Archive for March, 2011

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

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…

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

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.

REVIEW
• 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

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…

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.

and

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.

REVIEW
• 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

2. Hidden premises

Often, premises are implicit or hidden in an argument. This means they are not mentioned but are assumed – either knowingly or unknowingly – by the speaker or writer. Reconsider example 5 from the previous post:

5. Life on Earth is in deadly peril. 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. If there were less carbon dioxide in the atmosphere, the greenhouse effect would not be running out of control.

In fact, this argument only works if we assume the truth of another premise, namely, that ‘When the greenhouse effect runs out of control, life on earth is in deadly peril’. Often, hidden or implicit premises like these are not mentioned simply because they are obvious.
Here’s another argument with an implicit premise:

6. H2O is abundant on Earth. 70% of the world’s surface area is water.

The implicit premise is that H2O and water are the same thing, but why mention something in an argument which no one finds controversial? It would be tiresome to have to mention everything presupposed by an argument’s conclusion when most of it is not in dispute. That being said, however, it is essential that the philosophy student realises that hidden premises are particularly important precisely because in complex arguments on controversial issues, there are often implicit premises that may not be recognised by one or more parties. What is more, it is these that often turn out to be the very premises which are responsible for the controversy. Here is a good example:

7. Murder is always wrong. Even though the state sanctions capital punishment, clearly capital punishment is wrong.

The claim in this argument cannot be established unless one agrees to the hidden premise that ‘capital punishment is murder’. Whether one agrees with the argument’s claim that ‘capital punishment is wrong’ will turn precisely on this hidden – and controversial – premise rather than on the two uncontroversial stated ones.

If you think carefully about any argument, you will almost always find a hidden premise. This is because speakers and writers usually have a common background with their audience, so that some information is unnecessary to mention. However, you should always think about hidden premises and weigh up whether they are significant or not. The argument in 7. has many implicit premises including, for example, ‘the State has the power to punish people’. However, this is not significant because it is irrelevant to establishing the claim – it makes no difference to the argument one way or the other.

REVIEW
• All philosophically-interesting arguments contain hidden premises
• Hidden premises need to be recognized and assessed as to their significance

Try Exercise 2 to test your understanding of this post, or continue reading

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