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

REVIEW
• 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

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