39 1.10.1 The soundness of inference rules ..................... 39 1.10.2 Soundness and Completeness -
Introduction to Abstract Algebra (Math 113)
Introduction to Abstract Algebra (Math 113) Alexander Paulin, with edits by David Corwin FOR FALL 2019 MATH 113 002 ONLY Contents 1 Introduction 4 1.1 What is Algebra? . 4 1.2 Sets . 6 1.3 Functions . 9 1.4 Equivalence Relations . 12 2 The Structure of + and × on Z 15 2.1 Basic Observations . 15 2.2 Factorization and the Fundamental Theorem of Arithmetic . 17 2.3 Congruences . 20 3 Groups 23 1 3.1 Basic Definitions . 23 3.1.1 Cayley Tables for Binary Operations and Groups . 28 3.2 Subgroups, Cosets and Lagrange's Theorem . 30 3.3 Generating Sets for Groups . 35 3.4 Permutation Groups and Finite Symmetric Groups . 40 3.4.1 Active vs. Passive Notation for Permutations . 40 3.4.2 The Symmetric Group Sym3 . 43 3.4.3 Symmetric Groups in General . 44 3.5 Group Actions . 52 3.5.1 The Orbit-Stabiliser Theorem . 55 3.5.2 Centralizers and Conjugacy Classes . 59 3.5.3 Sylow's Theorem . 66 3.6 Symmetry of Sets with Extra Structure . 68 3.7 Normal Subgroups and Isomorphism Theorems . 73 3.8 Direct Products and Direct Sums . 83 3.9 Finitely Generated Abelian Groups . 85 3.10 Finite Abelian Groups . 90 3.11 The Classification of Finite Groups (Proofs Omitted) . 95 4 Rings, Ideals, and Homomorphisms 100 2 4.1 Basic Definitions . 100 4.2 Ideals, Quotient Rings and the First Isomorphism Theorem for Rings . 105 4.3 Properties of Elements of Rings . 109 4.4 Polynomial Rings . 112 4.5 Ring Extensions . 115 4.6 Field of Fractions . -
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 Peripatetic Program in Categorical Logic: Leibniz on Propositional Terms
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 65 THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS MARKO MALINK Department of Philosophy, New York University and ANUBAV VASUDEVAN Department of Philosophy, University of Chicago Abstract. Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such as reductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titled Specimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule of reductio ad absurdum in a purely categor- ical calculus in which every proposition is of the form AisB. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule of reductio ad absurdum,but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic ¬ known as RMI→ . From its beginnings in antiquity up until the late 19th century, the study of formal logic was shaped by two distinct approaches to the subject. The first approach was primarily concerned with simple propositions expressing predicative relations between terms. -
Arxiv:1310.3297V1 [Math.AG] 11 Oct 2013 Bertini for Macaulay2
Bertini for Macaulay2 Daniel J. Bates,∗ Elizabeth Gross,† Anton Leykin,‡ Jose Israel Rodriguez§ September 6, 2018 Abstract Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation. The Macaulay2 package Bertini provides an interface to Bertini, making it possible to access the core run modes of Bertini in Macaulay2. With these run modes, users can find approximate solutions to zero-dimensional systems and positive-dimensional systems, test numerically whether a point lies on a variety, sample numerically from a variety, and perform parameter homotopy runs. 1 Numerical algebraic geometry Numerical algebraic geometry (numerical AG) refers to a set of methods for finding and manipulating the solution sets of systems of polynomial equations. Said differently, given f : CN → Cn, numerical algebraic geometry provides facilities for computing numerical approximations to isolated solutions of V (f) = z ∈ CN |f(z)=0 , as well as numerical approximations to generic points on positive-dimensional components. The book [7] provides a good introduction to the field, while the newer book [2] provides a simpler introduction as well as a complete manual for the software package Bertini [1]. arXiv:1310.3297v1 [math.AG] 11 Oct 2013 Bertini is a free, open source software package for computations in numerical alge- braic geometry. The purpose of this article is to present a Macaulay2 [3] package Bertini that provides an interface to Bertini. This package uses basic datatypes and service rou- tines for computations in numerical AG provided by the package NAGtypes. -
First-Order Logic
Chapter 10 First-Order Logic 10.1 Overview First-Order Logic is the calculus one usually has in mind when using the word ‘‘logic’’. It is expressive enough for all of mathematics, except for those concepts that rely on a notion of construction or computation. However, dealing with more advanced concepts is often somewhat awkward and researchers often design specialized logics for that reason. Our account of first-order logic will be similar to the one of propositional logic. We will present ² The syntax, or the formal language of first-order logic, that is symbols, formulas, sub-formulas, formation trees, substitution, etc. ² The semantics of first-order logic ² Proof systems for first-order logic, such as the axioms, rules, and proof strategies of the first-order tableau method and refinement logic ² The meta-mathematics of first-order logic, which established the relation between the semantics and a proof system In many ways, the account of first-order logic is a straightforward extension of propositional logic. One must, however, be aware that there are subtle differences. 10.2 Syntax The syntax of first-order logic is essentially an extension of propositional logic by quantification 8 and 9. Propositional variables are replaced by n-ary predicate symbols (P , Q, R) which may be instantiated with either variables (x, y, z, ...) or parameters (a, b, ...). Here is a summary of the most important concepts. 101 102 CHAPTER 10. FIRST-ORDER LOGIC (1) Atomic formulas are expressions of the form P c1::cn where P is an n-ary predicate symbol and the ci are variables or parameters.