Understanding Logical Connectives: a Comparative Study of Language Influence
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UNIT-I Mathematical Logic Statements and Notations
UNIT-I Mathematical Logic Statements and notations: A proposition or statement is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. Connectives: Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation ¬p “not p” Conjunction p q “p and q” Disjunction p q “p or q (or both)” Exclusive Or p q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p q “p if and only if q” Truth Tables: Logical identity Logical identity is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is true and a value of false if its operand is false. The truth table for the logical identity operator is as follows: Logical Identity p p T T F F Logical negation Logical negation is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is false and a value of false if its operand is true. -
Introduction to Logic
IntroductionI Introduction to Logic Propositional calculus (or logic) is the study of the logical Christopher M. Bourke relationship between objects called propositions and forms the basis of all mathematical reasoning. Definition A proposition is a statement that is either true or false, but not both (we usually denote a proposition by letters; p, q, r, s, . .). Computer Science & Engineering 235 Introduction to Discrete Mathematics [email protected] IntroductionII ExamplesI Example (Propositions) Definition I 2 + 2 = 4 I The derivative of sin x is cos x. The value of a proposition is called its truth value; denoted by T or I 6 has 2 factors 1 if it is true and F or 0 if it is false. Opinions, interrogative and imperative sentences are not Example (Not Propositions) propositions. I C++ is the best language. I When is the pretest? I Do your homework. ExamplesII Logical Connectives Connectives are used to create a compound proposition from two Example (Propositions?) or more other propositions. I Negation (denoted or !) ¬ I 2 + 2 = 5 I And (denoted ) or Logical Conjunction ∧ I Every integer is divisible by 12. I Or (denoted ) or Logical Disjunction ∨ I Microsoft is an excellent company. I Exclusive Or (XOR, denoted ) ⊕ I Implication (denoted ) → I Biconditional; “if and only if” (denoted ) ↔ Negation Logical And A proposition can be negated. This is also a proposition. We usually denote the negation of a proposition p by p. ¬ The logical connective And is true only if both of the propositions are true. It is also referred to as a conjunction. Example (Negated Propositions) Example (Logical Connective: And) I Today is not Monday. -
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. -
Folktale Documentation Release 1.0
Folktale Documentation Release 1.0 Quildreen Motta Nov 05, 2017 Contents 1 Guides 3 2 Indices and tables 5 3 Other resources 7 3.1 Getting started..............................................7 3.2 Folktale by Example........................................... 10 3.3 API Reference.............................................. 12 3.4 How do I................................................... 63 3.5 Glossary................................................. 63 Python Module Index 65 i ii Folktale Documentation, Release 1.0 Folktale is a suite of libraries for generic functional programming in JavaScript that allows you to write elegant modular applications with fewer bugs, and more reuse. Contents 1 Folktale Documentation, Release 1.0 2 Contents CHAPTER 1 Guides • Getting Started A series of quick tutorials to get you up and running quickly with the Folktale libraries. • API reference A quick reference of Folktale’s libraries, including usage examples and cross-references. 3 Folktale Documentation, Release 1.0 4 Chapter 1. Guides CHAPTER 2 Indices and tables • Global Module Index Quick access to all modules. • General Index All functions, classes, terms, sections. • Search page Search this documentation. 5 Folktale Documentation, Release 1.0 6 Chapter 2. Indices and tables CHAPTER 3 Other resources • Licence information 3.1 Getting started This guide will cover everything you need to start using the Folktale project right away, from giving you a brief overview of the project, to installing it, to creating a simple example. Once you get the hang of things, the Folktale By Example guide should help you understanding the concepts behind the library, and mapping them to real use cases. 3.1.1 So, what’s Folktale anyways? Folktale is a suite of libraries for allowing a particular style of functional programming in JavaScript. -
Theorem Proving in Classical Logic
MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report. -
Learning Dependency-Based Compositional Semantics
Learning Dependency-Based Compositional Semantics Percy Liang∗ University of California, Berkeley Michael I. Jordan∗∗ University of California, Berkeley Dan Klein† University of California, Berkeley Suppose we want to build a system that answers a natural language question by representing its semantics as a logical form and computing the answer given a structured database of facts. The core part of such a system is the semantic parser that maps questions to logical forms. Semantic parsers are typically trained from examples of questions annotated with their target logical forms, but this type of annotation is expensive. Our goal is to instead learn a semantic parser from question–answer pairs, where the logical form is modeled as a latent variable. We develop a new semantic formalism, dependency-based compositional semantics (DCS) and define a log-linear distribution over DCS logical forms. The model parameters are estimated using a simple procedure that alternates between beam search and numerical optimization. On two standard semantic parsing benchmarks, we show that our system obtains comparable accuracies to even state-of-the-art systems that do require annotated logical forms. 1. Introduction One of the major challenges in natural language processing (NLP) is building systems that both handle complex linguistic phenomena and require minimal human effort. The difficulty of achieving both criteria is particularly evident in training semantic parsers, where annotating linguistic expressions with their associated logical forms is expensive but until recently, seemingly unavoidable. Advances in learning latent-variable models, however, have made it possible to progressively reduce the amount of supervision ∗ Computer Science Division, University of California, Berkeley, CA 94720, USA. -
Reasoning with Ambiguity
Noname manuscript No. (will be inserted by the editor) Reasoning with Ambiguity Christian Wurm the date of receipt and acceptance should be inserted later Abstract We treat the problem of reasoning with ambiguous propositions. Even though ambiguity is obviously problematic for reasoning, it is no less ob- vious that ambiguous propositions entail other propositions (both ambiguous and unambiguous), and are entailed by other propositions. This article gives a formal analysis of the underlying mechanisms, both from an algebraic and a logical point of view. The main result can be summarized as follows: sound (and complete) reasoning with ambiguity requires a distinction between equivalence on the one and congruence on the other side: the fact that α entails β does not imply β can be substituted for α in all contexts preserving truth. With- out this distinction, we will always run into paradoxical results. We present the (cut-free) sequent calculus ALcf , which we conjecture implements sound and complete propositional reasoning with ambiguity, and provide it with a language-theoretic semantics, where letters represent unambiguous meanings and concatenation represents ambiguity. Keywords Ambiguity, reasoning, proof theory, algebra, Computational Linguistics Acknowledgements Thanks to Timm Lichte, Roland Eibers, Roussanka Loukanova and Gerhard Schurz for helpful discussions; also I want to thank two anonymous reviewers for their many excellent remarks! 1 Introduction This article gives an extensive treatment of reasoning with ambiguity, more precisely with ambiguous propositions. We approach the problem from an Universit¨atD¨usseldorf,Germany Universit¨atsstr.1 40225 D¨usseldorf Germany E-mail: [email protected] 2 Christian Wurm algebraic and a logical perspective and show some interesting surprising results on both ends, which lead up to some interesting philosophical questions, which we address in a preliminary fashion. -
A Computer-Verified Monadic Functional Implementation of the Integral
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 411 (2010) 3386–3402 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A computer-verified monadic functional implementation of the integral Russell O'Connor, Bas Spitters ∗ Radboud University Nijmegen, Netherlands article info a b s t r a c t Article history: We provide a computer-verified exact monadic functional implementation of the Riemann Received 10 September 2008 integral in type theory. Together with previous work by O'Connor, this may be seen as Received in revised form 13 January 2010 the beginning of the realization of Bishop's vision to use constructive mathematics as a Accepted 23 May 2010 programming language for exact analysis. Communicated by G.D. Plotkin ' 2010 Elsevier B.V. All rights reserved. Keywords: Type theory Functional programming Exact real analysis Monads 1. Introduction Integration is one of the fundamental techniques in numerical computation. However, its implementation using floating- point numbers requires continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the end-user by providing a library of exact analysis in which the computer handles the error estimates. For high assurance we use computer-verified proofs that the implementation is actually correct; see [21] for an overview. It has long been suggested that, by using constructive mathematics, exact analysis and provable correctness can be unified [7,8]. Constructive mathematics provides a high-level framework for specifying computations (Section 2.1). -
Edinburgh Research Explorer
Edinburgh Research Explorer Propositions as Types Citation for published version: Wadler, P 2015, 'Propositions as Types', Communications of the ACM, vol. 58, no. 12, pp. 75-84. https://doi.org/10.1145/2699407 Digital Object Identifier (DOI): 10.1145/2699407 Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: Communications of the ACM General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 28. Sep. 2021 Propositions as Types ∗ Philip Wadler University of Edinburgh [email protected] 1. Introduction cluding Agda, Automath, Coq, Epigram, F#,F?, Haskell, LF, ML, Powerful insights arise from linking two fields of study previously NuPRL, Scala, Singularity, and Trellys. thought separate. Examples include Descartes’s coordinates, which Propositions as Types is a notion with mystery. Why should it links geometry to algebra, Planck’s Quantum Theory, which links be the case that intuitionistic natural deduction, as developed by particles to waves, and Shannon’s Information Theory, which links Gentzen in the 1930s, and simply-typed lambda calculus, as devel- thermodynamics to communication. -
A Multi-Modal Analysis of Anaphora and Ellipsis
University of Pennsylvania Working Papers in Linguistics Volume 5 Issue 2 Current Work in Linguistics Article 2 1998 A Multi-Modal Analysis of Anaphora and Ellipsis Gerhard Jaeger Follow this and additional works at: https://repository.upenn.edu/pwpl Recommended Citation Jaeger, Gerhard (1998) "A Multi-Modal Analysis of Anaphora and Ellipsis," University of Pennsylvania Working Papers in Linguistics: Vol. 5 : Iss. 2 , Article 2. Available at: https://repository.upenn.edu/pwpl/vol5/iss2/2 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/pwpl/vol5/iss2/2 For more information, please contact [email protected]. A Multi-Modal Analysis of Anaphora and Ellipsis This working paper is available in University of Pennsylvania Working Papers in Linguistics: https://repository.upenn.edu/pwpl/vol5/iss2/2 A Multi-Modal Analysis of Anaphora and Ellipsis Gerhard J¨ager 1. Introduction The aim of the present paper is to outline a unified account of anaphora and ellipsis phenomena within the framework of Type Logical Categorial Gram- mar.1 There is at least one conceptual and one empirical reason to pursue such a goal. Firstly, both phenomena are characterized by the fact that they re-use semantic resources that are also used elsewhere. This issue is discussed in detail in section 2. Secondly, they show a striking similarity in displaying the characteristic ambiguity between strict and sloppy readings. This supports the assumption that in fact the same mechanisms are at work in both cases. (1) a. John washed his car, and Bill did, too. b. John washed his car, and Bill waxed it. -
Logical Vs. Natural Language Conjunctions in Czech: a Comparative Study
ITAT 2016 Proceedings, CEUR Workshop Proceedings Vol. 1649, pp. 68–73 http://ceur-ws.org/Vol-1649, Series ISSN 1613-0073, c 2016 K. Prikrylová,ˇ V. Kubon,ˇ K. Veselovská Logical vs. Natural Language Conjunctions in Czech: A Comparative Study Katrin Prikrylová,ˇ Vladislav Kubon,ˇ and Katerinaˇ Veselovská Charles University in Prague, Faculty of Mathematics and Physics Czech Republic {prikrylova,vk,veselovska}@ufal.mff.cuni.cz Abstract: This paper studies the relationship between con- ceptions and irregularities which do not abide the rules as junctions in a natural language (Czech) and their logical strictly as it is the case in logic. counterparts. It shows that the process of transformation Primarily due to this difference, the transformation of of a natural language expression into its logical representa- natural language sentences into their logical representation tion is not straightforward. The paper concentrates on the constitutes a complex issue. As we are going to show in most frequently used logical conjunctions, and , and it the subsequent sections, there are no simple rules which analyzes the natural language phenomena which∧ influence∨ would allow automation of the process – the majority of their transformation into logical conjunction and disjunc- problematic cases requires an individual approach. tion. The phenomena discussed in the paper are temporal In the following text we are going to restrict our ob- sequence, expressions describing mutual relationship and servations to the two most frequently used conjunctions, the consequences of using plural. namely a (and) and nebo (or). 1 Introduction and motivation 2 Sentences containing the conjunction a (and) The endeavor to express natural language sentences in the form of logical expressions is probably as old as logic it- The initial assumption about complex sentences contain- self. -
Introduction to Deductive Logic
Introduction to Deductive Logic Part I: Logic & Language There is no royal road to logic, and really valuable ideas can only be had at the price of close attention. Charles Saunders Peirce INTRODUCTION TO DEDUCTIVE LOGIC PART I: LOGIC AND LANGUAGE I. INTRODUCTION What is Logic? . 4 What is an argument? . 5 Three Criteria for Evaluating Arguments . 6 II. ARGUMENT ANALYSIS – PART 1 Preliminary Issues . 9 Statements and Truth Value Premises and Conclusions . 9 Statements vs Sentences 10 Simple vs Compound Statements 12 Recognizing premises and conclusions . 13 Indicator Language. 15 TWO TYPES OF ARGUMENTS Inductive Reasoning/Argument . 18 Deductive Reasoning/Argument . 19 III. ANALYSIS – PART II FORMALIZING ARGUMENT Preliminary Issue: Formal vs. Informal Argument . 24 Simple and Compound Statements . 25 Types of Compound Statements . 26 Formation Rules for Sentential . 26 DEFINING LOGICAL OPERATORS . 29 Negation . 29 Conjunction . 30 Disjunction . 31 Conditional/ Material Implication . 32 Bi-conditional . 33 APPLICATION: TRANSLATING USING LOGICAL OPERATORS . 35 TRUTH TABLE RULES/SUMMARY OF LOGICAL OPERATORS . 38 TRUTH TABLE APPLICATIONS . 39 Application I: Substitute and Calculate Method . 39 Application II: Calculating with Unknown Values . 41 Application III: Full Truth Table Method . 42 A Short Cut Technique for statements. 44 Truth Table Test for Logical Equivalence . 45 Application IV: Using Truth Tables to Facilitate Reasoning . 47 THE NEED FOR PREDICATE . 50 Using Predicate to Represent Categorical Statements . 52 Preliminary Issues– Grammar . 53 Formation Rules for Predicate . 55 Expressing Quantity . 56 APPLICATION: Symbolizing Categorical Statements using Predicate . 56 Negating Standard Form Categorical Statements. 57 Summary Chart for Standard Form Categorical Statements . 59 Extending Predicate . 61 A Universe of Discourse .