Posts Tagged ‘ universal statement ’

5. Inference & Validity

We have seen that an argument is a language function that must consist of a claim and at least one premise. However, not just any old claim and any premise make for an argument. The premise(s) and claim must bear a certain relationship to each other in order to form an argument. This relationship is captured by the notion of inference. In philosophy, as well as science and mathematics, we often use a kind of formal or deductive reasoning to support our claims. In order for the support to be considered good, we must be sure that the claims we derive from our premises are valid. It is now time to give formal consideration to the two related notions of inference and validity.

Suppose both these sentences are true:

1. All philosophers are clever.
2. Bertrand is a philosopher.

It follows that

3. Bertrand is clever.

On the other hand, suppose these sentences are true

1. All philosophers are clever.
3. Bertrand is clever.

It does not follow that

2. Bertrand is a philosopher.

Whether it is expressed as a claim or a premise, sentence 2 above may or may not be true; however, if it is true, its truth is not a logical conclusion from premises 1 and 3, since 1 and 3 can both be true in the case when Bertrand is a philosopher and in the case when Bertrand is not a philosopher (say, when Bertrand is an architect).

An argument is defined as valid only if the conclusion cannot be false when the premises are true. Consider:

2. Bertrand is a philosopher.
3. Bertrand is clever.

Does the conclusion

1. All philosophers are clever.

follow from the premises? The answer is clearly ‘no’. Whenever the conclusion does not logically follow from the premises of an argument, we say it is invalid. However, whether the conclusion ‘follows’ (is valid) or not isn’t always easy to tell in anything except the simplest of arguments like these. Real arguments are a lot more complex; consequently, we need a reliable procedure that will help us to decide the validity of an argument.

Fortunately, such a procedure exists. Indeed, there is more than one, but the simplest and oldest was invented by the ancient Greek philosopher Aristotle around 300BC.

Aristotle realised that whether an argument is valid or invalid cannot be determined by knowing the truth of its premises, as the invalid examples above clearly show. Rather, its validity depends on what kind of sentence we use, and what order the terms in the sentence are arranged. Consider again:

1. All philosophers are clever.
2. Bertrand is a philosopher.

The form of these sentences is importantly different. Both are subject-predicate, which means that they contain a noun which names a thing (the subject of the sentence) and a verb-phrase that ascribes some property to that object (the predicate of the subject). Nonetheless, the sentence 1 is a universal statement whereas sentence 2. is not.

In other words, sentence 1 says that everything of a certain kind has a certain property. It has the form ‘All As are B’. As we have seen earlier, this form can be expressed in ordinary language in a number of ways:

All swans are white.
Every bachelor is an unmarried man.
Nobody lives without suffering (= all living people suffer).
Anyone who cheats will fail the exam ( = all cheats fail the exam).

Sentence 2, however, is quite different. It is a particular statement. Particular statements are the opposite of universals in that they say that some (one or more but not all) things of a certain kind have a certain property. They have the form ‘This A is a B’. Examples of particular statements are:

Some teachers are kind.
Ludwig is a kind teacher.
That man is a total jerk!
This room is cold.

Apart from being either universal or particular, a statement can be either positive or negative. All the statements above are positive: they say that some property belongs to some object. Negative statements deny that a property belongs to some object. For example:

No swans are black (universal negative statement).
Ludwig is not kind (particular negative statement).

According to Aristotle, every statement in an argument has one of the following four forms:

U+ Universal affirmative:
All As are B (e.g. All philosophers are wealthy)
U- Universal negative:
No As are B (e.g., No philosophers are wealthy)
P+ Particular affirmative:
Some A is an B (e.g., Some philosophers are wealthy)
P- Particular negative:
Some A is not a B (e.g., Some philosophers are not wealthy)

Most importantly, the actual subject and predicate are irrelevant to determining which of the four forms of proposition any given statement belongs to. Whether we are talking about unicorns or philosophers, swans or spacemen, the form of the proposition depends only on whether it is being asserted or denied and whether universally or particularly.

Accordingly, the four forms of proposition can be represented in symbolic form, using S for subject, and P for predicate, as:

U+ All S are P
U- No S are P
P+ Some S are P
P- Some S are not P

• an argument is defined as valid only if the conclusion cannot be false when the premises are true
• validity of an argument depends on the form of the sentences in the premises (universals, particular, positive, negative) and their relationship to each other

Try Exercise 5 to test your understanding of this post, or continue reading


4. Two kinds of argument

There are many ways premises can support their claims. In this post we’re going to look at two common forms widely used in academic and popular writing: inductive generalisations and arguments from authority. To begin, first read Sarah Graham’s short article “Effects of Smoking May Be Passed Down Through Generations”.

Look at the structure of each paragraph in Graham’s article. Paragraph 1 claims that the dangers of smoking for pregnant women may be greater than previously believed. The support for this claim is that research has found that the grandchildren of women who smoked when they were pregnant are more likely (double, in fact) to have asthma.

By the lights of logic, this is an informal but very common way of arguing: the writer tries to persuade you of the argument’s truth by citing an authority that you should believe: recent research. Hence, this kind of argument is called ‘Argument from authority’.

Other arguments from authority are

1. Jesus was the Son of God. How do I know? It says so in the Bible.
2. Darwin’s principle of natural selection is true for sure. It must be because it is accepted by modern science.
3. Pam is cheating on her boyfriend. How do I know? She told me so.
4. An argument’s definitely not the same as an explanation. I don’t know why, but that’s what it says in this text book.
5. Wittgenstein famously said that ‘the world is all that is the case’.
Therefore, we know that anything that isn’t the case in not part of the world.

Arguments from authority are only as good as the reliability of the authority being quoted. As an independent and critical thinker, you should never wholly accept a claim just on the basis of an argument from authority. If you do, you have to accept that others might disagree with you simply because they do not trust that authority. Good critical thinkers must judge an argument for themselves, and not believe it just because others tell them to.

Paragraph 2 of Graham’s article claims that if a woman smokes while she is pregnant, both her children and grandchildren may be more likely to have asthma as a result. The support for this claim comes from three premises:

Premise 1:

children of women who smoked while pregnant were 1.5 times as likely to develop asthma as the offspring of nonsmokers were.

Premise 2:

If both the mother and grandmother smoked during pregnancy, the risk increased to 2.6 times that of children of nonsmokers.

Premise 3:

Most surprising, even when a mother did not smoke while she was pregnant her child had nearly double the risk of developing asthma as a child from a smoke-free home if her mother had smoked during pregnancy.

This form of argument is known as ‘Argument from induction’. Inductive arguments are based on evidence, observation and past experience. Most scientific arguments are inductive. If the premises are true, you have good reason to believe the claim, but arguments from induction are never certain – they are only probable (look at the language used in the claim: ‘may be more likely’). It is possible that further evidence could undermine the argument (just check the history of science to see!).

Some philosophers have been so upset by this observation that they refuse to accept any argument based on induction. Referring to the problem of induction, the 18th Century Scottish philosopher David Hume famously complained that the past was not a reliable guide to the future.

While inductive reasoning cannot rule out the logical possibility that things might change, it remains true that, in practice, we could neither live our daily lives nor do science if we did not place regular faith in inductive generalisations.

Other arguments from induction are

1. Every swan we have ever seen is white. Therefore, all swans are white.
2. The sun always comes up in the morning. Therefore, the sun will come up tomorrow morning.

You will notice that 1 and 2, both contain universal statements. These are statements that have the logical form ‘All As are B’ (like “all swans are white” in 1), or which can easily be parsed into that form (as “The sun always comes up in the morning” can be rephrased as “All mornings are mornings with sunrises”).

Most universal statements are inductive generalisations, meaning we think that they are true because we have observed instances of them many times. Nonetheless, be careful because some universal statements are not generalisations based on evidence. Consider, for example,

‘All unicorns have only one horn.’

This universal statement is not an inductive generalisation. It is a semantic tautology; that is, a statement that is true by definition: the meanings of the terms make the proposition as a whole necessarily true (you may remember we saw another kind of tautology in post 1: What is an argument).

Other universal statements like this are

‘Every bachelor is an unmarried man.’
‘No object can be coloured both red and green all over at the same time.’ (this is a logical impossibility ruled out by the meanings of “coloured all over” and “at the same time”).

For some universal statements, it remains unclear whether they are semantic tautologies or inductive generalisations. For instance, are the universal statements

‘Every event has a cause.’ and  ‘Nothing can be in two places at the same time.’

inductive generalisations, or simply true because of the meaning of the terms in the sentence? Philosophers and quantum physicists seem unable to agree, but for now it is enough that we recognise that universal statements usually express either inductive generalisations or semantic tautologies.

• premises can provide different kinds of support for a claim. Two kinds of support are ‘argument from authority’ and ‘argument from induction’
• universal statements have the form ‘All As are B’. Some universal statements are true because of the meaning of ‘A’ and ‘B’, while others are inductive generalisations based on past experience

Try Exercise 4 to test your understanding of this post, or continue reading

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