Chapter 1 Arguments

Logic has many uses, as mention in the preface. What we will be focussing on here is its use in evaluating arguments; sorting the good from the bad.

In everyday language, we sometimes use the word ‘argument’ to talk about belligerent shouting matches. If you and a friend have an argument in this sense, things are not going well between the two of you. Logic is not concerned with such teeth-gnashing and hair-pulling. They are not arguments, in our sense; they are just disagreements.

An argument, as we will understand it, is something more like this:

Either the butler or the gardener did it.
The butler didn’t do it.
∴

The gardener did it.

We have here a series of sentences. The three dots on the third line of the argument are read ‘therefore.’ They indicate that the final sentence expresses the conclusion of the argument. The two sentences before that are the premises of the argument. If you believe the premises, and you think the conclusion follows from the premises—that the argument, as we will say, is valid—then this (perhaps) provides you with a reason to believe the conclusion.

This is the sort of thing that logicians are interested in. We will say that an argument is any collection of premises, together with a conclusion.

This Part discusses some basic logical notions that apply to arguments in a natural language like English. It is important to begin with a clear understanding of what arguments are and of what it means for an argument to be valid. Later we will represent English-language arguments in a formal language.

In the example just given, we used individual sentences to express both of the argument’s premises, and we used a third sentence to express the argument’s conclusion. Many arguments are expressed in this way, but a single sentence can contain a complete argument. Consider:

The butler has an alibi; so they cannot have done it.

This argument has one premise followed by a conclusion.

Many arguments start with premises, and end with a conclusion, but not all of them. The argument with which this section began might equally have been presented with the conclusion at the beginning, like so:

The gardener did it. After all, it was either the butler or the gardener. And the butler didn’t do it.

Equally, it might have been presented with the conclusion in the middle:

The butler didn’t do it. Accordingly, it was the gardener, given that it was either the gardener or the butler.

When approaching an argument, we want to know whether or not the conclusion follows from the premises. So the first thing to do is to separate out the conclusion from the premises. As a guide, these words are often used to indicate an argument’s conclusion:

so, therefore, hence, thus, accordingly, consequently

For this reason, they are sometimes called conclusion indicator words .

By contrast, these expressions are premise indicator words , as they often indicate that we are dealing with a premise, rather than a conclusion:

since, because, given that

But in analysing an argument, there is no substitute for a good nose.

1.1 Sentences

To be perfectly general, we can define an argument as a series of sentences. The sentences at the beginning of the series are premises. The final sentence in the series is the conclusion. If the premises are true and the argument is a good one, then you have a reason to accept the conclusion.

In logic, we are only interested in sentences that can figure as a premise or conclusion of an argument, i.e., sentences that can be true or false. So we will restrict ourselves to sentences of this sort, and define a sentence as a sentence that can be true or false.

You should not confuse the idea of a sentence that can be true or false with the difference between fact and opinion. Often, sentences in logic will express things that would count as facts— such as ‘Rudolf Carnap was born in Ronsdorf’ or ‘Simone de Beauvoir liked taking walks’. They can also express things that you might think of as matters of opinion—such as, ‘Rhubarb is tasty’. In other words, a sentence is not disqualified from being part of an argument because we don’t know if it is true or false, or because its truth or falsity is a matter of opinion. If it is the kind of sentence that could be true or false it can play the role of premise or conclusion.

Also, there are things that would count as ‘sentences’ in a linguistics or grammar course that we will not count as sentences in logic.

Questions

In a grammar class, ‘Are you sleepy yet?’ would count as an interrogative sentence. Although you might be sleepy or you might be alert, the question itself is neither true nor false. For this reason, questions will not count as sentences in logic. Suppose you answer the question: ‘I am not sleepy.’ This is either true or false, and so it is a sentence in the logical sense. Generally, questions will not count as sentences, but answers will.

‘What is this course about?’ is not a sentence (in our sense). ‘No one knows what this course is about’ is a sentence.

Imperatives

Commands are often phrased as imperatives like ‘Wake up!’, ‘Sit up straight’, and so on. In a grammar class, these would count as imperative sentences. Although it might be good for you to sit up straight or it might not, the command is neither true nor false. Note, however, that commands are not always phrased as imperatives. ‘You will respect my authority’ is either true or false—either you will or you will not—and so it counts as a sentence in the logical sense.

Exclamations

‘Ouch!’ is sometimes called an exclamatory sentence, but it is neither true nor false. We will treat ‘Ouch, I hurt my toe!’ as meaning the same thing as ‘I hurt my toe.’ The ‘ouch’ does not add anything that could be true or false.

Practice exercises

At the end of some chapters, there are exercises that review and explore the material covered in the chapter. There is no substitute for actually working through some problems, because learning logic is more about developing a way of thinking than it is about memorizing facts.

So here’s the first exercise. Highlight the phrase which expresses the conclusion of each of these arguments:

1.

It is sunny. So I should take my sunglasses.
2.

It must have been sunny. I did wear my sunglasses, after all.
3.

No one but you has had their hands in the cookie-jar. And the scene of the crime is littered with cookie-crumbs. You’re the culprit!
4.

Miss Scarlett and Professor Plum were in the study at the time of the murder. Reverend Green had the candlestick in the ballroom, and we know that there is no blood on his hands. Hence Colonel Mustard did it in the kitchen with the lead pipe. Recall, after all, that the gun had not been fired.