Propositional Logic – An Introduction

127 Pages·2009·1.18 MB·English

by  Guy Davies

PROPOSITIONAL LOGIC
– An Introduction
GUY DAVIES
LOVE EKENBERG
¨
JOHAN THORBIORNSON
The authors, in brief
Guy DaviesholdsaPh.D.incomputersciencefromtheKTHRoyalInstitute
of Technology, Stockholm, and is also affiliated with Stockholm University.
Love EkenbergholdsaPh.D.incomputerscienceandaPh.D.inmathemat-
ics from Stockholm Universtity and is full Professor at Stockholm University,
KTH Royal Institute of Technology as well as Mid-Sweden University.
Johan Thorbi¨ornsonholdsaPh.D. inmathematicsandis Associate Profes-
sor in Mathematics at the Royal Institute of Technology, Stockholm, as well
as Director of Resource Centre for Net-Based Education at the KTH Royal
Institute of Technology.
Supplementary materials are available at:
http://sites.google.com/site/logicbasicsbeyond/
Other works by the same authors:
Davies-Ekenberg-Thorbi¨ornson: Logic - Basics and Beyond, ISBN 978–91–
978450–1–4
Edition 1, first print run, 2009
(cid:176)c 2009 Guy Davies, Love Ekenberg, Johan Thorbi¨ornson
SINE METU, Valhallav¨agen 82, Stockholm, Sweden, www.sinemetu.se
Order and information: www.sinemetu.se
ISBN 978–91–978450–2–1
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Preface
Purpose
This small book in propositional logic is written for everybody who would like
to get introduced in that realm of human cognition that most differentiates
us from all other creatures we know of in the universe. It is written to touch
upon this faculty as the ultimate embodiment of thought in an age when the
superficialandirrationalisnotonlyrife,buthasbecomeadominatingcultural
expression of intellectual laziness. The danger of this culture forwarns the de-
generationofcivilisationintoasocietywhereunschooledmindsappreciatethe
opinion of celebrity ignorants, military bullies, and the values of pop-up reli-
gions more than academia’s systematically scrutinised wisdom. Such a culture
stimulates the emergence of narrow mindedness, prejudice, and foolishness,
feeding totalitarianism and oppression.
We hereby join the battle that may ensure a tolerable existence for future
generations,byexpoundingthevirtuesofcriticalthinkinginitspurestembod-
iment. We hope that this book will show how deeper meaning lurks beneath
the skin of every man and woman. Anyone can harness the tools that can
empower us to escape the relentless sea of puerile mass-culture. Allow us to
persuade you with the allure of truth and the means to discover it.
Audience
The book is for anybody who would wish to strive to learn more of our innate
faculty of reason. In practice the book can be used for basic undergraduate
studies in logic, while still observing both a formal as well as a philosophical
perspective. For more advanced studies, with a special focus on applications
in systems science, we strongly recommend Logic – Basics and Beyond, by
the same authors. Read this as intellectual ‘entertainment’ with a view to
glimpsingthepowerandbeautyofthoughtorsimplytounderstandtheformal
culmination of centuries of cultural history.
Free use and printing
This book is free to copy and distribute for noncommercial use. Details are
given with reference to Creative Commons Licence in the cover page of the
book.
For those who want to read the material in printed version, a significantly
extendedbookbythesameauthors,Logic – Basics and Beyond, ISBN 978-91-
978450-1-4,www.sinemetu.se,canbeboughtinprintedversion.Thisbooknot
only includes the chapters from Propositional Logic – an introduction but also
covers propositional and predicate logic, set theory and functions, complexity
theory and algorithm analysis as well as modal logic and model theory. It can
6
be used for basic and also for more advanced studies in logic with a special
focus on applications in systems science. The material there is sufficient for
2–3coursesinlogicandcanbedividedintwobasicpartsandamoreadvanced
part. In this book, you will also find a large number of exercises together with
solutions for a large number of selected problems.
Courses
The material is sufficient for a very basic course in elementary logic. The book
Logic – Basics and Beyond by the same authors provides considerably more
material for further studies in logic.
Reading
Reading order is fairly straightforward. It is basically a brief overview of basic
conceptsandmethodsofclassicalpropositionallogic.Textsinlogicsometimes
become very technical and the natural beauty and usability of it is then lost.
We have tried to avoid that by providing a substantial intuition for the issues
involved. We also include some important meta-logical perspectives.
Maths
Some of the sections in the book assume some acquaintanceship with pre-
university mathematics, but these are mostly illustrations rather than an in-
tegral part of the logic, and are not necessary in order to assimilate the main
ideas of the book. Mathematical sections especially those marked with an
asterisk can be passed over without loss of understanding the material as a
whole.
Reading strategies
The book contains a large number of exercises and answers. Solutions as well
as additional material are to be found on the web page
http://sites.google.com/site/logicbasicsbeyond/
Werecommendthereaderfirsttotrytofindthesolutionindependentlybefore
seeking a solution. Effort, even when unsuccessful, focuses attention on the
key difficulties. Do not look at the solutions too early to encourage your own
thinking, rather than ... that’s right ... instant gratification.
Acknowledgments
The authors are insignificant amoebas living in the context of intellectual
giants.Wewouldespeciallyliketoextendourgratitudetothefollowingpeople
who have been of great significance in the creation of this book: professors,
0.0. PREFACE 7
doctors, inspirers, and friends ... our families, Veselka Boeva, Torkel Franz´en,
Paul Johannesson, Per-Erik Malmn¨as, Thomas Oakland, Elisabeth Ohlson
Wallin, Petra O¨stergren, Vide Jansson, Lars Asker, Karl Karlander, Fidel,
Ernesto, Selima, Google, Kazuo Koike, and the Cheshire Cat.
Go forth
Enjoy your adventure into logic, and all the activities that will distract you
along the way.
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Contents
Preface 5
Chapter 1. Introduction 10
1. The History of Logic 10
2. So What is Logic? 12
3. Content Disposition 16
Chapter 2. Introduction to Sentence Logic 18
1. Negation, implication and equivalence 20
2. The Connectives and, or 27
3. False Hypotheses and Bogus Solutions* 32
Chapter 3. The Language SL 40
1. Alphabet 41
2. The Syntax for Sentences in SL 41
3. The Meaning of a Sentence 43
4. The Expressive Power of Connectives 50
5. The Semantics of SL 52
6. Information Content of a Sentence 55
Chapter 4. Deductions and Arguments 58
1. Logical Consequence 59
2. Incomplete Arguments 66
3. Some Important Logical Relationships 71
Chapter 5. Rule Systems 75
1. Axiomatic Systems 76
2. Semantic Tableaux 80
3. The Resolution Method 89
4. Conjunctive Normal Form 89
5. Deductions with the Resolution Method 93
6. Natural Deduction 98
7. A note on sequent calculus * 107
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Chapter 6. Soundness and Completeness 111
1. Soundness and Completeness for sentence logic 112
Solutions to Exercises 121
Index 122
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CHAPTER 1
Introduction
Most rational lines of thought and calculations build on certain more or less
clearly expressed assumptions and conditions. When reasoning or arguing in
a way that can be called rational with the purpose of persuading others about
the plausibility of an argument, it is important to be able to see the intercon-
nectionbetweenassumptionsandconditionsinordertobeabletoassesswhat
conclusions these can lead to. It is also important to be able to recognise an
incorrect argument, and to understand what rational reasoning actually can
tell us.
In this book we will be analysing some of the fundamental types of state-
ments and rules that are used in rational argument. We will also introduce
certain symbols that are commonly used when studying statements and con-
texts that arguments can contain. Concepts are generally introduced in an
informal way first, offering the gist or intuition the ideas, before these are
formally presented.
1. The History of Logic
Inquiry into methods of formal deduction is called logic. Logic has been
studied since antiquity (by Aristotle and others), but has really only been
intensively developed during the 20th century. Logic gained a new role after
interest arose in constructing a formal system that could support attempts to
prove mathematical theorems in a way that would be absolutely irrefutable.
Mathematical proofs are often taken for granted, but what actually charac-
terisesthem?G.H.Hardywrotethat:“Amathematicalproofshouldresemble
asimpleandclear-cutconstellation,notascatteredclusterintheMilkyWay”1
Irrespective of whether or not this is helpful, it is in no way obvious what a
mathematicalargumentshouldlooklikeinordertoserveasaproof.Duringthe
1G.H. Hardy, A Mathematician’s Apology, Cambridge University Press, 1992.