Posts Tagged ‘ inference ’

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’

that

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…

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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

3.1 Assessing arguments

Analysing an argument in terms of its claims and premises in itself tells us little. What it does do, however, is make it easier for us to judge or weigh the argument in at least two ways. First, we can examine each premise to decide whether it should be accepted. Second, we can begin to examine the relationship between the premises and the claim. In other words, we can question whether the premises which we are willing to accept make the claim believable or whether the premises could be accepted and the claim still rejected.

This second notion, the relationship between the premises and claim, is called ‘inference’, and we will be looking at it a lot more closely in subsequent posts. Although there are some specific techniques to help us examine inference, seeing it is, in large part, a natural skill of rational creatures, and we all do it unconsciously everyday in many ordinary ways. It is possible without any formal training whatsoever to make a judgment about whether you find a claim believable or not based on premises. What requires practice, and usually some degree of tuition, is the means by which to articulate our response.

To practice assessing an argument, re-read the argument in exercise 3 and your answers, then try exercise 3.1 here. Only after that should you engage with the answer key and notes.

Alternatively, continue reading the book…

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