7.3 Resisting scientism

We have much to learn from science, and it is one of our chief windows onto the world and defences against ignorance. However, scientific writing often leaves much to be desired. There is nothing wrong with the passage below, but what exactly – if anything – does it prove?

Exercise 7.3

It follows from relativity theory that one cannot speak of an event occurring at precisely the same time for different observers. Each observer’s time frame is relative to himself. Imagine an observatory on Jupiter looking at an observatory on Earth. In each observatory, an astronomer looks through his eyeglass at the other at, it seems, exactly the same time. Since light takes 35 minutes to travel from Jupiter to Earth, the event on Jupiter in which the astronomer looks through his telescope must have taken place 35 minutes before the astronomer on Earth observes the event. Equally, the same applies to the astronomer on Jupiter. It is tempting to think there is some absolute position in which the two events could be observed as simultaneous, but this is exactly the possibility ruled out by relativity theory. Space and time are not independent dimensions, but form a four-dimensional unity, ‘space-time’, in which every event can only be recorded relative to a local time frame.

Type of argument: _____________________________________________
Claim: _________________________________________________________

It is a good / bad argument because __________________________________ ______________________________________________________________________

When you are ready, check out the answer and explanations here.

Alternatively, continue reading

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7.2 Resisting obscurantism

Read the following text, and answer the questions that follow. Pay careful attention to the questions and whether the text supports or fails to support the assertion in each case.

Exercise 7.2

There are numerous theories about the origins of a person’s sexual orientation; most scientists today agree that sexual orientation is most likely the result of a complex interaction of environmental, cognitive and biological factors. In most people, sexual orientation is shaped at an early age. There is also considerable recent evidence to suggest that biology, including genetic or inborn hormonal factors, plays a significant role in a person’s sexuality. In summary, it is important to recognise that there are probably many reasons for a person’s sexual orientation and the reasons may be different for many people.

a. This argument concludes that sexual orientation is largely determined by one factor.
True / False?
b. This passage tells us which factors determine one’s sexual orientation.
True / False
c. The author gives her opinion as to the most likely cause of a person’s sexual orientation.
True / False
d. The author tells us which is the best theory about the origins of sexual orientation.
True / False

When you’re ready, check the answer and explanation here, or continue reading

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7.1 Resisting statistics

The argument in the following exercise looks fairly straightforward at first blush, but statistical data should always be treated with caution. Try to analyse the argument in terms of its claim and premises. Does all the information add up to the conclusion, or is there reason to think the conclusion should be doubted?

Exercise 7.1

Living in the US makes you fat.* A team at Northwestern University in Chicago investigated whether there were changes in the waistlines of immigrants, most of whom come from countries with lower rates of obesity than the US, after moving to one of the world’s fattest countries. The team pulled data from the 2000 National Health Interview Survey, which is based on detailed interviews with 32,374 adult US residents. The Northwestern team found that after a year or less in the US, only 8 per cent of immigrants were obese. But after 15 years of American life, the figure rose to 19 percent – approaching the 22 per cent of US-born survey respondents who were obese. The 15-year link with obesity held for immigrant whites, Latinos and Asians, but not for foreign-born blacks. Significant weight gain did not appear until immigrants had lived in the US for at least 10 years. The team says this threshold may reflect how long it takes new residents to adopt the high-calorie diet and sedentary lifestyle that can make US-born Americans so large.

Type of argument: _________________________________________
Claim: _________________________________________________________

It is a good / bad argument because __________________________________

*This text has been adapted for non-commercial, educational purposes from here.

When you are ready, check out the answer here, or continue reading

7. Resisting persuasion

Analysing the validity of deductive arguments can be very useful. However, arguments can take many forms. We have seen three already: deductive, inductive, and argument from authority. Moreover, any given text often combines two or more forms of argument, and it is important to be aware of the competing demands mixed arguments make on the critical reader. The very best persuasive writers try to pummel their readers into submission with simultaneous assaults on several levels. The good critical reader needs to pull the different strategies apart and weigh them individually, resisting the temptation to be convinced by nothing other than sheer weight and variety of the assault.

As we have said, a claim that depends entirely on an argument from authority should be treated with caution, since you cannot weigh the argument yourself, you have to trust the authority. Moreover, whether the premises support the claim or not is more difficult with inductive arguments. Deciding whether an inductive argument supports the claim is a skill that you can only acquire with practice. It requires, amongst other things, the ability to imagine other possible explanations that would account for the evidence equally as well as the claim being offered.

In this post and the ones that will shortly follow, you will find a mixture of inductive, deductive and arguments from authority from a variety of sources, not just purely philosophical ones. Consider each carefully, and try to answer the questions. Here’s is the first. Study the answer key and explanatory notes before moving on to the next argument.

For the argument presented below, say what type of argument it is (deductive, inductive, argument from authority or a combination) and identify the claim. Then say WHY you think the premises either support or do not support the conclusion.

Exercise 7

Pythagoras proved that the squares on the two sides of a right-angled triangle are equal to the square on the hypotenuse. This can be seen by considering the figure below.

Square 1 and square 2 contain four triangles altogether. The large square 3 (top right), is the square on the hypotenuse c. It is the same size as the large square 4 (bottom left), which also contains four triangles. Therefore, the square on the hypotenuse c, is equal to the squares of the other two sides, a and b. Pythagoras sums this up in his famous equation, a2 + b2 = c2.

Type of argument: ___________________________________________
Claim: ________________________________________________________

It is a good/bad argument because _____________________________

When you’re done, you can check the surprising answer here. 🙂

Alternatively, continue reading the book…

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)
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
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)
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’


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

• 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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