Sense and Syllogism: Logic in Life

By Aparna Tulpule

'Sense and Syllogism: Logic in Life' introduces logic to the general reader. This book is for you if you have never studied logic, if you believe you have no use for logic, if you have no idea what logic is. Just a few essential ideas in logic are presented, and those are made simple and interesting through examples from life and literature. If all goes well, you should finish the book a little sharper and more alert than before.

• Language: English
• Publisher: Aparna Tulpule
• Release date: Jul 16, 2018
• ISBN: 9781386993230

Book preview

Sense and Syllogism

Logic in Life

Aparna Tulpule

Copyright © 2018 Aparna Tulpule

All rights reserved.

I believe that intellectual property rights are just and reasonable as are other property rights, specifically, rights to one’s work. If you do not believe in copyright and do not wish to pay for this book, please do not read the book at all, but do not buy a pirated copy. If you buy a pirated copy, you support criminal activity, possibly organised crime.

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Price: ₹ 100 (ebook)

Table of Contents

Introduction

What Is Logic?

About the Book

Guess Who?

Part I Basic Concepts

Ch 1 Truth and Truth Relation

Propositions

Parts of a Proposition

Negation

Negative Thinking

Visual Assertions

Propositional Functions

Truth Relation

Logic and Rationality

Ch 2 Classes

Classification and Definition

Venn Diagrams

Narrow and Broad Definitions

Negative

Laws of Logic

Clear or Fuzzy

Categories of Thought

Fallacies

Ch 3 Hypothetical Relationship

Hypothetical Proposition

Symbolic Expression

Other Forms

No Reverse

Negation

Material Equivalence

Hypotheticals in Everyday Use

Understanding Hypotheticals

Hypothetical Nature of Inference

Part II Drawing Conclusions

Ch 4 Categorical Syllogism: One Class

Symbolic Expressions

Inferences

Negative and Complement

Ch 5 Categorical Syllogism: Two Classes

Types of Categorical Propositions

Inferences

Distribution of Terms

Negation

Other Forms

Liar’s Paradox

Ch 6 Categorical Syllogism: Three Classes

Inferences

Universal positive and universal positive

Universal positive and universal negative

Universal positive and particular positive

Universal positive and particular negative

Universal negative and particular positive

Rules of Validity

Other Forms

Four or More Classes

Universal and Particular Propositions

Positive and Negative Propositions

Ch 7 Compound Propositions and Syllogisms

Symbolic Expression

Conjunction

Alternative

Disjunction

Truth Tables

Ch 8 Hypothetical Syllogism

Hypothetical and Its Constituents

Pure Hypothetical Syllogism

Mixed Syllogism

Ch 9 Hypothetical and Compound Syllogisms

Conjunction

Alternative

Disjunction

Confounding Compounds

Dilemma

Part III Uncertain Conclusions

Ch 10 Inductive Inferences

Ch 11 Classes and Uncertain Conclusions

Analogy

Generalisation

Problems of Generalisation

Probability

Joint and Conditional Probability

Ch 12 Hypothetical Relationship: Causation

Necessary and Sufficient Conditions

Cause as Necessary and Sufficient Conditions

Hypothesis

Scientific Method

Hypothesis Formulation and Testing

Fallacies

Part IV Reasoning, the Fourth R

Ch 13 Verbal Reasoning

Assessing Inferences

Opinion Versus Argument

Relevance

Omissions

Facts First or Theory

Objective or Subjective

Majority and Consensus

Countering Arguments

Fallacies

About Thinking

Ch 14 Mathematical Reasoning

‘Some’ and Proportions

9/11 Numbers Game

Population Statistics

Measurement

Business of Uncertainty

Normal Distribution

Some Cautions

Conclusion

Bibliography

Author’s Note

Acknowledgements

Introduction

Detection is, or ought to be, an exact science, and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid. (Sherlock Holmes in ‘The Sign of Four’ by Arthur Conan Doyle)

Writers of crime fiction construct clues, red herrings and evidence into a logical plot. Conan Doyle must have had a logical mind. But he also believed in spiritual mediums and communication with the dead. How could a logical person be superstitious? Isn’t that a contradiction? But perhaps one can be logical without being rational — know the rules of logic without living by them. Besides, people are not consistent in their beliefs, so Conan Doyle’s personal belief in spiritual mediums need not be reconciled with his fictional creation’s belief in logic.

The previous paragraph is an example of logical reasoning, which comes naturally to you and me. Why then do we need to work at reasoning? The next two examples taken from newspapers show why. Read the analyses to appreciate why we should think critically about what we read.

The increase in the number of unopposed elected members shows this rule has led to creation of better consensus among people. (quote in a newspaper report, ‘this rule’ refers to the new requirement of educational qualifications for panchayat candidates in Rajasthan and Haryana.)

Do unopposed elections — only one candidate contesting in one constituency — indicate consensus? Yes, if that candidate represents the views and wishes of most people. No, if other candidates who want to contest elections cannot do so because of the new rule. Which is it?

Now think of the rule: the requirement of minimum educational qualifications for contesting panchayat elections. This rule disqualifies some people, among whom may be potential candidates with popular support. And are these candidates likely to be competent? Why not, if they missed out on education only because of poverty or social backwardness?

An increase in unopposed elections after the imposition of minimum education is a warning sign that willing and able candidates are being kept out and as a result, voters do not have their choice of representative. Unopposed elections here mean the constriction of democracy, not consensus.

The next example is a headline for an advertisement for a computer training institute.

Over 16,000 [of our students] found jobs in the last year.

This claim suggests that students of this institute have great success in finding jobs, presumably better than students of other institutes. Think of the questions to ask before accepting this claim.

Did all 16,000 students complete their education from the institute last year? The number could include students who passed earlier and either took jobs late or moved from one job to another. How many students passed last year and what proportion of the total are these 16,000 students? What is the average salary drawn by these 16,000 students and how does it compare to the industry average? How do the salaries compare to the fees paid to the institute? In other words, does the education get students average or above-average salaries? Are students earning a good return on investment? How does the institute perform on these measures compared to others offering similar education?

Without this kind of critical examination, the tremendous amount of information available today is mere noise. When you read with a critical eye, everything yields more meaning. That’s not an exaggeration; logic is everywhere. It is at the heart of all the sciences, both natural and social. The three Rs of learning — reading, writing and arithmetic — are incomplete without reasoning. Logic shapes casual conversation as well as learned seminars. It rules family councils as well as business negotiations. All fiction, not just detective fiction, has to ‘make sense’ to be enjoyable, with events moving logically to a conclusion. Even entertainment — puzzles, riddles and jokes — depends on logic.

In the story of the elephant and the seven blind men, there is an omniscient narrator who tells the reader that the animal is an elephant. In reality, reality is the elephant, we are the blind, and there’s no storyteller. What we have are observations and logic, imagination and judgement. Let’s learn one of these now.

What Is Logic?

Logic is the science of reasoning and defines rules for valid argument. (In this book, forget the common meaning of ‘argument’ as disagreement or quarrel.) An argument is made up of some information and the conclusion drawn from it. Logic defines what conclusions can be validly drawn from various types of information. For example, given that metals expand on being heated and that iron is a metal, we argue that iron will expand on being heated.

For a formal definition of logic, let us go to a dictionary and a textbook.

Logic: Reasoning conducted or assessed according to strict principles of validity (The New Oxford Dictionary of English, 3rd ed., 2010)

The study of logic is the study of the methods and principles used in distinguishing correct from incorrect reasoning. (Introduction to Logic by Copi, 1968)

Why is the study of valid inferences a science in itself? Human knowledge is built by drawing new conclusions from what is already known. It is a large and complex structure of observations, experiments and theories about all kinds of things — from heated metals to cooling stars. The links holding it all together must be strong, and those links are the concern of logic. Logical connections tie together disparate facts into organised knowledge. In daily life, logic is what tells us that if it rained last night, we will have a cooler morning with a chance of potholes.

Humour depends on logic, though warped logic, as in these jokes passed around in emails.

Used Cars: Why go elsewhere to be cheated? Come here first. (Classified advertisement)

Dog for sale: Eats anything and is fond of children. (Classified advertisement)

Customer: I have a long-distance modem. (Microsoft Helpdesk conversation)

Tech Support: What does the screen say now?

Customer: It says, ‘Hit ENTER when ready’.

Tech Support: Well?

Customer: How do I know when it’s ready? (Microsoft Helpdesk conversation)

Customer: I want to buy a dress to put on around the house.

Sales assistant: Yes, Madam. How large is your house? (Crazy replies)

Kindly aunt: Now that you're married, you should have some insurance.

Man: But why? My wife isn't dangerous. (Crazy replies)

The next two examples are older and the indirect writing demands a bit of thought.

The Prefect ... had the fashion of calling everything ‘odd’ that was beyond his comprehension, and thus lived amid an absolute legion of ‘oddities’. (‘The Purloined Letter’ by Edgar Allan Poe)

Charles Henry Twain … was a zealous and distinguished missionary. He converted sixteen thousand South Sea Islanders, … His poor flock loved him very, very dearly; and when his funeral was over they got up in a body (and came out of the restaurant) with tears in their eyes, and saying one to another that he was a good, tender missionary, and they wished they had some more of him. (‘Burlesque Autobiography’ in A Treasury of Mark Twain)1

We ‘get the joke’ in the warped reasoning because we are inherently logical. Why study logic if it comes naturally? Because not all reasoning is as easy or as straightforward as these jokes. Because there are certain mistakes that we are prone to make. Because in some areas of life, a mistake costs far more than not getting a joke. But the best reason is that logic is very interesting, even amusing.

Are you convinced that logic is good for you? You shouldn’t be. You have nothing more than the say-so of one writer at this point. Start your study with healthy scepticism and decide for yourself if logic really is as good as claimed.

About the Book

This is an introductory book; if you have never heard of logic, this book is for you. It presents the basics of logic in simple language and demonstrates logical analysis with plenty of examples from routine experience, current affairs and popular books.

Part I explains three basic and essential concepts — truth versus validity, classes or categories of things and hypothetical relationships. Classes and hypotheticals are central ideas in the organisation of this book. Most of these ideas are familiar to you because they are in common use, though there are some finer points to note. If anything seems abstract or difficult, press on and it will become clearer as you read further.

The main business of logic is drawing conclusions; therefore, inferences form the bulk of this book. Inferences are grouped by classes and hypotheticals. Part II presents deductive inferences and Part III inductive inferences.

Part IV presents examples of how logic works in life. When you have finished the book and begun to look at everything with a critical eye, you will gather many more examples of your own.

If you are wondering whether logic could really be easy to learn, read the mini-story below. It illustrates all the topics covered in this book. You will recognise the reasoning in the story as the kind of thinking we all do, and easily.

Guess Who?

The man sitting at the side of the busy street looked lost and couldn’t answer even the simplest questions. Almost everybody who stopped to help, or just to gawk, had a theory about the man.

‘He must have lost his memory, like they show in films. He has amnesia.’

‘He must be drunk. He has that dazed look in his eyes.’

‘He must be possessed. His eyes look haunted.’

‘No, no, his eyes are filled with peace. He is a pious man. Maybe he went into a trance while meditating and couldn’t wake up.’

‘How can you talk such superstitious rubbish? He has to be a terrorist, can’t you see? He must have planted a bomb nearby and is waiting for the bang.’

‘Then he wouldn’t be sitting here. And if you really thought so, you wouldn’t be standing here.’

‘He must be mad. Nobody in his senses would sit on the side of the road like this.’

‘He must have taken drugs. These days everyone takes drugs, even the educated and well-off.’

‘He is definitely educated and well-off going by his clothes. Not a hawker or a beggar, this one.’

‘Kaay zaala? Bara nahi waatat? ... No, he didn’t understand that. Not a Marathi-speaking person then.’

‘But he said he didn’t understand in rather good Hindi. Definitely a north Indian. Aap ka naam kya hai, bhaisaab? Kahan se hai?’

‘Don’t remember? Don’t worry. Check your pockets; maybe there’s some ID card.’

‘Oh no, his wallet and everything is gone. He must have been mugged.’

‘Maybe he was hit on the head. That’s why he can’t remember anything.’

‘He doesn’t look injured. Or even ill. He looks quite well, actually.’

‘Look at his shoulders and forearms. He is quite muscular and fit.’

‘He must be working out in a gym. Too well-dressed to be doing manual labour.’

‘Exactly. He is not a poor, casual worker with no one to miss him when he doesn’t return home.’

‘But if he doesn’t have a fixed routine, no one may be expecting him, no one may miss him for days.’

‘I am putting his photograph on social media. Somebody in his circle must be using the internet.’

‘If his is a rags-to-middle class story, his family may not be educated or in cyber space.’

‘Poor man. He is like an orphan; he has loved ones somewhere but knows nothing about them. Completely alone.’

‘Completely free. He is like a child, innocent and washed clean of past sins and memories of pain.’

‘Don’t get poetic. Aadhaar biometrics will soon identify him and reunite him with his loved ones and his past sins.’

Part I Basic Concepts

Logic is the science of reasoning. Can it tell us why the stock market crashed? No, it can’t. Economics tells us that. Can logic explain why the spacecraft crashed? No, it can’t. Physics tells us that. In general, the sciences investigate truth, and logic makes rules to form links between truths to construct knowledge. This important aspect of logic is explained in the first chapter. Logic is not about truth but about truth relations.

What exactly is the stock market? Who are investors? What kind of people are they? What is speculation? We need to define terms if we are to understand and analyse anything. Classification and definition are explained in the second chapter.

The third basic concept essential to learning logic is the hypothetical relationship ‘if … then’. If the monsoon is good, stock prices will go up. If you make money in the stock market, you must pay capital gains tax. This familiar sentence form is at the heart of logic. You will learn why in the third chapter.

Ch 1 Truth and Truth Relation

What’s the difference between the following statements? ‘I saw the man throw a stone at the bus.’ ‘I saw the man pick up a stone, so he must have thrown it at the bus.’ The first is a statement of fact, which can be true or false. The second is an inference or argument, which can be valid or invalid.

An inference asserts one thing on the basis of another; it is a truth relation. The inference in this example is: If it is true that the man picked up a stone, it must be true that he threw a stone at the bus. This inference is, in fact, invalid. True, the man picked up the stone but only to hammer a nail, hold down a map in a breeze, exercise his forearm or whatever else you can imagine.

Perhaps the best known logical argument is: ‘All men are mortal. Socrates is a man. Therefore, Socrates is mortal.’ It is equally valid to argue: ‘All men are immortal. Socrates is a man. Therefore, Socrates is immortal.’ In the second example, a false statement is a valid conclusion because it follows from the premises. How can that be?

Truth and truth relation (validity) are two different things. Truth does not guarantee validity nor does validity guarantee truth. An argument is a truth relation and is logically valid if the conclusion follows from the premises — whenever the premises are true, the conclusion must be true. ‘Garbage in, garbage out’ applies to argumentation. Starting from false premises, you can derive a false statement as a valid conclusion as is sometimes demonstrated in television debates. Worse, a true statement can be an invalid conclusion. ‘All men are mortal. Socrates is a philosopher. Therefore, Socrates is mortal.’ Even though all the statements are true, the conclusion is invalid.2

‘He must have lost his memory, like they show in films. He has amnesia.’

‘Maybe he was hit on the head. That’s why he can’t remember anything.’

In Guess Who, the public makes guesses while the truth is unknown. If a man has a head injury, he may suffer memory loss or other problems in mental functioning. This is a truth relation. Whether the amnesiac actually has a head injury is unknown.

Propositions

Because logic deals with truth relations, it can only work with statements that are capable of being true or false. Such statements are called propositions. Propositions make assertions about things. The following are propositions.

This is a wooden chair.

Alsatians are guard dogs.

Biographies contain some truth.

I hope the plumber knows what he is doing.

This company designs beautiful cars.

It may seem that everything we say is either true or false and therefore is a proposition. But promises, commands, resolutions, questions and exclamation are not capable of being true or false. Examples of what is not a proposition will clarify what is a proposition.

Take the chair. (command)

You should walk your dog every day. (command)

I will always speak the truth. (promise)

Can’t we find a good plumber, for god’s sake? (question)

Wow! What a beautiful car. (exclamation)

‘This is a wooden chair’ asserts a quality (wooden) of the chair. It is true if the chair really is made of wood. If the chair is made of steel, or if the object is a table, the proposition is false. ‘Take the chair’ does not say anything about any quality of the chair or of anything else and can’t be true or false. Therefore, it is not a proposition. Compare the other four pairs of statements to see why the first set are propositions and the second are not. Notice that an exclamation implies a proposition, and sometimes, so does a question.

Here are some more examples, beginning with the amnesiac story.

‘Somebody ought to help the poor man,’ says a bystander looking at the man sitting at the roadside. The statement is a wish or a request and not a proposition. ‘Amnesia is just a trick in the fiction writer’s bag’ is a proposition.

‘A rose by any other name smells as sweet’ is a proposition. A celebrity’s endorsement of a perfume may seem outside the scope of logic. Isn’t it just a case of consumers being swayed by a popular face? But ‘a perfume by the celebrity’s name smells sweeter’ is a proposition and capable of logical analysis — not to say financial analysis.

‘Be logical’ is a command,