Introducing Logic E-Book

Bill Mayblin

• Herausgeber: Icon Books
• Kategorie: Geisteswissenschaft
• Serie: Graphic Guides
• Sprache: Englisch

Seitenzahl: 124

Veröffentlichungsjahr: 2014

Beschreibung

Logic is the backbone of Western civilization, holding together its systems of philosophy, science and law. Yet despite logic's widely acknowledged importance, it remains an unbroken seal for many, due to its heavy use of jargon and mathematical symbolism.This book follows the historical development of logic, explains the symbols and methods involved and explores the philosophical issues surrounding the topic in an easy-to-follow and friendly manner. It will take you through the influence of logic on scientific method and the various sciences from physics to psychology, and will show you why computers and digital technology are just another case of logic in action.

Leseprobe

Published by Icon Books Ltd, Omnibus Business Centre, 39-41 North Road, London N7 9DP email: [email protected]

ISBN: 978-184831-761-1

Text and illustrations copyright © 2013 Icon Books Ltd

The author and artist have asserted their moral rights.

Originating editor: Richard Appignanesi

No part of this book may be reproduced in any form, or by any means, without prior permission in writing from the publisher.

Contents

Cover

Title Page

Copyright

What is Logic?

Studying Sentences

The Square off Oppositions

The Syllogism

Connective Logic

Leibniz’s Law

The Reductio ad Absurdum

A “New Organon”

Frege’s Quantifiers

The Context Principle

Propositional Calculus

Cantor’s Set Theory

The Usefulness of Connectives

The Russell Paradox

The Fatal Flaw

The Problem of Surface Grammar

Russell’s System

Wittgenstein’s Logical Pictures

Carnap and the Vienna Circle

The Tolerance Principle

Hilbert’s Proof Theory

The Arrival of Gödel

Gödel’s Incompleteness Theorem

The Connections to Proof Theory

Wittgenstein’s Table of Logical Connectives

Wittgenstein’s Truth Tables

Discovering Tautologies

The Logic Gates of Digital Electronics

A Vending Machine

Turing and the “Enigma Code”

Euclid’s Axiomatic Method

Leibniz’s Proof Method

Abuse of Contradiction

Rules for Connectives

Sensitivity to Grammar

Predicate Calculus

Model Theoretic Semantics

Hilbert’s Recursion Model

Finite Rules for Infinite Production

Simple Instructions

Proof Theory and Formal Language

Tarski’s Truth Conditions

Formal Semantics in Practice

Constructing a Soap Opera

Prolog to an AI Soap Opera

Turing’s recipe for AI

The Problem of Paradoxes

Can Paradoxes be Avoided?

Theory of Types

Tarski’s Solution to the Liar

The Unexorcised Paradox

Gödel’s Incompleteness Theorem

The Consequences of Gödel’s Theorem

The “Halting Problem”

The Limit of Gödel’s Proof

Zeno’s Movement Paradox

An Infinite Sum

A Convergence on Limits

How Much is a “Heap”?

The Challenge to Sets

Undermining Logic

The Fiction of Vague Words

What Do Words “Mean”?

Fuzzy Logic

Fuzzy Heaps

Can Logic Escape Paradox?

Non-Classical Logics: Intuitionism

The Devil’s Argument

Intuitionistic Logic

Intuitionism versus the Reductio Method

The Intuitionistic Fad

Addressing Some Old Problems

The Value of Possible

Truth Values as Numbers

The Possible and Non-Contradiction

From Classical to Fuzzy Logic

Electronic “Possible” States

The Fuzzy Logic Search Engine

The Fuzzy Logical Machine

Logic in the Quantum World

The Distributive Law of Quantum Logic

Logic by Experiment

Logic and Science

The Copernican Revolution

Galileo’s Revolution

Methods of Deduction and Induction

Problems with Induction

Hume’s Fork

Nomological Deduction

Induction by Generalization

Laws or Empirical Predictions

The Raven Paradox

A Problem of Cause and Effect

Popper’s Answer to Hempel

Popper’s Disconfirmation Theory

The Probability of Viable Theory

Quine’s “Web of Belief”

Alterations to the “Web”

Insufficient Evidence

Quine’s Relativism

Feyerabend’s Denial of Scientific Method

Davidson’s Reply to Quine

The Presentation of Truth

Hard-edged Truth versus Relativism

Cognitive Science and Logic

Chomsky’s Universal Grammar

Noun and Verb Categories

Recursive Rules of Grammar

The X-bar Theory

A Logical Theory

Problems of Syntax and Semantics

Complex Grammatical Structures

Problems with “Universal” Grammar

The Symbolic Brain Model

Training a Neural Net

Pattern Recognition

The Rational Behaviour Model

Practical Reason

What is Consciousness?

The Place of Logic

Wittgenstein’s Change of View

Further Reading

Index

What is Logic?

Nothing is more natural to conversation than argument. We try to convince the person we are arguing with that we are right, that our conclusion follows from something that they will accept. It would be no good if we could not tell when one thing followed from another. What is often passed off in conversation as an argument does not fit the bill.

ALL PROPERTY IS THEFT.SO ALL THEFT IS PROPERTY.THEREFORE THESE PEARLS ARE MINE.

This is clearly rubbish because there is nothing to link the truth of the conclusion to the truth of the supporting claims. What we need is to ensure that the truth of the supporting claims is preserved by the argument. Logic is quite simply the study of truth-preserving arguments.

Studying Sentences

The Greek philosopher Aristotle (384–322 BC) first gave us the idea of a tool (organon) to argue convincingly. This study included grammar, rhetoric and a theory of interpretation, as well as logic. The first thing Aristotle does is discuss sentences.

SENTENCES COME IN THREE TYPES …IN EACH OF THESE TYPES WE ARE SAYING THAT SOMETHING OR SOME THINGS ARE OF A CERTAIN KIND.

1. Singular: Socrates is a man.2. Universal: Every man is mortal.3. Particular: Some men are mortal.

The objects we talk about (e.g., nouns like Socrates and tables; abstract nouns like walking; and pronouns like someone and everyone) Aristotle calls the subject of the sentence.

What we say about the subject of the sentence (e.g., verbs like is eating and has fallen; adjectives like is difficult; and nouns like man in things like “Socrates is a man”) Aristotle called the predicate.

The Square off Oppositions

Aristotle noticed that the truth of some subject-predicate sentences has an effect on the truth of other subject-predicate sentences.

THE FOLLOWING SENTENCES STAND IN DEFINITE RELATIONS TO EACH OTHER. I CALL THIS MY SQUARE OF OPPOSITIONS.

Sentences 1 and 2 cannot both be true.

The diagonal statements 1 and 4 are known contradictories. As long as there are men, one of them has to be true but never both – the truth of one guarantees that the other is false.

The same is true for the diagonal statements 2 and 3.

Sentences 1 and 3 can both be true. If 1 is true then 3 must be true, but 3 being true does not mean that 1 must be true.

Similarly with 2 and 4. The same relation holds between “All men are mortal” and “Socrates is mortal”.

The Syllogism

Using the square of oppositions, Aristotle noticed a mysterious fact. Take a sentence like “Socrates is a man”. If an argument of three statements is built where the subject of the first statement is the predicate of the second (call these the premises) and the third statement is composed of the remaining terms (call this the conclusion), then the truth of the conclusion is guaranteed by the truth of the premises.

THIS SCHEMA I CALL A SYLLOGISM. WE CAN USE IT TO SEE WHY ONE ARGUMENT IS TRUE AND ANOTHER FALSE.

Aristotle forgot conditional statements that have more than one predicate, e.g.,

“If Socrates is a man, then Socrates is mortal”.

We now have two reasons why the argument “Arsenal are in London, therefore Arsenal will win the cup” is false. The first comes from what is actually said. There is no way that the facts that I support Arsenal and that Arsenal are in London are enough to guarantee that Arsenal will win the cup. But there is also the formal reason that the predicate of the first premise is not the subject of the second.

YES, BUT THIS IS VALID … 1. If I support Arsenal, then they will win the cup. 2. I do support Arsenal, so ... 3. Arsenal will win the cup.IT IS STILL FALSE BECAUSE VALIDITY ONLY GRARANTEES THE TRUTH OF THE CONCLUSION IF THE PREMISES ARE TRUE. IN YOUR EXAMPLE THE PREMISES ARE FALSE SO THE CONCLUSION REMAINS FALSE.SO WHAT GOOD DOES THIS FORMALIZATION DO US?YOU’LL SEE.

Connective Logic

About a hundred years later, Chrysippus of Soli (c.280–c.206 BC) changed the focus of logic from single subject-predicate statements to complex statements such as: “Socrates is a man and Zeno is a man.” This was a major achievement. It was said, “If the gods used logic, it would be the logic of Chrysippus”. As we shall see, the same is true of us humans, but it took us a couple of millennia to catch on.

WITH WORDS LIKE “AND”, “OR” AND “IF…THEN…”, DIFFERENT STATEMENTS CAN BE JOINED TOGETHER AND THE TRUTH OF THE WHOLE WILL DEPEND EXCLUSIVELY ON THE TRUTH OF THE PARTS.

Each of these connectives has a unique way of combining the truth of the parts into the truth of the whole.

For example the “or” connective and only the “or” connective can be used in the following way.

Either Muhammad will go to the mountain or the mountain will go to Muhammad. Muhammad did not go to the mountain, therefore the mountain went to Muhammad.USING MY DEFINITIONS FOR CONNECTIVES, I CAN SHOW HOW TO DERIVE VARIOUS STATEMENTS WHOSE TRUTH IS ALWAYS GUARANTEED BY THE TRUTH OF THE INITIAL STATEMENT. Chrysippus had no real impact on the history of logic for at least the next 1,500 years, not least because his writings were lost and his ideas known only by second-hand reports, but also because Aristotle became the darling of the Catholic Church.

Leibniz’s Law

For the next 2,000 years, logicians came up with an ever increasing number of syllogisms, some including more than two premises. The logician was a kind of alchemist playing around with concepts to get valid arguments. Eventually, a method in this madness was provided by Gottfried Leibniz (1646–1716).

Leibniz came up with the idea of treating statements like equations in algebra. Equations use the equality sign “=” to say that two sides must have the same numerical value,

Leibniz introduced the equality sign in logic to show that “a” is identical with “b”.

TWO THINGS ARE IDENTICAL IF EVERYTHING THAT CAN BE SAID OF THE ONE CAN BE SAID OF THE OTHER.IF YOU CAN SAY EXACTLY THE SAME OF TWO THINGS, THEN THEY ARE IDENTICAL.

This has been known ever since as Leibniz’s Law. He analysed it into two inseparable claims, “a is b” and “b is a”, which he took to mean that “all a’s are b’s” and “all b’s are a’s”,

e.g., “All bachelors are unmarried men and all unmarried men are bachelors.”

Clearly if a is identical with b, then we can replace the symbol “a” in any statement with the symbol “b”, whilst preserving the truth value of the statement. For example: “Socrates is an unmarried man, an unmarried man is the same as a bachelor, so Socrates is a bachelor.”

This is important because it allows us to assess the truth value of a potentially infinite number of sentences using a manageable number of steps. Leibniz had four.

2. If “a is b” and “b is c” then “a is c”

e.g., “All men are mortal, Socrates is a man, therefore Socrates is mortal.”

Saying “a is b” is the same as saying that “all a’s are b”.

SO THIS HAS EXACTLY THE SAME FORM AS MY FIRST SYLLOGISM!AH, BUT THERE ARE STEPS 3 AND 4 ...

From these simple laws, Leibniz could prove every possible syllogism. Instead of Aristotle’s square of oppositions, Leibniz came up with the first real truth theory – deriving conclusions from pre-established laws by substituting identical symbols (synonyms) with each other.

The Reductio ad Absurdum

Leibniz’s preferred method of proof is an immensely important tool much loved by logicians and philosophers ever since. He called it reductio ad absurdum.

The “reductio” is a very simple yet fantastically powerful tool. It has been used extensively since Leibniz invented it. It is well illustrated by example.

SO, WHERE WERE YOU ON THE NIGHT OF THE 25TH?I DIDN’T DO THE ROBBERY. I WAS PLAYING BASKETBALL.OK, MAYBE THAT’S TRUE. LET’S SEE IF IT COULD BE TRUE.HEY, DOC. COULD THE SUSPECT HAVE BEEN PLAYING BASKETBALL ON THE 25TH?NO WAY. HE’D BROKEN HIS ARM THE DAY BEFORE.IT APPEARS YOU HAD A BROKEN ARM THAT NIGHT.SO WHAT?SO, YOU COULDN’T HAVE BEEN PLAYING BASKETBALL WITH A BROKEN ARM! YOUR STORY LEADS TO A CONTRADICTION, BUDDY. YOU’RE LYING.

In the reductio method we assume a statement to be true and see what conclusions we can draw from it. If, when drawing these conclusions, we get a contradiction, we know that the initial statement is false, because contradictions are always false.

SOME PEOPLE DON’T LIKE MY WONDERFUL NEW METHOD BECAUSE IT ASSUMES THAT EVERY SENTENCE IS EITHER TRUE OR FALSE, AND OFFERS NO SUPPORT FOR THIS ASSUMPTION.

The great advantage of the reductio method is that it allows us to tell if a statement is true, even if we do not know how to construct a proof for it. We can tell a statement is true by showing that its negation leads to a contradiction.

A “New Organon”

“For my invention used reason in its entirety and is, in addition, a judge of controversy, an interpreter of notions, a balance of probabilities, a compass which will guide us over the ocean of experiences, an inventory of things, a table of thoughts, a microscope for scrutinising things, a telescope for predicting distant things, a general calculus, an innocent magic, a non-chimerical cabal, a script which all will read in their own language and which will lead the way for the true religion everywhere it goes.”

Letter from Leibniz to the Duke of Hanover, 1679

THIS IS A REVOLUTION. ARISTOTLE’S OLD ORGANON IS DEAD, INSTEAD I GIVE YOU A “NEW ORGANON”. IT IS A NEW WAY OF THINKING ABOUT THE WORLD AND ABOUT LOGIC.LOGIC IS NO LONGER A TOOL FOR CONVINCING ARGUMENTS BUT RATHER A SYSTEM OF RULES OF THOUGHT, SO THAT EVEN GOD’S THOUGHT IS NECESSARILY LOGICAL. EVEN HE COULD NOT CREATE A WORLD WHERE A CONTRADICTION IS TRUE.

Perhaps unsurprisingly, the Church dubbed him a heretic. But the idea of necessary rules of thought proved to have a lasting influence on Western philosophers like Kant, Hegel, Marx and Russell.

WE ALL TRIED TO GIVE AN ACCOUNT OF WHAT THIS ESSENTIAL LOGIC OF THOUGHT MIGHT BE.IT SHOULD BE NOTED, HOWEVER, THAT LEIBNIZ’S SYSTEM IS NOT AN ORGANON (TOOL) AT ALL. IT IS A CANON OR CODE OF LAWS THAT ORIGINATES IN THINKING BUT THAT NECESSARILY APPLIES TO THE WORLD.

Frege’s Quantifiers

The Encyclopaedia of Philosophy says that modern logic began in 1879 with the publication of Gottlob Frege’s Begriffsschrift. It introduces a propositional calculus which combines Leibniz’s proof theory with an account of logical connectives. So we did finally get to Chrysippus.

But the most significant of Frege’s new inventions was the quantifier. Quantifiers are words like: “all”, “some”, “many” and “most”. They allow us to say things about groups of objects, e.g., “Some men are bald.