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…

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