Logic: A Complete Introduction: Teach Yourself

Siu-Fan Lee

• Publisher: Teach Yourself
• Year: 2017
• Print ISBN: 9781473608436
• eBook ISBN: 9781473608443
• Topic: Philosophy
• Subtopic: Logic in Philosophy

Understand Logic is a comprehensive introduction to this fascinating though sometimes challenging subject. As well as looking at logic in theoretical terms the book considers its everyday uses and demonstrates how it has genuine practical applications. It will take you step by step through the most difficult concepts and is packed with exercises to help you consolidate your learning at every stage. Covering everything from syllogistic logic to logical paradoxes and even looking at logic in Alice in Wonderland, this is the only guide you will ever need.

1

What is logic?

In this chapter you will learn about:

• what logic is

• what an argument is

• deduction and induction

• truth and validity

• the study of logic

‘It used to be said that God could create anything except what would be contrary to the laws of logic – The truth is that we could not say what an “illogical” world would look like.’

Ludwig Wittgenstein (1921), Tractatus Logico-Philosphicus, 3.031

1.1 What is logic?

Logic is the study of the methods and principles used to distinguish between good and bad reasoning. It is a normative discipline, in the sense that it does not survey and describe how we actually reason (which is the job of the psychologist) but what we should do in reasoning.

Reasoning concerns arguments. In the first part of this chapter we explain what an argument is made up of and how to identify an argument. Then we introduce the distinction between two basic types of reasoning: deduction and induction. Since we often talk of an argument being true or valid (indeed it is technically wrong to describe an argument as true, we shall explain later), we discuss the relation between truth and validity in Section 1.4. Finally, we look at certain misconceptions about logic and reasoning to allay any unnecessary fears about logic.

1.2 What is an argument?

STRUCTURE

An argument is a structure that comprises a conclusion, namely, a proposition that one wants to uphold, and some premises, as reasons to support the belief in the conclusion. Logic is about whether and how a conclusion follows from the premises. What it means for a conclusion (a certain proposition B) to ‘follow’ from a premise (another proposition A) is sometimes cashed out as whether whenever the premise (A) is true, the conclusion (B) is also true. Arguments come in many forms in real life; sometimes they are easy to spot but sometimes they are covert. We thus need first of all some training to heighten our awareness of and ability to identify arguments.

An argument contains one and only one conclusion. In fact, an argument is individuated by its conclusion. If we want to make more than one conclusion, then there must be more than one argument. An argument can be represented in the following standard form: the premises are listed on the top followed by the conclusion at the end and each sentence is numbered for easy reference. We also use a line to separate the premises and conclusion.

Premise 1

Premise 2

…

Conclusion

A conclusion is supposed to be supported by the premises. There may be many premises in an argument. Indeed, there is no limit to the number of premises. However, is there a minimum number of premises in an argument? If so, what is it? One, two, or three? Surprisingly, formally, the answer to the minimum number of premises required is zero! This is surprising because premises are reasons to support a conclusion and allowing zero premises would mean that the conclusion is supported by nothing, and that seems to directly contradict the goal of reasoning, namely, to accept an argument only if one can provide a reason to believe in it. These are legitimate points. Yet when logicians allow an argument to have no premises, it merely means that nothing goes against the conclusion, rather than that the conclusion is supported by no reason. We regard such a conclusion as self-evident. The following are some examples of self-evident truths.

Example (1): Everything is identical to itself.

Example (2): Something is the case or it is not the case.

Example (3): It cannot be true that something is both the case and not the case at the same time.

Example (4): An object cannot be red and green all over at the same time.

Self-evident truths are obvious truths that do not need anything to support them. Equally, it can also be said that everything supports them because nothing counts as a reason to reject them, and they do not contradict with anything. Later we introduce the idea that an argument is valid if it preserves truth from the premises to the conclusion. Given this definition, because a conclusion containing a self-evident truth is always true, any statement serving as its premise, if true, will lead to the truth of the conclusion. Thus, every argument supporting a self-evident truth is a valid argument!

Examples (1)–(3) are so general and intuitive that their general forms are accepted as laws of logic. The three logical laws are named respectively:

Example (1) The Law of Identity

Example (2) The Law of Excluded Middle

Example (3) The Law of Contradiction

Example (4) is an example of a necessary truth in metaphysics.

Some philosophers even claim that contingent truths can be self-evident truths as long as they are very intuitive and hardly anything counts to refute them. For example, first person perceptual truths, such as (5), are self-evident.

Example (5): This is my hand (while I hold it up and look at it closely in normal perceptual circumstances) and this is another (waving it).

Self-evident truths are special cases. Most arguments are not like that but contain a conclusion and at least one premise.

More examples of an argument are given below when we introduce the techniques in identifying an argument.

Key idea: Self-evidence

Is self-evidence just the same as having nothing to go against it? How do you take it?

The question of self-evidence is more complicated than it looks. Self-evident truths are intuitive. It is difficult to argue against widespread intuition, especially when many other judgements are based on it. Consider what I am to do if I really doubt that these hands are mine? I will also need to doubt many other things that we so take for granted in our lives. Yet we do know that sometimes what we believe is true may turn out to be false; for example, I could mistake a robot dog seen in the distance for a real dog. How to distinguish various types of intuition and more importantly, how to justify those very general and foundational ones? These are not easy questions!

A philosopher Ludwig Wittgenstein once described the situation as: ‘If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to say: “This is simply what I do.”’ (Ludwig Wittgenstein (1958), Philosophical Investigations, section 217)

Moreover, should we treat self-evidence the same as having nothing to go against it? When we do so, don’t we already assume that a proposition is either true or false, and then nothing is both true or false? Yet these laws of logic (the Law of Excluded Middle and the Law of Contradiction respectively) are exactly some of the self-evident truths that we claim to exist. So we would be defining self-evident truth using some self-evident truths. Isn’t that begging the question?

For our very question is that although we do not usually challenge self-evident truths, it seems hard to explain why self-evident truths cannot be challenged. Indeed, some laws of logic are challenged under different systems of logic. The three logical laws stated above are the laws for classical logic. There are non-classical logic systems that try to answer these questions in a different way. For instance, paraconsistent logic does not accept the Law of Contradiction – some paraconsistent logicians claim that there are tr...

Table of contents

• Cover 
• Title
• Contents 
• Meet the author
• Introduction
• 1 What is logic?
• 2 Meaning
• 3 Informal fallacies
• 4 Categorical logic
• 5 Propositional logic
• 6 Predicate logic
• Epilogue
• Notes
• Glossary
• Taking it further
• Solutions
• Copyright