Posts Tagged ‘ argument ’

5.1 More on validity

Though regimenting all statements into four basic types is a useful tool, we still need rules that show how these types combine into valid arguments. Aristotle recognised that if the U+ statement

U+: ‘all unmarried men are bachelors’

is true then it follows that the P+ statement

P+: ‘some unmarried men are bachelors’

must also be true. There is no way that the first statement could be true and the second one false at the same time. Since the second one must be true if the first one is true, any inference from U+ to P+ is valid. On the other hand, it cannot be validly inferred from

P+: ‘some bachelor is unmarried’


U+: all bachelors are unmarried

even though it is, as a matter of fact in this instance, true. That it is not valid can be seen by considering what happens when we substitute ‘girls’ and ‘blonde hair’ for ‘bachelors’ and ‘unmarried’. Clearly, if all girls have blonde hair, some girls do, but the converse does not apply. An argument form must remain valid regardless of the content. If changing the content can change the truth or falsity of the conclusion, the argument cannot be valid. Therefore, we cannot validly infer from a P+ statement to a U+ statement.

Clearly, if a U+ statement is true, the corresponding P- statement must be false. For example, if

U+: ‘all unmarried men are bachelors’

is true, then

P- : ‘some unmarried men are not bachelors’

must be false. Both statements cannot be true simultaneously, for they are contradictory. Therefore, we can validly infer from the truth of any U+ statement that the corresponding P- statement is false. Similarly, if a U- statement is true, for example,

U- : ‘nothing is red and green all over at the same time’

then the particular negative is also true:

P- : ‘something is not red and green all over at the same time’.

These simple relations between true statements can be captured by using the symbolic forms we introduced above, such that we can define the following inference rules (IR):

IR1: From ‘All S are P’ we can infer ‘Some S is P’ U+ → P+
IR2: From ‘Some S is P’ we cannot infer ‘All S are P’ P+ ≠ U+
IR3: From ‘No S are P’ we can infer ‘Some S is not P’ U- → P-
IR4: From ‘Some S is not P’ we cannot infer ‘No S are P’ P- ≠ U-

If we consider the universal negative statement ‘no man is an island’ we can infer another universal negative, ‘no island is a man’. Using our simple symbolism, we can add the inference rule

IR5: From ‘No S are P’ we can infer ‘No P are S’ U- → (U-)

This property of terms in a statement is called the converse relation. It also holds with particular affirmatives, so that if ‘Some philosophers are wise men’ is true, then so is ‘Some wise men are philosophers’:

IR6: From ‘Some S is P’ we can infer ‘Some P is S’ P+ → (P+)

However, the negative particular does not licence the converse: it does not follow from ‘Some philosophers are not wise men’ to ‘Some wise men are not philosophers’. Thus,

IR7: From ‘Some S is not P’ we cannot infer ‘Some P is not S’ P- ≠ (P-)

We can also say that

IR8: From IR2 and IR4, it is never valid to infer any universal statement from a particular statement (in fact this is just David Hume’s problem of induction again!). P ≠ U

Finally, the converse relation between subject and predicate becomes very interesting when we consider what happens with the universal affirmative, ‘All S are P’. It turns out that reasoning the converse – ‘All P are S’ – gives rise to a common fallacy that was well known to the ancient Greeks and which can often be heard committed in debates today, from the local pub to local politicians:

Consider the universal affirmative
U+: ‘All unmarried men are bachelors.’

Clearly, the converse is also true,
(U+): ‘All bachelors are unmarried men.’

However, now suppose that the universal affirmative

U+: ‘Every one entering the country illegally is an immigrant.’

is true. The converse

(U+): ‘Every immigrant is entering the country illegally.’

is clearly not a valid inference since many immigrants arrive with permission or through the proper channels. Since the converse of a universal affirmative is sometimes true and sometimes false, it cannot be considered a form of valid inference. Hence,

IR9: From ‘All S are P’ we cannot infer ‘All P are S’ U+ ≠ (U+)

Before we see how these rules can help us to analyse real arguments, try exercise 5.1 to test your understanding of this post.

Alternatively, continue reading the book…


3. More about claims and premises

We said earlier that an argument must have at least one claim and one premise. However, some arguments contain only hidden premises. Consider the argument:

8. The best way to avoid getting AIDS is to simply not have sex.

This is an argument with a single, hidden premise, namely, ‘having sex is the main way of getting AIDS.’ As we have said, when the premise is, or seems to be, uncontroversial, premises may not be stated. It is assumed in 8. that everyone agrees with the hidden premise and that the claim can be inferred from it. Whether you accept the conclusion or not will depend partly on whether you accept the hidden premise; however, you could accept the premise and still argue that there is a better way to avoid getting AIDS. As we shall see in later posts, words, like ‘best’ carry a lot of implicit content.

Be aware that sometimes something that looks like an argument is not an argument at all, but an explanation or description. You must be clear about the differences between these functions. As we have said, an argument must have a claim and at least one premise (even if hidden), but another way to characterise arguments is in terms of their function.

Thinking along these lines, we can think of an argument as a way of persuading someone that a statement is true or correct. Usually, an argument tries to convince someone to believe something new or different whereas a language function like explanation is an answer to a ‘Why?’ question – in other words, a request for further information or clarification. Explanations are commonly used to make something that is already accepted clearer or more understandable. Consider the difference between

9. Why does the Sun come up in the morning? (you already believe it, but you want to know how it happens). Explanation: The sun appears to come up in the morning because the Earth revolves around the Sun. If you are standing on the Earth, the Sun would appear to move in relation to your position.


10. Why should I believe that the Sun comes up in the morning? (You have lived your whole life in a cave and have never seen the sun). Argument: The sun comes up in the morning. If you were to go outside of your cave, you would observe a bright orange globe that rises over the horizon at the start of each day.

• An argument must have a claim and at least one premise
• One, more, or all the premises of an argument could be implicit
• Arguments whose premises are entirely implicit are usually uncontroversial
• Arguments can be distinguished from other language functions by thinking about their purpose: the purpose of an argument is to convince somebody of something they do not yet believe.

Try Exercise 3 to test your understanding of this and earlier posts, or continue reading

1. What is an argument?

Much of the business of philosophy is concerned with arguments – their construction, analysis, defence and refutation – but how do we determine exactly what is and what is not an argument? How do we distinguish arguments from other linguistic and logical functions such as explanations, descriptions, exhortations and the like? How do we assess whether any particular argument is worthy of assent or deserving of dismissal? How do we determine whether an argument’s acceptance amounts to something significant or is merely a nod to the trivial?

The answers to such questions lie in a loose body of techniques and skills called ‘critical thinking.’ They are ‘loose’ in the sense that although there are some very definite and universally accepted techniques involved in assessing the logical validity and soundness of arguments, there is a large dose of creative or imaginative skill involved too. In the 1960s, Edward de Bono coined the term ‘lateral thinking’ to refer to the ability to come up with novel ways of looking at and solving problems. Like other creative arts such as painting, writing, or playing musical instruments, skill at both logical and lateral thinking can be learned.

What is an argument?
In order to answer the questions posed above, it is necessary to first understand what makes a series of sentences an argument, rather than say an explanation, description or other language function. We can determine whether a passage contains an argument by asking whether it contains any claims that are being supported or defended. The sentences that provide – or are intended to provide – the support to a claim are called the premises of the argument. All arguments, by definition, must contain at least one premise and one claim.

Consider the following:

1. Tomorrow’s lecture will be on Kant. It’s the last lecture of the semester, and last year this professor chose Kant as his topic for his final lecture.

The speaker is making an ordinary prediction about what will happen at some future time (in this case, ‘tomorrow’). The prediction, however, takes the form of an argument, which is simply to say that the prediction amounts to a claim with some reason or evidence (a premise) given in support of it. Formally, it can be broken down in this way:

: Tomorrow’s lecture will be on Kant.
Premise: The same professor gave a lecture on Kant at the same point in time in last year’s course.

What links the claim to the premise is an inference. Inference in ordinary language is often signalled by a connecting word after the claim such as ‘because’, or if the premises are stated before the claim, ‘so’ or ‘therefore’:

i. Tomorrow’s lecture will be on Kant because it’s the last lecture of the semester and Dr Burke chose Kant as the topic for his final lecture last year.

ii. It’s the last lecture of the semester, and last year Dr Burke chose Kant as the topic for his final lecture. Therefore, tomorrow’s lecture will be on Kant.

However, when we break down or analyse an argument’s structure, we usually leave the connecting words out. In part, this is because the strength of the inference, its validity, will be something that we want to test when we evaluate the argument, and it helps to reduce the argument to its simplest form in order to do this. We will be discussing inference and validity at length later on.

For now, compare 1 above, with 2 below:

2. I’m really bored. Every week is just the same. Study, study and more study.

Our teenage angst might well be used as a prelude to an argument – a persuasive appeal for a study break or extra allowance might easily follow – but barely stated, it offers nothing to trouble the disinterested parent simply because sentence 2 fails to amount to an argument. The speaker is not here trying to establish or support a claim of being bored. Rather, he is reporting that he is bored and offering an explanation for it.

One way to think of the difference between a claim and other language functions such as reports, descriptions and explanations is to say that a claim must, at least in principle, be capable of being either true or false. The statement ‘I’m really bored’, so long as it is not spoken in jest or insincerity, does not seem capable of being false in quite the same way that the claim in sentence 1 is capable of being false. That is to say, ordinarily, we would not expect a speaker to be mistaken or to turn out to be wrong or ignorant about their own immediate feelings.

The sentence in 3 below also fails to be an argument, but for a different reason:

3. If God exists, then atheism is false.

This kind of sentence is called a closed conditional. It is a statement of the relationship between a finite number of possibilities (in this case, two), where all the possibilities are covered, so the conditional sentence remains logically true no matter what is, as a matter of fact, the case. We say it is closed and remains logically true because even in a world where God were proven not to exist, it would still be true that if God had existed or ever would exist in that world, then atheism would be false. Thus, the sentence 3. is true regardless of whether God exists or not (later, we will learn that 3. is an example of a wider species of statements called ‘tautologies‘).

Open conditionals are ‘if…then’ statements that leave some options out of the statement:

4. If God exists, Jesus was the son of God.

It could be the case that God exists, but still false that Jesus was his son; hence, the conditional is said to be open. Even so, an open conditional barely stated does not amount to an argument if it contains no premise or supporting reason to justify its belief or acceptance. A claim without any premises or supporting reasons for its acceptance is called an assertion and is much like our report of teenage ennui in sentence 2, which is to say it fails to be philosophically interesting.

Conditionals, open or closed, do not on their own make arguments, but they can make claims and premises. Sentence 4 could be a claim if it were backed up with some supporting reason.

Here’s an example of an argument that use conditionals in both its premises:

5. Life on Earth is in peril. If we had not burned so much fossil fuel in the late 20th century, there would not have been so much carbon dioxide in the atmosphere. If there were less carbon dioxide in the atmosphere, the greenhouse effect would not be running out of control.

The form of the argument is:

Claim: Life on Earth is in peril.
Premise 1: If we had not burnt so much fossil fuel in the late 20th century, there would not have been so much carbon dioxide in the atmosphere.
Premise 2: If there were less carbon dioxide in the atmosphere, the greenhouse effect would not be running out of control.

• An argument must contain both a claim and one or more premises
• A claim without any premises is called an assertion

Try Exercise 1 to test your understanding of this post, or continue reading

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