La interculturalidad: más allá del etnocentrismo y relativismo cultural

One very common form of deductive argument that it is important to know about is called the syllogism. Aristotle constructed a system of evaluating syllogistic arguments as deductively valid or invalid, and this system was the backbone of logic in schools and universities for over two thousand years. A syllogism is a particular type of argument that always has two premises and a single conclusion, and all three statements are what are called categorical propositions. A categorical proposition is prefaced by the term ‘all’ or the term ‘some’. The following argument is a syllogism.

All stunt pilots are daredevils. Some stunt pilots are accountants.

Therefore, some accountants are daredevils.

A categorical proposition is made up of four components, the quantifier, the subject term, the copula, and the predicate term. A quantifier is of one of two types: the universal quantifier ‘all’ or the particular (existential) quantifier ‘some’. A term is a word that stands for a class of individuals, called the ‘extension’ of that class. For example, the term ‘stunt pilots’ stands for the class of stunt pilots. A copula is a form of the verb ‘is’ or ‘are’ that joins one term to another. The subject term stands for a class

3. Syllogisms 55

said to belong, or not to belong, to another class, denoted by the predicate term. In the example above, ‘Some accountants are daredevils’ is a cate- gorical proposition, because it can be paraphrased as ‘Some accountants are individuals who are daredevils’. Every syllogism must be made up of exactly three propositions, and each of these propositions must be a categorical proposition.

The negative expression ‘no’ is also allowed as a quantifier. So uni- versal negative generalizations, such as ‘No stunt pilots are cowardly’, are categorical propositions. To express negative existential statements, the particle ‘not’ is attached to the copula. For example, the sentence ‘Some stunt pilots do not have life insurance’ is a categorical proposi- tion of the negative existential type. This admission of negative as well as positive versions of the categorical propositions gives the theory of syllogistic inference a good generality. A syllogism, as a particular type of argument, however, is narrowly defined. There must be exactly three terms in it, and each term must occur exactly twice. One, the middle term, must occur once in each premise. The others, the end terms, must occur once in one of the premises only, and once in the conclusion. So the following inference is a syllogism, because it meets all of the above requirements.

All ducks are birds that have webbed feet. All mallards are ducks.

Therefore, all mallards are birds that have webbed feet.

There are only four types of propositions recognized as categorical propo- sitions in syllogistic reasoning. Where F and G are variables for terms, these four types are represented below.

A: All F are G: Universal Affirmative I: Some F are G: Particular Affirmative E: No F are G: Universal Negative O: Some F are not G: Particular Negative

A universal proposition, such as ‘All men are mortal’ makes an assertion about each and every individual referred to by the subject term (in this case ‘men’). A particular proposition asserts literally that at least one thing is both an F and a G. Unlike conversational English, where saying ‘Some

F are G’ suggests that there is more than one thing that is both an F and

affirmative proposition true is one thing that has both properties F and

G. The A proposition is the contradictory opposite of the O, and the I

proposition is the contradictory opposite of the E.

Now that we understand A, I, E, and O propositions, an easy method of evaluating syllogistic inferences by means of diagrams, called Venn diagrams, can be set out. Since the word ‘all’ in syllogistic reasoning is taken to mean ‘all without exception’ (a strict generalization), a syllo- gism is structurally correct only when it is deductively valid. According to the definition of deductive validity, an inference is deductively valid if and only if it is logically impossible for the premises to be true and the conclusion false. In other words, to say that a syllogism is deduc- tively valid is to say that if the premises are true, then the conclusion must be true too. Deductive inference, as noted above, is a kind of nec- essary inference, meaning that if the premises are true, no option is left but that the conclusion must be true as well. The mallards example is a valid syllogism because the premises are inconsistent with the oppo- site of the conclusion, ‘Some mallards are birds that do not have webbed feet’.

Let’s test the mallards example above for validity using a Venn dia- gram. We let D stand for ducks, B for birds with webbed feet, and M for mallards. Now we construct a Venn diagram with three intersecting cir- cles, where each circle represents one term.

D

M B

By convention, the middle term, the one occurring in both premises, is represented by the middle circle. Then we put each premise on the dia- gram, using shading to represent an empty class. Both premises have been marked on the diagram, using shading. Then we ask whether the

3. Syllogisms 57

conclusion also has to be represented on the diagram. It does, so we con- clude that the argument is valid.

Next, let’s test the stunt pilots argument above for validity.

S

A

X

D

As above, both premises have been represented on the diagram. Then we have to ask whether the conclusion is also true, according to the diagram. We see that it is, so we conclude that the argument is valid.

Now let’s test another syllogism for validity. All neurotics exhibit deviant behavior.

All obsessive-compulsives exhibit deviant behavior. Therefore, all neurotics are obsessive-compulsives.

Let’s put both premises of this argument on the following Venn diagram.

D

N O

By examining the diagram, it can be seen that even though both premises are true, it is possible that the conclusion is false. Thus, this argument is invalid.

Note, however, that even though an argument is deductively valid, it is possible for one or more of the premises to be false. For example, consider the following argument.

All crocodiles are friendly.

All friendly animals make good pets. Therefore all crocodiles make good pets.

This argument is deductively valid. If both premises are true, the conclu- sion has to be true too. It would be inconsistent to assert the following three statements.

All crocodiles are friendly.

All friendly animals make good pets. Not all crocodiles make good pets.

Even though the argument is deductively valid, as shown by the inconsis- tency of the three statements above, the first premise is false. The impor- tant point illustrated here is that just because an argument is valid, it doesn’t mean that the premises are true.

As can be illustrated by another syllogism, it is even possible to have a valid argument in which not only all the premises are false, but the conclusion is false too. Consider the following argument.

All crocodiles have six legs.

All animals with six legs make friendly pets. All crocodiles make friendly pets.

Even though this argument is deductively valid, both premises and the conclusion are false statements.

There are three general lessons that can be drawn. One is that a valid argument is not necessarily one that is a good argument in every respect. It may have false premises or even a false conclusion. Another lesson is that validity of an argument has to do with the link between the premises and the conclusion. A third lesson is that there are two distinct ways to attack a deductive argument. One is to show that the argument is not valid. This way is to attack the link between the premises and the conclusion, arguing that the conclusion does not follow from the premises. The other way is to attack one or more of the premises individually, arguing that it is not true. In the example above, the first premise can be attacked as a generalization that does not hold.

4. Complex Propositions 59

EXERCISE 2.3

Use Venn diagrams to determine whether the following syllogistic argu- ments are valid or invalid.

(a) All chameleons are lizards. All lizards are cold-blooded animals. Therefore, all chameleons are cold-blooded animals.

(b) All ducks are birds. No birds like chocolate sauce. Therefore, no ducks like chocolate sauce.

(c) All manufactured foods have a prolonged shelf life. Some foods that have a prolonged shelf life are made from unhealthy ingredi- ents. Therefore, some manufactured foods are made from unhealthy ingredients.

(d) Some budgies are birds that exhibit aggressive behaviors. Some birds that exhibit aggressive behaviors are dangerous to humans. Therefore, some budgies are dangerous to humans.

(e) Some mining companies are organizations that are concerned about the environment. Some organizations that are concerned about the environment approve of Greenpeace. Therefore, some mining com- panies are organizations that approve of Greenpeace.