Notes on Substitution in First–Order Logic
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Relative Interpretations and Substitutional Definitions of Logical
Relative Interpretations and Substitutional Definitions of Logical Truth and Consequence MIRKO ENGLER Abstract: This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that. Keywords: substitutional definitions of logical consequence, Quine, meta- theory of logical consequence 1 Introduction This paper investigates the applicability of relative interpretations in a substi- tutional account of logical truth and consequence. We introduce definitions of logical truth and consequence in terms of relative interpretations and demonstrate that they are extensionally equivalent to the model-theoretic definitions as developed in (Tarski & Vaught, 1956) for any first-order lan- guage (including equality). The benefit of such a definition is that it could be given in a meta-theoretic framework that only requires arithmetic as ax- iomatized by PA. Furthermore, we investigate how intensional constraints on logical truth and consequence force us to extend our framework to an arithmetical meta-theory that itself interprets set-theory. We will argue that such an arithmetical framework still might be in favor over a set-theoretical one. The basic idea behind our definition is both to generate and evaluate substitution instances of a sentence ' by relative interpretations. A relative interpretation rests on a function f that translates all formulae ' of a language L into formulae f(') of a language L0 by mapping the primitive predicates 0 P of ' to formulae P of L while preserving the logical structure of ' 1 Mirko Engler and relativizing its quantifiers by an L0-definable formula. -
Pst-Plot — Plotting Functions in “Pure” L ATEX
pst-plot — plotting functions in “pure” LATEX Commonly, one wants to simply plot a function as a part of a LATEX document. Using some postscript tricks, you can make a graph of an arbitrary function in one variable including implicitly defined functions. The commands described on this worksheet require that the following lines appear in your document header (before \begin{document}). \usepackage{pst-plot} \usepackage{pstricks} The full pstricks manual (including pst-plot documentation) is available at:1 http://www.risc.uni-linz.ac.at/institute/systems/documentation/TeX/tex/ A good page for showing the power of what you can do with pstricks is: http://www.pstricks.de/.2 Reverse Polish Notation (postfix notation) Reverse polish notation (RPN) is a modification of polish notation which was a creation of the logician Jan Lukasiewicz (advisor of Alfred Tarski). The advantage of these notations is that they do not require parentheses to control the order of operations in an expression. The advantage of this in a computer is that parsing a RPN expression is trivial, whereas parsing an expression in standard notation can take quite a bit of computation. At the most basic level, an expression such as 1 + 2 becomes 12+. For more complicated expressions, the concept of a stack must be introduced. The stack is just the list of all numbers which have not been used yet. When an operation takes place, the result of that operation is left on the stack. Thus, we could write the sum of all integers from 1 to 10 as either, 12+3+4+5+6+7+8+9+10+ or 12345678910+++++++++ In both cases the result 55 is left on the stack. -
What Is Abstract Algebraic Logic?
What is Abstract Algebraic Logic? Umberto Rivieccio University of Genoa Department of Philosophy Via Balbi 4 - Genoa, Italy email [email protected] Abstract Our aim is to answer the question of the title, explaining which kind of problems form the nucleus of the branch of algebraic logic that has come to be known as Abstract Algebraic Logic. We give a concise account of the birth and evolution of the field, that led from the discovery of the connections between certain logical systems and certain algebraic structures to a general theory of algebraization of logics. We consider first the so-called algebraizable logics, which may be regarded as a paradigmatic case in Abstract Algebraic Logic; then we shift our attention to some wider classes of logics and to the more general framework in which they are studied. Finally, we review some other topics in Abstract Algebraic Logic that, though equally interesting and promising for future research, could not be treated here in detail. 1 Introduction Algebraic logic can be described in very general terms as the study of the relations between algebra and logic. One of the main reasons that motivate such a study is the possibility to treat logical problems with algebraic methods. This means that, in order to solve a logical problem, we may try to translate it into algebraic terms, find the solution using algebraic techniques, and then translate back the result into logic. This strategy proved to be fruitful, since algebra is an old and well{established field of mathematics, that offers powerful tools to treat problems that would possibly be much harder to solve using the logical machinery alone. -
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. -
The Substitutional Analysis of Logical Consequence
THE SUBSTITUTIONAL ANALYSIS OF LOGICAL CONSEQUENCE Volker Halbach∗ dra version ⋅ please don’t quote ónd June óþÕä Consequentia ‘formalis’ vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [...] Iohannes Buridanus, Tractatus de Consequentiis (Hubien ÕÉßä, .ì, p.óóf) Zf«±§Zh± A substitutional account of logical truth and consequence is developed and defended. Roughly, a substitution instance of a sentence is dened to be the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category. In particular atomic formulae can be replaced with any formulae containing. e denition of logical truth is then as follows: A sentence is logically true i all its substitution instances are always satised. Logical consequence is dened analogously. e substitutional denition of validity is put forward as a conceptual analysis of logical validity at least for suciently rich rst-order settings. In Kreisel’s squeezing argument the formal notion of substitutional validity naturally slots in to the place of informal intuitive validity. ∗I am grateful to Beau Mount, Albert Visser, and Timothy Williamson for discussions about the themes of this paper. Õ §Z êZo±í At the origin of logic is the observation that arguments sharing certain forms never have true premisses and a false conclusion. Similarly, all sentences of certain forms are always true. Arguments and sentences of this kind are for- mally valid. From the outset logicians have been concerned with the study and systematization of these arguments, sentences and their forms. -
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. -
Explanations
© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell. -
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. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
12 Propositional Logic
CHAPTER 12 ✦ ✦ ✦ ✦ Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. In more recent times, this algebra, like many algebras, has proved useful as a design tool. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. A third use of logic is as a data model for programming languages and systems, such as the language Prolog. Many systems for reasoning by computer, including theorem provers, program verifiers, and applications in the field of artificial intelligence, have been implemented in logic-based programming languages. These languages generally use “predicate logic,” a more powerful form of logic that extends the capabilities of propositional logic. We shall meet predicate logic in Chapter 14. ✦ ✦ ✦ ✦ 12.1 What This Chapter Is About Section 12.2 gives an intuitive explanation of what propositional logic is, and why it is useful. The next section, 12,3, introduces an algebra for logical expressions with Boolean-valued operands and with logical operators such as AND, OR, and NOT that Boolean algebra operate on Boolean (true/false) values. This algebra is often called Boolean algebra after George Boole, the logician who first framed logic as an algebra. We then learn the following ideas. ✦ Truth tables are a useful way to represent the meaning of an expression in logic (Section 12.4). ✦ We can convert a truth table to a logical expression for the same logical function (Section 12.5). ✦ The Karnaugh map is a useful tabular technique for simplifying logical expres- sions (Section 12.6). -
Logical Consequence: a Defense of Tarski*
GREG RAY LOGICAL CONSEQUENCE: A DEFENSE OF TARSKI* ABSTRACT. In his classic 1936 essay “On the Concept of Logical Consequence”, Alfred Tarski used the notion of XU~@~ZC~~NIto give a semantic characterization of the logical properties. Tarski is generally credited with introducing the model-theoretic chamcteriza- tion of the logical properties familiar to us today. However, in his hook, The &ncel>f of Lugid Consequence, Etchemendy argues that Tarski’s account is inadequate for quite a number of reasons, and is actually incompatible with the standard mod&theoretic account. Many of his criticisms are meant to apply to the model-theoretic account as well. In this paper, I discuss the following four critical charges that Etchemendy makes against Tarski and his account of the logical properties: (1) (a) Tarski’s account of logical consequence diverges from the standard model-theoretic account at points where the latter account gets it right. (b) Tarski’s account cannot be brought into line with the model-thcorctic account, because the two are fundamentally incompatible. (2) There are simple countcrcxamples (enumerated by Etchemcndy) which show that Tarski’s account is wrong. (3) Tarski committed a modal faIlacy when arguing that his account captures our pre-theoretical concept of logical consequence, and so obscured an essential weakness of the account. (4) Tarski’s account depends on there being a distinction between the “log- ical terms” and the “non-logical terms” of a language, but (according to Etchemendy) there are very simple (even first-order) languages for which no such distinction can be made. Etchemcndy’s critique raises historica and philosophical questions about important foun- dational work. -
On Pocrims and Hoops
On Pocrims and Hoops Rob Arthan & Paulo Oliva August 10, 2018 Abstract Pocrims and suitable specialisations thereof are structures that pro- vide the natural algebraic semantics for a minimal affine logic and its extensions. Hoops comprise a special class of pocrims that provide al- gebraic semantics for what we view as an intuitionistic analogue of the classical multi-valuedLukasiewicz logic. We present some contribu- tions to the theory of these algebraic structures. We give a new proof that the class of hoops is a variety. We use a new indirect method to establish several important identities in the theory of hoops: in particular, we prove that the double negation mapping in a hoop is a homormorphism. This leads to an investigation of algebraic ana- logues of the various double negation translations that are well-known from proof theory. We give an algebraic framework for studying the semantics of double negation translations and use it to prove new re- sults about the applicability of the double negation translations due to Gentzen and Glivenko. 1 Introduction Pocrims provide the natural algebraic models for a minimal affine logic, ALm, while hoops provide the models for what we view as a minimal ana- arXiv:1404.0816v2 [math.LO] 16 Oct 2014 logue, LL m, ofLukasiewicz’s classical infinite-valued logic LL c. This paper presents some new results on the algebraic structure of pocrims and hoops. Our main motivation for this work is in the logical aspects: we are interested in criteria for provability in ALm, LL m and related logics. We develop a useful practical test for provability in LL m and apply it to a range of prob- lems including a study of the various double negation translations in these logics.