Chapter Three What Mathematical Logic Says
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Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Some First-Order Probability Logics
Theoretical Computer Science 247 (2000) 191–212 www.elsevier.com/locate/tcs View metadata, citation and similar papers at core.ac.uk brought to you by CORE Some ÿrst-order probability logics provided by Elsevier - Publisher Connector Zoran Ognjanovic a;∗, Miodrag RaÄskovic b a MatematickiÄ institut, Kneza Mihaila 35, 11000 Beograd, Yugoslavia b Prirodno-matematickiÄ fakultet, R. Domanovicaà 12, 34000 Kragujevac, Yugoslavia Received June 1998 Communicated by M. Nivat Abstract We present some ÿrst-order probability logics. The logics allow making statements such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. We describe the corresponding probability models. We give a sound and complete inÿnitary axiomatic system for the most general of our logics, while for some restrictions of this logic we provide ÿnitary axiomatic systems. We study the decidability of our logics. We discuss some of the related papers. c 2000 Elsevier Science B.V. All rights reserved. Keywords: First order logic; Probability; Possible worlds; Completeness 1. Introduction In recent years there is a growing interest in uncertainty reasoning. A part of inves- tigation concerns its formal framework – probability logics [1, 2, 5, 6, 9, 13, 15, 17–20]. Probability languages are obtained by adding probability operators of the form (in our notation) P¿s to classical languages. The probability logics allow making formulas such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. Probability models similar to Kripke models are used to give semantics to the probability formulas so that interpreted formulas are either true or false. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”. -
Open-Logic-Enderton Rev: C8c9782 (2021-09-28) by OLP/ CC–BY
An Open Mathematical Introduction to Logic Example OLP Text `ala Enderton Open Logic Project open-logic-enderton rev: c8c9782 (2021-09-28) by OLP/ CC{BY Contents I Propositional Logic7 1 Syntax and Semantics9 1.0 Introduction............................ 9 1.1 Propositional Wffs......................... 10 1.2 Preliminaries............................ 11 1.3 Valuations and Satisfaction.................... 12 1.4 Semantic Notions ......................... 14 Problems ................................. 14 2 Derivation Systems 17 2.0 Introduction............................ 17 2.1 The Sequent Calculus....................... 18 2.2 Natural Deduction......................... 19 2.3 Tableaux.............................. 20 2.4 Axiomatic Derivations ...................... 22 3 Axiomatic Derivations 25 3.0 Rules and Derivations....................... 25 3.1 Axiom and Rules for the Propositional Connectives . 26 3.2 Examples of Derivations ..................... 27 3.3 Proof-Theoretic Notions ..................... 29 3.4 The Deduction Theorem ..................... 30 1 Contents 3.5 Derivability and Consistency................... 32 3.6 Derivability and the Propositional Connectives......... 32 3.7 Soundness ............................. 33 Problems ................................. 34 4 The Completeness Theorem 35 4.0 Introduction............................ 35 4.1 Outline of the Proof........................ 36 4.2 Complete Consistent Sets of Sentences ............. 37 4.3 Lindenbaum's Lemma....................... 38 4.4 Construction of a Model -
Axiomatic Systems for Geometry∗
Axiomatic Systems for Geometry∗ George Francisy composed 6jan10, adapted 27jan15 1 Basic Concepts An axiomatic system contains a set of primitives and axioms. The primitives are Adaptation to the object names, but the objects they name are left undefined. This is why the primi- current course is in tives are also called undefined terms. The axioms are sentences that make assertions the margins. about the primitives. Such assertions are not provided with any justification, they are neither true nor false. However, every subsequent assertion about the primitives, called a theorem, must be a rigorously logical consequence of the axioms and previ- ously proved theorems. There are also formal definitions in an axiomatic system, but these serve only to simplify things. They establish new object names for complex combinations of primitives and previously defined terms. This kind of definition does not impart any `meaning', not yet, anyway. See Lesson A6 for an elaboration. If, however, a definite meaning is assigned to the primitives of the axiomatic system, called an interpretation, then the theorems become meaningful assertions which might be true or false. If for a given interpretation all the axioms are true, then everything asserted by the theorems also becomes true. Such an interpretation is called a model for the axiomatic system. In common speech, `model' is often used to mean an example of a class of things. In geometry, a model of an axiomatic system is an interpretation of its primitives for ∗From textitPost-Euclidean Geometry: Class Notes and Workbook, UpClose Printing & Copies, Champaign, IL 1995, 2004 yProf. George K. -
Juhani Pallasmaa, Architect, Professor Emeritus
1 CROSSING WORLDS: MATHEMATICAL LOGIC, PHILOSOPHY, ART Honouring Juliette Kennedy University oF Helsinki, Small Hall Friday, 3 June, 2016-05-28 Juhani Pallasmaa, Architect, ProFessor Emeritus Draft 28 May 2016 THE SIXTH SENSE - diffuse perception, mood and embodied Wisdom ”Whether people are fully conscious oF this or not, they actually derive countenance and sustenance From the atmosphere oF things they live in and with”.1 Frank Lloyd Wright . --- Why do certain spaces and places make us Feel a strong aFFinity and emotional identification, while others leave us cold, or even frighten us? Why do we feel as insiders and participants in some spaces, Whereas others make us experience alienation and ”existential outsideness”, to use a notion of Edward Relph ?2 Isn’t it because the settings of the first type embrace and stimulate us, make us willingly surrender ourselves to them, and feel protected and sensually nourished? These spaces, places and environments strengthen our sense of reality and selF, whereas disturbing and alienating settings weaken our sense oF identity and reality. Resonance with the cosmos and a distinct harmonious tuning were essential qualities oF architecture since the Antiquity until the instrumentalized and aestheticized construction of the industrial era. Historically, the Fundamental task oF architecture Was to create a harmonic resonance betWeen the microcosm oF the human realm and the macrocosm oF the Universe. This harmony Was sought through proportionality based on small natural numbers FolloWing Pythagorean harmonics, on Which the harmony oF the Universe Was understood to be based. The Renaissance era also introduced the competing proportional ideal of the Golden Section. -
Mathematical Logic
Copyright c 1998–2005 by Stephen G. Simpson Mathematical Logic Stephen G. Simpson December 15, 2005 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 http://www.math.psu.edu/simpson/ This is a set of lecture notes for introductory courses in mathematical logic offered at the Pennsylvania State University. Contents Contents 1 1 Propositional Calculus 3 1.1 Formulas ............................... 3 1.2 Assignments and Satisfiability . 6 1.3 LogicalEquivalence. 10 1.4 TheTableauMethod......................... 12 1.5 TheCompletenessTheorem . 18 1.6 TreesandK¨onig’sLemma . 20 1.7 TheCompactnessTheorem . 21 1.8 CombinatorialApplications . 22 2 Predicate Calculus 24 2.1 FormulasandSentences . 24 2.2 StructuresandSatisfiability . 26 2.3 TheTableauMethod......................... 31 2.4 LogicalEquivalence. 37 2.5 TheCompletenessTheorem . 40 2.6 TheCompactnessTheorem . 46 2.7 SatisfiabilityinaDomain . 47 3 Proof Systems for Predicate Calculus 50 3.1 IntroductiontoProofSystems. 50 3.2 TheCompanionTheorem . 51 3.3 Hilbert-StyleProofSystems . 56 3.4 Gentzen-StyleProofSystems . 61 3.5 TheInterpolationTheorem . 66 4 Extensions of Predicate Calculus 71 4.1 PredicateCalculuswithIdentity . 71 4.2 TheSpectrumProblem . .. .. .. .. .. .. .. .. .. 75 4.3 PredicateCalculusWithOperations . 78 4.4 Predicate Calculus with Identity and Operations . ... 82 4.5 Many-SortedPredicateCalculus . 84 1 5 Theories, Models, Definability 87 5.1 TheoriesandModels ......................... 87 5.2 MathematicalTheories. 89 5.3 DefinabilityoveraModel . 97 5.4 DefinitionalExtensionsofTheories . 100 5.5 FoundationalTheories . 103 5.6 AxiomaticSetTheory . 106 5.7 Interpretability . 111 5.8 Beth’sDefinabilityTheorem. 112 6 Arithmetization of Predicate Calculus 114 6.1 Primitive Recursive Arithmetic . 114 6.2 Interpretability of PRA in Z1 ....................114 6.3 G¨odelNumbers ............................ 114 6.4 UndefinabilityofTruth. 117 6.5 TheProvabilityPredicate . -
Mathematical Logic and Foundations of Mathematics
Mathematical Logic and Foundations of Mathematics Stephen G. Simpson Pennsylvania State University Open House / Graduate Conference April 2–3, 2010 1 Foundations of mathematics (f.o.m.) is the study of the most basic concepts and logical structure of mathematics as a whole. Among the most basic mathematical concepts are: number, shape, set, function, algorithm, mathematical proof, mathematical definition, mathematical axiom, mathematical theorem. Some typical questions in f.o.m. are: 1. What is a number? 2. What is a shape? . 6. What is a mathematical proof? . 10. What are the appropriate axioms for mathematics? Mathematical logic gives some mathematically rigorous answers to some of these questions. 2 The concepts of “mathematical theorem” and “mathematical proof” are greatly clarified by the predicate calculus. Actually, the predicate calculus applies to non-mathematical subjects as well. Let Lxy be a 2-place predicate meaning “x loves y”. We can express properties of loving as sentences of the predicate calculus. ∀x ∃y Lxy ∃x ∀y Lyx ∀x (Lxx ⇒¬∃y Lyx) ∀x ∀y ∀z (((¬ Lyx) ∧ (¬ Lzy)) ⇒ Lzx) ∀x ((∃y Lxy) ⇒ Lxx) 3 There is a deterministic algorithm (the Tableau Method) which shows us (after a finite number of steps) that particular sentences are logically valid. For instance, the tableau ∃x (Sx ∧∀y (Eyx ⇔ (Sy ∧ ¬ Eyy))) Sa ∧∀y (Eya ⇔ Sy ∧ ¬ Eyy) Sa ∀y (Eya ⇔ Sy ∧ ¬ Eyy) Eaa ⇔ (Sa ∧ ¬ Eaa) / \ Eaa ¬ Eaa Sa ∧ ¬ Eaa ¬ (Sa ∧ ¬ Eaa) Sa / \ ¬ Sa ¬ ¬ Eaa ¬ Eaa Eaa tells us that the sentence ¬∃x (Sx ∧∀y (Eyx ⇔ (Sy ∧ ¬ Eyy))) is logically valid. This is the Russell Paradox. 4 Two significant results in mathematical logic: (G¨odel, Tarski, . -
A Mathematical Theory of Computation?
A Mathematical Theory of Computation? Simone Martini Dipartimento di Informatica { Scienza e Ingegneria Alma mater studiorum • Universit`adi Bologna and INRIA FoCUS { Sophia / Bologna Lille, February 1, 2017 1 / 57 Reflect and trace the interaction of mathematical logic and programming (languages), identifying some of the driving forces of this process. Previous episodes: Types HaPOC 2015, Pisa: from 1955 to 1970 (circa) Cie 2016, Paris: from 1965 to 1975 (circa) 2 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 3 / 57 Why types? Modern programming languages: control flow specification: small fraction abstraction mechanisms to model application domains. • Types are a crucial building block of these abstractions • And they are a mathematical logic concept, aren't they? 4 / 57 We today conflate: Types as an implementation (representation) issue Types as an abstraction mechanism Types as a classification mechanism (from mathematical logic) 5 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 6 / 57 The quest for a \Mathematical Theory of Computation" How does mathematical logic fit into this theory? And for what purposes? 7 / 57 Prehistory 1947 8 / 57 Goldstine and von Neumann [. ] coding [. ] has to be viewed as a logical problem and one that represents a new branch of formal logics. Hermann Goldstine and John von Neumann Planning and Coding of problems for an Electronic Computing Instrument Report on the mathematical and logical aspects of an electronic computing instrument, Part II, Volume 1-3, April 1947. -
Math 310 Class Notes 1: Axioms of Set Theory
MATH 310 CLASS NOTES 1: AXIOMS OF SET THEORY Intuitively, we think of a set as an organization and an element be- longing to a set as a member of the organization. Accordingly, it also seems intuitively clear that (1) when we collect all objects of a certain kind the collection forms an organization, i.e., forms a set, and, moreover, (2) it is natural that when we collect members of a certain kind of an organization, the collection is a sub-organization, i.e., a subset. Item (1) above was challenged by Bertrand Russell in 1901 if we accept the natural item (2), i.e., we must be careful when we use the word \all": (Russell Paradox) The collection of all sets is not a set! Proof. Suppose the collection of all sets is a set S. Consider the subset U of S that consists of all sets x with the property that each x does not belong to x itself. Now since U is a set, U belongs to S. Let us ask whether U belongs to U itself. Since U is the collection of all sets each of which does not belong to itself, thus if U belongs to U, then as an element of the set U we must have that U does not belong to U itself, a contradiction. So, it must be that U does not belong to U. But then U belongs to U itself by the very definition of U, another contradiction. The absurdity results because we assume S is a set. -
Axiomatic Foundations of Mathematics Ryan Melton Dr
Axiomatic Foundations of Mathematics Ryan Melton Dr. Clint Richardson, Faculty Advisor Stephen F. Austin State University As Bertrand Russell once said, Gödel's Method Pure mathematics is the subject in which we Consider the expression First, Gödel assigned a unique natural number to do not know what we are talking about, or each of the logical symbols and numbers. 2 + 3 = 5 whether what we are saying is true. Russell’s statement begs from us one major This expression is mathematical; it belongs to the field For example: if the symbol '0' corresponds to we call arithmetic and is composed of basic arithmetic question: the natural number 1, '+' to 2, and '=' to 3, then symbols. '0 = 0' '0 + 0 = 0' What is Mathematics founded on? On the other hand, the sentence and '2 + 3 = 5' is an arithmetical formula. 1 3 1 1 2 1 3 1 so each expression corresponds to a sequence. Axioms and Axiom Systems is metamathematical; it is constructed outside of mathematics and labels the expression above as a Then, for this new sequence x1x2x3…xn of formula in arithmetic. An axiom is a belief taken without proof, and positive integers, we associate a Gödel number thus an axiom system is a set of beliefs as follows: x1 x2 x3 xn taken without proof. enc( x1x2x3...xn ) = 2 3 5 ... pn Since Principia Mathematica was such a bold where the encoding is the product of n factors, Consistent? Complete? leap in the right direction--although proving each of which is found by raising the j-th prime nothing about consistency--several attempts at to the xj power. -
Warren Goldfarb, Notes on Metamathematics
Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines .