demonstrate logical problem solving and error detection techniques

Demonstrate Logical Problem Solving and Error Detection Techniques

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• Gibbardian Collapse and Trivalent Conditionals Gibbardian Collapse and Trivalent Conditionals Paul Égré* Lorenzo Rossi† Jan Sprenger‡ Abstract This paper discusses the scope and significance of the so-called triviality result stated by Allan Gibbard for indicative conditionals, showing that if a conditional operator satisfies the Law of Import-Export, is supraclassical, and is stronger than the material conditional, then it must collapse to the material conditional. Gib- bard’s result is taken to pose a dilemma for a truth-functional account of indicative conditionals: give up Import-Export, or embrace the two-valued analysis. We show that this dilemma can be averted in trivalent logics of the conditional based on Reichenbach and de Finetti’s idea that a conditional with a false antecedent is undefined. Import-Export and truth-functionality hold without triviality in such logics. We unravel some implicit assumptions in Gibbard’s proof, and discuss a recent generalization of Gibbard’s result due to Branden Fitelson. Keywords: indicative conditional; material conditional; logics of conditionals; triva- lent logic; Gibbardian collapse; Import-Export 1 Introduction The Law of Import-Export denotes the principle that a right-nested conditional of the form A → (B → C) is logically equivalent to the simple conditional (A ∧ B) → C where both antecedentsare united by conjunction. The Law holds in classical logic for material implication, and if there is a logic for the indicative conditional of ordinary language, it appears Import-Export ought to be a part of it. For instance, to use an example from (Cooper 1968, 300), the sentences “If Smith attends and Jones attends then a quorum *Institut Jean-Nicod (CNRS/ENS/EHESS), Département de philosophie & Département d’études cog- arXiv:2006.08746v1 [math.LO] 15 Jun 2020 nitives, Ecole normale supérieure, PSL University, 29 rue d’Ulm, 75005, Paris, France. [Show full text]
• Glossary for Logic: the Language of Truth Glossary for Logic: The Language of Truth This glossary contains explanations of key terms used in the course. (These terms appear in bold in the main text at the point at which they are first used.) To make this glossary more easily searchable, the entry headings has ‘::’ (two colons) before it. So, for example, if you want to find the entry for ‘truth-value’ you should search for ‘:: truth-value’. :: Ambiguous, Ambiguity : An expression or sentence is ambiguous if and only if it can express two or more different meanings. In logic, we are interested in ambiguity relating to truth-conditions. Some sentences in natural languages express more than one claim. Read one way, they express a claim which has one set of truth-conditions. Read another way, they express a different claim with different truth-conditions. :: Antecedent : The first clause in a conditional is its antecedent. In ‘(P ➝ Q)’, ‘P’ is the antecedent. In ‘If it is raining, then we’ll get wet’, ‘It is raining’ is the antecedent. (See ‘Conditional’ and ‘Consequent’.) :: Argument : An argument is a set of claims (equivalently, statements or propositions) made up from premises and conclusion. An argument can have any number of premises (from 0 to indefinitely many) but has only one conclusion. (Note: This is a somewhat artificially restrictive definition of ‘argument’, but it will help to keep our discussions sharp and clear.) We can consider any set of claims (with one claim picked out as conclusion) as an argument: arguments will include sets of claims that no-one has actually advanced or put forward. [Show full text]
• 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. [Show full text]
• 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. [Show full text]
• 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. [Show full text]
• 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. [Show full text]
• False Dilemma Wikipedia Contents False dilemma Wikipedia Contents 1 False dilemma 1 1.1 Examples ............................................... 1 1.1.1 Morton's fork ......................................... 1 1.1.2 False choice .......................................... 2 1.1.3 Black-and-white thinking ................................... 2 1.2 See also ................................................ 2 1.3 References ............................................... 3 1.4 External links ............................................. 3 2 Affirmative action 4 2.1 Origins ................................................. 4 2.2 Women ................................................ 4 2.3 Quotas ................................................. 5 2.4 National approaches .......................................... 5 2.4.1 Africa ............................................ 5 2.4.2 Asia .............................................. 7 2.4.3 Europe ............................................ 8 2.4.4 North America ........................................ 10 2.4.5 Oceania ............................................ 11 2.4.6 South America ........................................ 11 2.5 International organizations ...................................... 11 2.5.1 United Nations ........................................ 12 2.6 Support ................................................ 12 2.6.1 Polls .............................................. 12 2.7 Criticism ............................................... 12 2.7.1 Mismatching ......................................... 13 2.8 See also [Show full text]
• 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. [Show full text]
• 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 . [Show full text]
• Hardware Abstract the Logic Gates References Results Transistors Through the Years Acknowledgements The Practical Applications of Logic Gates in Computer Science Courses Presenters: Arash Mahmoudian, Ashley Moser Sponsored by Prof. Heda Samimi ABSTRACT THE LOGIC GATES Logic gates are binary operators used to simulate electronic gates for design of circuits virtually before building them with-real components. These gates are used as an instrumental foundation for digital computers; They help the user control a computer or similar device by controlling the decision making for the hardware. A gate takes in OR GATE AND GATE NOT GATE an input, then it produces an algorithm as to how The OR gate is a logic gate with at least two An AND gate is a consists of at least two A NOT gate, also known as an inverter, has to handle the output. This process prevents the inputs and only one output that performs what inputs and one output that performs what is just a single input with rather simple behavior. user from having to include a microprocessor for is known as logical disjunction, meaning that known as logical conjunction, meaning that A NOT gate performs what is known as logical negation, which means that if its input is true, decision this making. Six of the logic gates used the output of this gate is true when any of its the output of this gate is false if one or more of inputs are true. If all the inputs are false, the an AND gate's inputs are false. Otherwise, if then the output will be false. Likewise, are: the OR gate, AND gate, NOT gate, XOR gate, output of the gate will also be false. [Show full text]
• Set Notation and Concepts Appendix Set Notation and Concepts “In mathematics you don’t understand things. You just get used to them.” John von Neumann (1903–1957) This appendix is primarily a brief run-through of basic concepts from set theory, but it also in Sect. A.4 mentions set equations, which are not always covered when introducing set theory. A.1 Basic Concepts and Notation A set is a collection of items. You can write a set by listing its elements (the items it contains) inside curly braces. For example, the set that contains the numbers 1, 2 and 3 can be written as {1, 2, 3}. The order of elements do not matter in a set, so the same set can be written as {2, 1, 3}, {2, 3, 1} or using any permutation of the elements. The number of occurrences also does not matter, so we could also write the set as {2, 1, 2, 3, 1, 1} or in an infinity of other ways. All of these describe the same set. We will normally write sets without repetition, but the fact that repetitions do not matter is important to understand the operations on sets. We will typically use uppercase letters to denote sets and lowercase letters to denote elements in a set, so we could write M ={2, 1, 3} and x = 2 as an element of M. The empty set can be written either as an empty list of elements ({})orusing the special symbol ∅. The latter is more common in mathematical texts. A.1.1 Operations and Predicates We will often need to check if an element belongs to a set or select an element from a set. [Show full text]
• An Introduction to Formal Methods for Philosophy Students An Introduction to Formal Methods for Philosophy Students Thomas Forster February 20, 2021 2 Contents 1 Introduction 13 1.1 What is Logic? . 13 1.1.1 Exercises for the first week: “Sheet 0” . 13 2 Introduction to Logic 17 2.1 Statements, Commands, Questions, Performatives . 18 2.1.1 Truth-functional connectives . 20 2.1.2 Truth Tables . 21 2.2 The Language of Propositional Logic . 23 2.2.1 Truth-tables for compound expressions . 24 2.2.2 Logical equivalence . 26 2.2.3 Non truth functional connectives . 27 2.3 Intension and Extension . 28 2.3.1 If–then . 31 2.3.2 Logical Form and Valid Argument . 33 2.3.3 The Type-Token Distinction . 33 2.3.4 Copies . 35 2.4 Tautology and Validity . 36 2.4.1 Valid Argument . 36 2.4.2 V and W versus ^ and _ .................... 40 2.4.3 Conjunctive and Disjunctive Normal Form . 41 2.5 Further Useful Logical Gadgetry . 46 2.5.1 The Analytic-Synthetic Distinction . 46 2.5.2 Necessary and Sufficient Conditions . 47 2.5.3 The Use-Mention Distinction . 48 2.5.4 Language-metalanguage distinction . 51 2.5.5 Semantic Optimisation and the Principle of Charity . 52 2.5.6 Inferring A-or-B from A . 54 2.5.7 Fault-tolerant pattern-matching . 54 2.5.8 Overinterpretation . 54 2.5.9 Affirming the consequent . 55 3 4 CONTENTS 3 Proof Systems for Propositional Logic 57 3.1 Arguments by LEGO . 57 3.2 The Rules of Natural Deduction . 57 3.2.1 Worries about reductio and hypothetical reasoning . [Show full text]

NQF 5 SD 115367:  Demonstrate logical problem solving and error detection techniques

U NIT STANDARD NUMBER    :           115367

L EVEL ON THE NQF               :           5

CREDITS                                 :           6

FIELD                                      :           Physical, Mathematical, Computer and Life Sciences

SUB FIELD                              :           Information Technology and Computer Sciences

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• TAP> Logical Problem Solving and Error Detection

TAP> Logical Problem Solving and Error Detection

Course Purpose People credited with this unit standard can describe different approaches to problem-solving. What will you learn People credited with this unit standard can describe different approaches to problem-solving. Use logical operations in descriptions of rules and relationships in …

Course Purpose

People credited with this unit standard can describe different approaches to problem-solving.

What will you learn

People credited with this unit standard can describe different approaches to problem-solving. Use logical operations in descriptions of rules and relationships in problem situations. Simplify Boolean expressions with Boolean Algebra and Karnaugh maps. Describe the basic concepts of error detection.

Duration: 1 Day

Featured course, issues affecting the management of a local area computer network.

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• NQF Level 4

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• Duration 3 days
• Skill level All levels
• Language English
• Students 1256
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Week 9 Methods | Demonstrate logical problem solving and error detection techniques | Day 45

Welcome to Week 9!

Your content is divided into two segments:

• C# Programming and,
• MICT SETA unit standard outcomes and assessments

In your C# programming component, you will focus on Methods.

Methods are ways that allow you as a programmer to give your code the capability to perform an action. For example, thin of the human body, it is made up of various parts i.e., hands, legs, arms etc. Hands allow you to wave. The concept of waving can be referred to as a method in programming.

A method (waving) is made possible by a property (hand). Without the hand you cannot wave. In C# programming, we make use of the various properties to make our code functional. You will learn this in detail in the next sessions.

In your MICT SETA unit standards you will focus on computer software development. You must be aware now that computer software products e.g., applications such as WhatsApp and Facebook, are created to meet a specific business need. In the case of our examples, this need is to enable communication between “friends” across the world at minimal cost. This is the business value.

You will not simply sit in front of a computer and start coding an application. You need to plan first. You need to have a business specification that you are addressing. All this is covered in detail in the following units.

Course Features

• Duration 7 hours
• Skill level All levels
• Language English
• Students 48
• Certificate No
• Assessments Yes

Emmanuel holds a Masters degree in Information Technology and has more than 10 years of experience in both academics and the software industry; with extensive experience in software development. Prior to joining redAcademy, Emmanuel held the position of Software Development Team Lead and Lecturer, working closely with students on Software Development projects.

During his career, Emmanuel has worked with students from different backgrounds, in both private and public institutions. This broad experience has further strengthened Emmanuel’s ability to engage a diverse group of students through his adaptive lecturing style. Emmanuel has a strong background in implementing and improving the full systems development cycle (SDLC), with the use of a wide range of technologies and frameworks.

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• redAcademy LMS, Windows PC / Apple PC / Linux PC, Microsoft Visual Studio, Microsoft Office Suite, Text Editor, Projector, White Board, Markers

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