DOCSLIB.ORG

1. the Prehistory 2. Arithmetizations of the Traditional Syllo- Gistic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. the Prehistory 2. Arithmetizations of the Traditional Syllo- Gistic Vladimir Sotirov ARITHMETIZATIONS OF SYLLOGISTIC `ala LEIBNIZ1 1. The Prehistory ÃLukasiewicz had every reason to suppose that Leibniz's winged Calculemus! had been connected with the Aristotelian syllogistic [4, x 34]. Indeed, after Louis Couturat's pioneer e®orts in commenting and publishing Leibniz's logical opuscula ([1], [2]). The basic idea of the arithmetization of syllogistic was to establish a correspondence between terms and suitable integers (the characteristic numbers of notions), so that the logical truth of a proposition would turn into an arithmetical truth of unsuccessful. The second one used a transla- tion of terms into pairs of co-prime numbers and was successful, as SÃlupecki proved ([8]; see also [4, x 34]). This second translation obviously was more so±sticated but the bigger trouble was in the paper we justify the viability of the earlier (and less complicated) Leibniz idea. Moreover, we propose two translations into arithmetic which are appropriate for syllogistic with all Boolean term operations. 2. Arithmetizations of the Traditional Syllo- gistic We will treat the Aristotelian syllogistic in the style that became canonical after ÃLukasiewicz's celebrated book [4]. For this purpose the language of the classical propositional calculus is extended by term variables (for short, terms) t1; t2; . together with two binary term relations: A and 1Supported by contract U{705/1997 with the Bulgarian Ministry of Education and Science. An extended version of the paper will appear in the special issue of the Journal of Applied Non-Classical Logics dedicated to the memory of George Gargov. 156 I. Syllogistic atoms are all formulae of the kind sAp or sIp with s and p being terms. A syllogism is any propositional formula with all propositional letters replaced by syllogistic atoms. The standard and the most intuitive semantics of the Aristotelian syllogistic is that in the theory of sets: if S and P are arbitrary non-empty sets, sAp (\Every s is a p") is translated into S ⊆ P , sIp (\Some s is a p") into S \ P 6= ; (briefly, SθP ), and the formal propositional connectives are replaced with the informal ones. Thus any syllogism is translated into a sentence about non-empty sets. If this sentence is true, i.e., if the expression so obtained is a set-theoretical tautology, the syllogism is said to be true. It is true in a given (non-empty) set U when any replacement of its terms with (non-empty) subsets of U gives a true sentence. This semantics we call Scholastic following Leibniz himself. It is characterized by (⊆; \ 6= ;). Another semantics in the theory of sets is possible; it will be named Leibnizian being (partially) accepted by him. When a non-empty set U is given, term variables are evaluated by subsets of U di®erent from U. If S and P are such sets, sAp is interpreted as S ¶ P , sIp as S [ P 6= U, and the formal propositional connectives are replaced by informal ones. This semantics is characterized by (¶; [ 6= U). A syllogism is said to be true in U when the sentence obtained after any replacement of all term variables with subsets of U (di®erent from U) is true. The syllogism is true when it is true in any set U. Obviously, both semantics are dual, and a syllogism is true in the Scholastic semantics i® it is true in the Leibnizian one. The following theorem expresses the decidability of syllogistic: Theorem 1. A syllogism with n term letters is true i® it is true in any set with no more than 2n elements. The theorem is similar to the statement of decidability of the monadic predicate calculus. The only-if part is obvious. For the if part, let us suppose that a syllogism is not true in the Scholastic semantics, and let T1; ... ;Tn be some non-empty sets rejecting it. Denote by U their union. Let us identify the elements of U following x ¼ y $ (8i)(x 2 Ti $ y 2 Ti). Denote by jxj the equivalence class of x, i.e., w 2 jxj $ w ¼ x. The factorization of U produces no more than 2n elements2 2The upper bound 2n can es obtain the proof for the Leibnizian semantics, the ¤ ¤ ¤ complements of T1 ; ... ;Tn to U may be taken. 157 The ¯rst interpretation of the traditional syllogistic in arithmetic we name Scholastic. Let a1; a2; . denote arbitrary integers greater than 1. Given a syllogism, replace tiAtj with aijaj (\ai is a divisor of aj"), tiItj with a new relation aiGaj (\ai and aj have a common divisor greater than 1", or: g.c.d. (ai; aj) > 1), and the formal propositional connectives with informal ones. In short, this interpretation is characterized by (j, g.c.d. > 1). Call the syllogism arithmetically true (in the Scholastic sense) if the sentence so obtained is an arithmetical truth. Theorem 2. (Adequacy of the Scholastic arithmetical interpre- tation): A syllogism is true i® it is arithmetically true (in the Scholastic sense). Let faig denote the set of all divisors of ai greater than 1. Then ai j aj is equivalent to faig ⊆ fajg, and ai G aj to faigθfajg. So, if the syllogism is true in the Scholastic semantics of sets, it will be arithmetically true as well. Conversely, if the syllogism is not true in the Scholastic ¤ ¤ semantics, and contains n term letters, let T1 ; ... ;Tn be the rejecting sets n provided by Theorem 1. Suppose all their elements are w1; ... ; wk (k · 2 ). ¤ Take p1; ... ; pk to be di®erent prime numbers. If the elements of Ti are ¤ wi1 ; ... ; wim (remember that Ti 6= ;), pose pi1 ... pim = ai. Any of the ¤ ¤ integers so obtained is greater than 1, and aijaj $ Ti ⊆ Tj ; aiGaj $ ¤ ¤ Ti θTj . Ergo, the integers a1; ... ; ak will make the syllogism arithmetically false. If the empty set is admitted to evaluate terms in the Scholastic seman- tics, a new syllogistic will be obtained which di®ers from the traditional one. For example, some of Aristotle's syllogisms like Bramalip stop being true. The law sIs introduced by Leibniz will not be true, either. The arithmetical interpretation can be modi¯ed to serve syllogistic with empty terms by admitting 1 as their number value together with a sole modi¯ca- tion in the proof of Theorem 2: 1 has to be added to the list of divisors of ai being the number corresponding to the empty term; in the second part ¤ of the proof, if Ti = ; then pose ai = 1. Further, for the arithmetical interpretation named Leibnizian, let u (the Universe number) be an arbitrary integer greater than 1, and let a1; a2; . be arbitrary its proper divisors, i.e., ai < u for any i (however, ai = 1 is permitted). Replace tiAtj by a relation ai=aj (\ai is divisible by aj"; this notation is to remind of the division ai=aj), and tiItj by a 158 relation ai L aj de¯ned as \the least common multiple of ai and aj is less than u", or: \there is a prime divisor of u dividing neither ai nor aj". Finally, formal propositional connectives are replaced with their informal analogues. In short, the Leibnizian arithmetical interpretation is charac- terized by (/, l.c.m. < u). The syllogism is said to be arithmetically true (in the Leibnizian sense) with respect to u if the sentence so obtained is an arithmetical truth. The syllogism is arithmetically true (in the Leibnizian sense) if it is arithmetically true, in the same sense, with respect to any u > 1. Theorem 3. (Adequacy of the Leibnizian arithmetical interpre- tation): A syllogism is true i® it is arithmetically true in the Leibnizian sense. For proving Theorem 3 a bridge will be thrown to Theorem 2. If a and b are arbitrary integers satisfying the conditions of the Leibniz inter- u u pretation with respect to a Universe number u, then a and b are integers u u satisfying the conditions of the Scholastic interpretation, and a = b $ ajb, u u a G b $ aLb. So, if a syllogism is arithmetically true in the Scholastic sense, it will be arithmetically true in the Leibnizian sense as well. Conversely, if the syllogism is arithmetically true in the Leibnizian sense, and a1; ... ; an are arbitrary integers satisfying the Scholastic con- u ditions (i.e., ai > 1 for any i), take u = l:c:m: (a1; ... ; an). Then all a u u i satisfy the conditions of the Leibnizian interpretation, and aijaj $ = , ai aj u u u u aiLaj $ G . Because the syllogism evaluated by ; ... ; was sup- ai aj a1 an posed to be true, the sentence obtained after evaluating it by a1; ... ; an will be true, too. Ergo, the syllogism is arithmetically true in the Scholastic sense. 3. Arithmetizations of Syllogistic with Term Negation In contrast to the traditional syllogistic, there are many systems contain- ing term negation, and none of them is canonized. To compare di®erent semantics and their correspondent axiomatics is not our aim; see, e.g., [6] and [7]. Here we will examine only the system that is nearest to Leibniz's view. 159 Expand the language of syllogistic by adding an operation of term negation ¡; then, if t is a term, ¡t (\non-t") is a term, too. The de¯nition of atoms is modi¯ed by permitting s and p to be arbitrary terms in sAp and in sIp as well. In both set-theoretical semantics, a universal set U is introduced. Terms are evaluated by subsets of U di®erent from ; and U. If a term t is evaluated by a set T , the value of ¡t is the complement of T to U.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Propositional Calculus
    CSC 438F/2404F Notes Winter, 2014 S. Cook REFERENCES The first two references have especially influenced these notes and are cited from time to time: [Buss] Samuel Buss: Chapter I: An introduction to proof theory, in Handbook of Proof Theory, Samuel Buss Ed., Elsevier, 1998, pp1-78. [B&M] John Bell and Moshe Machover: A Course in Mathematical Logic. North- Holland, 1977. Other logic texts: The first is more elementary and readable. [Enderton] Herbert Enderton: A Mathematical Introduction to Logic. Academic Press, 1972. [Mendelson] E. Mendelson: Introduction to Mathematical Logic. Wadsworth & Brooks/Cole, 1987. Computability text: [Sipser] Michael Sipser: Introduction to the Theory of Computation. PWS, 1997. [DSW] M. Davis, R. Sigal, and E. Weyuker: Computability, Complexity and Lan- guages: Fundamentals of Theoretical Computer Science. Academic Press, 1994. Propositional Calculus Throughout our treatment of formal logic it is important to distinguish between syntax and semantics. Syntax is concerned with the structure of strings of symbols (e.g. formulas and formal proofs), and rules for manipulating them, without regard to their meaning. Semantics is concerned with their meaning. 1 Syntax Formulas are certain strings of symbols as specified below. In this chapter we use formula to mean propositional formula. Later the meaning of formula will be extended to first-order formula. (Propositional) formulas are built from atoms P1;P2;P3;:::, the unary connective :, the binary connectives ^; _; and parentheses (,). (The symbols :; ^ and _ are read \not", \and" and \or", respectively.) We use P; Q; R; ::: to stand for atoms. Formulas are defined recursively as follows: Definition of Propositional Formula 1) Any atom P is a formula.
    [Show full text]
  • Foundations of Mathematics
    Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .
    [Show full text]
  • Lecture Notes for MATH 770 : Foundations of Mathematics — University of Wisconsin – Madison, Fall 2005
    Lecture notes for MATH 770 : Foundations of Mathematics — University of Wisconsin – Madison, Fall 2005 Ita¨ıBEN YAACOV Ita¨ı BEN YAACOV, Institut Camille Jordan, Universite´ Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex URL: http://math.univ-lyon1.fr/~begnac/ c Ita¨ıBEN YAACOV. All rights reserved. CONTENTS Contents Chapter 1. Propositional Logic 1 1.1. Syntax 1 1.2. Semantics 3 1.3. Syntactic deduction 6 Exercises 12 Chapter 2. First order Predicate Logic 17 2.1. Syntax 17 2.2. Semantics 19 2.3. Substitutions 22 2.4. Syntactic deduction 26 Exercises 35 Chapter 3. Model Theory 39 3.1. Elementary extensions and embeddings 40 3.2. Quantifier elimination 46 Exercises 51 Chapter 4. Incompleteness 53 4.1. Recursive functions 54 4.2. Coding syntax in Arithmetic 59 4.3. Representation of recursive functions 64 4.4. Incompleteness 69 4.5. A “physical” computation model: register machines 71 Exercises 75 Chapter 5. Set theory 77 5.1. Axioms for set theory 77 5.2. Well ordered sets 80 5.3. Cardinals 86 Exercises 94 –sourcefile– iii Rev: –revision–, July 22, 2008 CONTENTS 1.1. SYNTAX CHAPTER 1 Propositional Logic Basic ingredients: • Propositional variables, which will be denoted by capital letters P,Q,R,..., or sometimes P0, P1, P2,.... These stand for basic statements, such as “the sun is hot”, “the moon is made of cheese”, or “everybody likes math”. The set of propositional variables will be called vocabulary. It may be infinite. • Logical connectives: ¬ (unary connective), →, ∧, ∨ (binary connectives), and possibly others. Each logical connective is defined by its truth table: A B A → B A ∧ B A ∨ B A ¬A T T T T T T F T F F F T F T F T T F T F F T F F Thus: • The connective ¬ means “not”: ¬A means “not A”.
    [Show full text]
  • Axiomatization of Logic. Truth and Proof. Resolution
    H250: Honors Colloquium – Introduction to Computation Axiomatization of Logic. Truth and Proof. Resolution Marius Minea [email protected] How many do we need? (best: few) Are they enough? Can we prove everything? Axiomatization helps answer these questions Motivation: Determining Truth In CS250, we started with truth table proofs. Propositional formula: can always do truth table (even if large) Predicate formula: can’t do truth tables (infinite possibilities) ⇒ must use other proof rules Motivation: Determining Truth In CS250, we started with truth table proofs. Propositional formula: can always do truth table (even if large) Predicate formula: can’t do truth tables (infinite possibilities) ⇒ must use other proof rules How many do we need? (best: few) Are they enough? Can we prove everything? Axiomatization helps answer these questions Symbols of propositional logic: propositions: p, q, r (usually lowercase letters) operators (logical connectives): negation ¬, implication → parentheses () Formulas of propositional logic: defined by structural induction (how to build complex formulas from simpler ones) A formula (compound proposition) is: any proposition (aka atomic formula or variable) (¬α) where α is a formula (α → β) if α and β are formulas Implication and negation suffice! First, Define Syntax We define a language by its symbols and the rules to correctly combine symbols (the syntax) Formulas of propositional logic: defined by structural induction (how to build complex formulas from simpler ones) A formula (compound proposition) is: any
    [Show full text]
  • Lecture 4 1 Overview 2 Propositional Logic
    COMPSCI 230: Discrete Mathematics for Computer Science January 23, 2019 Lecture 4 Lecturer: Debmalya Panigrahi Scribe: Kevin Sun 1 Overview In this lecture, we give an introduction to propositional logic, which is the mathematical formaliza- tion of logical relationships. We also discuss the disjunctive and conjunctive normal forms, how to convert formulas to each form, and conclude with a fundamental problem in computer science known as the satisfiability problem. 2 Propositional Logic Until now, we have seen examples of different proofs by primarily drawing from simple statements in number theory. Now we will establish the fundamentals of writing formal proofs by reasoning about logical statements. Throughout this section, we will maintain a running analogy of propositional logic with the system of arithmetic with which we are already familiar. In general, capital letters (e.g., A, B, C) represent propositional variables, while lowercase letters (e.g., x, y, z) represent arithmetic variables. But what is a propositional variable? A propositional variable is the basic unit of propositional logic. Recall that in arithmetic, the variable x is a placeholder for any real number, such as 7, 0.5, or −3. In propositional logic, the variable A is a placeholder for any truth value. While x has (infinitely) many possible values, there are only two possible values of A: True (denoted T) and False (denoted F). Since propositional variables can only take one of two possible values, they are also known as Boolean variables, named after their inventor George Boole. By letting x denote any real number, we can make claims like “x2 − 1 = (x + 1)(x − 1)” regardless of the numerical value of x.
    [Show full text]