Posts Tagged ‘ syllogism ’

6. Syllogisms

Although understanding the internal relation between subject and predicate provides a powerful tool for codifying the rules by which one sentence can be validly inferred from another, arguments in public debates are far more complex than the simple immediate inferences used in the examples earlier posts. In order to develop a practical guide to sound reasoning, Aristotle developed his system to cover more complex arguments.

To see how we can start to understand natural arguments, consider the old anecdote of an elderly priest who warned his young successor of the stresses and disturbing nature of taking a confessional. By way of example, the elder priest recounted how the first confession he ever heard was a confession of murder. Sometime after the elderly priest’s departure, the new priest began meeting his congregation. A local businessman welcomed him to the community and commented that he had known the previous priest very well. “Indeed,” he remarked, “I was his very first penitent.”

Consider how the anecdote above might be laid out in formal terms. The young priest receives two pieces of information from separate sources

1. The elderly priest’s first confessor is a murderer
2. The local businessman was the elderly priest’s first confessor

From these, the young priest can validly conclude a third, and new piece of information that will be true just so long as premises 1 and 2 are true, namely

3. The local businessman is a murderer

As before, the terms in the argument can be replaced with symbols to reveal its logical form. Using M to represent ‘the elderly priest’s first confessor’, P for ‘a murderer’ and S for ‘the local dignitary’, gives:

1: M is P (the elderly priest’s first confessor is a murderer)
2: S is M (the local dignitary is the elderly priest’s first confessor)
therefore
3: S is P (the local dignitary is a murderer)

In fact, Aristotle argued that all arguments, no matter how lengthy or complex, could ultimately be analysed into the form of two premises and a conclusion, which he called a ‘syllogism’. Just as with statements, where the actual content or meaning of the terms involved is irrelevant to the validity of the inference, so the same is true of the syllogism.

Another syllogism already considered is

1: Bertrand is a philosopher
2: All philosophers are clever
therefore
3: Bertrand is clever

Substituting S for ‘Bertrand’, M for ‘philosopher’ and P for ‘clever’, the argument can be simply symbolised as

1: S is M (Bertrand is a philosopher)
2: M is P (All philosophers are clever)
therefore
3: S is P (Bertrand is clever)

One of the most important features of the syllogism is that in the two premises there are three terms, but only one of those terms is common to both premises. In the example above, it is the term ‘philosopher’. This common term – called the middle term and represented by M in both examples – is what licences the conclusion, which always features the other two terms and never the middle term. This is a most important discovery. For it means that we can work out whether any conclusion is the result of a valid inference by purely formal means:

If the term represented by M appears in both premises 1 and 2, we know that any conclusion with M in it will not be a valid argument.

Since the middle term can appear as either subject or predicate in either premise, there are four possible arrangements of the terms, known as “the four figures of the syllogism”. Notice that in the two examples above, the arrangement of terms is such that in both cases the M terms are diagonally opposed. In the first example, the M terms were diagonally opposed from top left to bottom right:

M is P (the elderly priest’s first confessor is a murderer)
S is M (the local dignitary is the elderly priest’s first confessor)
/S is P (the local dignitary is a murderer)

In the second example the diagonal opposition was reversed, with M appearing top right and bottom left:

S is M (Bertrand is a philosopher)
M is P (All philosophers are clever)
/S is P (Bertrand is clever)

There are two other possible variations, where the M terms appear either both on the left hand side or both on the right hand side. These variations make up the four possible figures of the syllogism. But since each premise in the syllogism can be universal or particular, affirmative or negative, it turns out that combined with the four figures there are 256 possible variations of syllogism (4 x 4 x 4 x 4). Of these, only nineteen turn out to be forms of valid inference. Despite there being many different kinds of syllogism, Aristotle showed how it was possible to investigate the structure of the syllogism without needing to list all the various kinds. Since only nineteen are valid, rather than expect his students to learn all 256 variations, Aristotle and his successors identified certain rules of syllogism that were common to all, and only, the valid nineteen.

In fact, there are many rules of syllogism and many names for the fallacies committed by using the remaining 237 invalid patterns, but we need not go into all those here. For present purposes, it will suffice to record one important pattern that Aristotle noticed about the 19 valid forms of syllogism. In any valid argument with a particular statement as a conclusion, the premises must contain one universal and one particular statement. Again, this means that any argument with a particular conclusion can quickly be ruled out as invalid if it contains two universal or two particular statements, without regard to the meaning or distribution of the terms.

This is a very useful conclusion, but it also raises a difference between natural language and logical symbolism that we must be wary of. In natural language, a statement like

‘The elderly priest’s first confessor is a murderer’ looks clearly like a particular statement. However, from a logical point of view, since the statement is true for all substitutions (i.e., no matter who was the first confessor that person was a murderer), it can be treated in logic as a universal:

‘Anyone who was the priest’s first confessor is a murderer.’

Try exercise 6 to test your understanding of this post, or continue reading

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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

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