Working with Logic
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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. -
Section 2.1: Proof Techniques
Section 2.1: Proof Techniques January 25, 2021 Abstract Sometimes we see patterns in nature and wonder if they hold in general: in such situations we are demonstrating the appli- cation of inductive reasoning to propose a conjecture, which may become a theorem which we attempt to prove via deduc- tive reasoning. From our work in Chapter 1, we conceive of a theorem as an argument of the form P → Q, whose validity we seek to demonstrate. Example: A student was doing a proof and suddenly specu- lated “Couldn’t we just say (A → (B → C)) ∧ B → (A → C)?” Can she? It’s a theorem – either we prove it, or we provide a counterexample. This section outlines a variety of proof techniques, including direct proofs, proofs by contraposition, proofs by contradiction, proofs by exhaustion, and proofs by dumb luck or genius! You have already seen each of these in Chapter 1 (with the exception of “dumb luck or genius”, perhaps). 1 Theorems and Informal Proofs The theorem-forming process is one in which we • make observations about nature, about a system under study, etc.; • discover patterns which appear to hold in general; • state the rule; and then • attempt to prove it (or disprove it). This process is formalized in the following definitions: • inductive reasoning - drawing a conclusion based on experi- ence, which one might state as a conjecture or theorem; but al- mostalwaysas If(hypotheses)then(conclusion). • deductive reasoning - application of a logic system to investi- gate a proposed conclusion based on hypotheses (hence proving, disproving, or, failing either, holding in limbo the conclusion). -
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. -
'The Denial of Bivalence Is Absurd'1
On ‘The Denial of Bivalence is Absurd’1 Francis Jeffry Pelletier Robert J. Stainton University of Alberta Carleton University Edmonton, Alberta, Canada Ottawa, Ontario, Canada [email protected] [email protected] Abstract: Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting. I. Introduction Let us being with a word about what our topic is not. There is a familiar kind of argument for an epistemic view of vagueness in which one claims that denying bivalence introduces logical puzzles and complications that are not easily overcome. One then points out that, by ‘going epistemic’, one can preserve bivalence – and thus evade the complications. James Cargile presented an early version of this kind of argument [Cargile 1969], and Tim Williamson seemingly makes a similar point in his paper ‘Vagueness and Ignorance’ [Williamson 1992] when he says that ‘classical logic and semantics are vastly superior to…alternatives in simplicity, power, past success, and integration with theories in other domains’, and contends that this provides some grounds for not treating vagueness in this way.2 Obviously an argument of this kind invites a rejoinder about the puzzles and complications that the epistemic view introduces. Here are two quick examples. First, postulating, as the epistemicist does, linguistic facts no speaker of the language could possibly know, and which have no causal link to actual or possible speech behavior, is accompanied by a litany of disadvantages – as the reader can imagine. -
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. -
Three Ways of Being Non-Material
Three Ways of Being Non-Material Vincenzo Crupi, Andrea Iacona May 2019 This paper presents a novel unified account of three distinct non-material inter- pretations of `if then': the suppositional interpretation, the evidential interpre- tation, and the strict interpretation. We will spell out and compare these three interpretations within a single formal framework which rests on fairly uncontro- versial assumptions, in that it requires nothing but propositional logic and the probability calculus. As we will show, each of the three intrerpretations exhibits specific logical features that deserve separate consideration. In particular, the evidential interpretation as we understand it | a precise and well defined ver- sion of it which has never been explored before | significantly differs both from the suppositional interpretation and from the strict interpretation. 1 Preliminaries Although it is widely taken for granted that indicative conditionals as they are used in ordinary language do not behave as material conditionals, there is little agreement on the nature and the extent of such deviation. Different theories tend to privilege different intuitions about conditionals, and there is no obvious answer to the question of which of them is the correct theory. In this paper, we will compare three interpretations of `if then': the suppositional interpretation, the evidential interpretation, and the strict interpretation. These interpretations may be regarded either as three distinct meanings that ordinary speakers attach to `if then', or as three ways of explicating a single indeterminate meaning by replacing it with a precise and well defined counterpart. Here is a rough and informal characterization of the three interpretations. According to the suppositional interpretation, a conditional is acceptable when its consequent is credible enough given its antecedent. -
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. -
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. -
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. -
Conundrums of Conditionals in Contraposition
Dale Jacquette CONUNDRUMS OF CONDITIONALS IN CONTRAPOSITION A previously unnoticed metalogical paradox about contrapo- sition is formulated in the informal metalanguage of propositional logic, where it exploits a reflexive self-non-application of the truth table definition of the material conditional to achieve semantic di- agonalization. Three versions of the paradox are considered. The main modal formulation takes as its assumption a conditional that articulates the truth table conditions of conditional propo- sitions in stating that if the antecedent of a true conditional is false, then it is possible for its consequent to be true. When this true conditional is contraposed in the conclusion of the inference, it produces the false conditional conclusion that if it is not the case that the consequent of a true conditional can be true, then it is not the case that the antecedent of the conditional is false. 1. The Logic of Conditionals A conditional sentence is the literal contrapositive of another con- ditional if and only if the antecedent of one is the negation of the consequent of the other. The sentence q p is thus ordinarily un- derstood as the literal contrapositive of: p⊃ :q. But the requirement presupposes that the unnegated antecedents⊃ of the conditionals are identical in meaning to the unnegated consequents of their contrapos- itives. The univocity of ‘p’ and ‘q’ in p q and q p can usually be taken for granted within a single context⊃ of: application⊃ : in sym- bolic logic, but in ordinary language the situation is more complicated. To appreciate the difficulties, consider the use of potentially equivo- cal terms in the following conditionals whose reference is specified in particular speech act contexts: (1.1) If the money is in the bank, then the money is safe. -
Immediate Inference
Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P.