Part II, Lesson Part II, Lesson FourFour

The Opposition of The Opposition of PropositionsPropositions

The Rules of Truth and The Rules of Truth and FalsityFalsity

IntroductionIntroduction

Opposition between propositions occurs when we relate two propositions to each other.

We have already seen how to distinguish the parts of a proposition (subject, predicate, and copula), as well as the use of words within the proposition (supposition and distribution).

Now we will consider the ways of relating one proposition to another.

Opposition of Opposition of PropositionsPropositions

In general, opposition between two propositions occurs when one affirms and the other denies the same predicate of the same subject.

Example:All dogs are cats.

No dogs are cats.These two propositions are said to be opposed because one affirms and the other denies “cats”of “dogs”.

If a different predicate is used (or a different subject), then there is no opposition between the two propositions; they are merely different.

Example:All dogs are carnivorous.No dogs are rational.

Because we are not affirming and denying thesame subject of the same predicate, these propositions are not opposed to each other inany way.

Furthermore, in order to have opposition between two propositions, not only must the same subject and same predicate be used in each, but also they must have the same meaning and the same supposition. Nor is it permissible to use equivocal or analogous words.

Kinds of OppositionKinds of Opposition

There are different ways of affirming and denying the same predicate of the same subject, which gives rise to different kinds of opposition between propositions.

The distinction of the kinds of opposition has to do with both the quality (affirmative or negative) and the quantity (universal, particular, indefinite, or singular) of the two propositions.

1. Contradictory 1. Contradictory OppositionOpposition

When there is contradictory opposition between two propositions, one denies absolutely everything that the other affirms. They are as opposed as can be.

Example:All men are honest.

Some men are not honest.

At first glance, we might be tempted to think that the contradictory of “All men are honest” is “No men are honest”, as it seems that they are more opposed than the two mentioned previously. Yet this is not so; in order to refute the truth of the proposition “All men are honest”, it would be enough to show that some men are not honest, or even that one man is not honest. One exception would disprove the truth of the universal affirmative proposition.

Thus, in order to contradict the proposition,

“All apples are red”, all we need to show is that “Some apples are not red”, or even “This apple is not red”.

The contradictory proposition of a universal affirmative proposition is a particular negative proposition (using the same subject and the same predicate, of course.)

The same applies in the case of a universal negative proposition, whose contradictory will be a particular (or singular) affirmative proposition that uses the same subject and the same predicate.

Example:No exam is difficult.will be contradicted by

Some exams are difficult.or even by

This exam is difficult.

Contradictory opposition is opposition in truth and falsity.

This means that whenever we know that one of the two propositions with this kind of opposition is true, the other must necessarily be false. It is impossible that both be true or that both be false.

All men are honest.

Some men are not honest.

If the first proposition is false, the second mustnecessarily be true.

All apples are red.

This apple is not red.

No exam is difficult.

Some exams are difficult.

This exam is difficult.

2. Contrary Opposition2. Contrary Opposition

Contrary opposition exists between two propositions when both have universal quantity but one affirms and the other denies its predicate of the subject.

Example:All men are honest.

No men are honest.

At first glance, it might appear that this is a more radical type of opposition than contradictory opposition because “all” and “none” are extremes.

However, contrary opposition is in fact not as great as contradictory opposition because the contraries are opposed only in truth.

That is, it is impossible for both propositions to be true, but both may be false.

To say that contrary opposition between propositions is an opposition only in truth is to say that when one of the contrary propositions is true, its contrary must necessarily be false.

But if we only know that one of the two contrarypropositions is false, we cannot by that fact alone know that its contrary is true; it could be true or it might also be false.

Sometimes when one contrary is false, the other is true.

Example:

No man is rational. (False)

All men are rational.(True)

But other times, the contrary of a false proposition is also false.

Example:

All men are honest.

No man is honest.

(False)

(False)

In other words, when one of the contrary propositions is false, the other may be true or it may be false. In this case its truth is unknown.

Contrary propositions do not have as absolute an opposition as is found between contradictory propositions.

Contradictory propositions are opposed in truth and in falsity, but contrary propositions are only opposed in truth.

Also, contrary propositions are both universal. With contradictory propositions, one is universal and the other is particular or singular. Thus, contradictory propositions differ in quality and quantity, whereas contrary propositions only differ in quality.

3. Sub-contrary 3. Sub-contrary OppositionOpposition

Two propositions are in sub-contrary opposition when they differ in quality but are both particular.

Example:Some dogs are black.

Some dogs are not black.

Propositions in sub-contrary opposition are opposed in falsity only. That is, if one is false, the other is necessarily true.

However, it may be that both are true, as in the example just given.

Example:Some dog is black.

Some dog is not black.

Example:

Some dogs are cats.(False)

Some dogs are not cats.(True)

Summary of OppositionSummary of Opposition

1. Contradictory Opposition: One proposition denies the other

absolutely. Opposition in truth and falsity. The propositions differ in both quality and quantity.

To have opposition between two propositions,they must use the same subject and the same predicate.

2. Contrary OppositionOpposition in truth only. Both propositions are universal (one is affirmative, the other negative.)

3. Sub-contrary OppositionOpposition in falsity only.Both propositions are particular (one is affirmative, the other is negative).

The Relation of Sub-The Relation of Sub-alternationalternation

There is another possible relation between two propositions that use the same subject and the same predicate, but this is not a relation of opposition.

This relation, called sub-alternation, occurs when the propositions differ in quantity but not in quality (which is why there is no opposition between them.)

Example:All men are brave.

Some men are brave.

When the universal proposition is true, its subalternate must also be true.

If all we know is that the particular is true, this tells us nothing about the truth of the universal.

But if the particular is false, the universal must also be false.

Since subalternation is not a kind of opposition, there is no opposition in truth or falsity.

Yet we can conclude from the truth of the universal to the truth of the particular, or from the falsity of the particular to the falsity of the universal.

Example:

All strawberries are sweet.

Some strawberries are sweet.

If the universal is true, the particular is necessarilytrue as well.

And if it is false that

Some children are not human beings.

then it must necessarily be false that

No children are human beings.

The Square of OppositionThe Square of Opposition

The Square of Opposition is a very useful visual aid to understanding the consequences of the various relations of opposition and sub-alternation of propositions using the same subject and the same predicate.

It uses vowels to represent the main types of propositions:

A stands for the universal affirmative.E stands for the universal negative.I stands for the particular affirmative.O stands for the particular negative.

The Square of OppositionThe Square of Opposition

All men are honest.All men are honest. No men are honest. No men are honest.

AA EE

II OOSome men are honest.Some men are honest. Some men are Some men are

not honest.not honest.

The lines of the Square represent the three types of opposition and the relation of subalternation.

AO and EI (the diagonals) represent the propositions in contradictory opposition.

AE represents the propositions in contrary opposition.

IO represents the propositions in sub-contrary opposition.

AI and EO represent the relation of sub-alternation.

The Rules of Truth and The Rules of Truth and Falsity in the Square of Falsity in the Square of

OppositionOpposition

To use the Square of Opposition, the propositions must use the same subject and the same predicate with the same meaning, the same supposition (personal or simple) and must respect the difference between true universal names and collective names.

1. Two propositions in contradictory opposition cannot simultaneously be true, nor simultaneously false.

If one is true, the other will be false, and if one is false, the other will be true.

2. Two propositions in contrary opposition cannot be simultaneously true.

When one is true, the other will be false, but if one is false, the other will be unknown.

3. Two propositions in sub-contrary opposition cannot be simultaneously false.

If one is false, the other must be true, but if one is true, the other is unknown.

4. In the relation of subalternation, when the universal is true, the particular must also be true, and when the particular is false, the universal must also be false.

If the particular is known to be true, this tells us nothing about the truth of the universal (its truth is unknown.)

Similarly, when the universal is known to be false, the particular is unknown.

This is sometimes summarized by saying that we can descend with truth and rise with falsehood.

In this entire discussion, we have been examining what can be concluded from the formal relationship between propositions. To have a starting point (to know that a proposition is true or false), we need knowledge from some science outside of Logic. Logic can help us arrive at valuable consequences from the formal relationship between propositions once we have that starting point.

We can be quite sure that these consequences follow from the mere fact that propositions are related in this way, no matter what the subject matter being discussed.

We are always forced to distinguish between the matter and form of the propositions we use, between the subject matter and the form we use to express our knowledge of it. Elementary Logic is necessarily a consideration of the form of our expressions. Knowledge of the subject matter comes from other branches of knowledge.

Whenever we use words we are necessarily considering the “subject matter” of the proposition, that is, what the proposition means as well as what form it is expressed in.

In order to avoid being distracted unnecessarily by the content of propositions, we could merely use letters in place of actual subjects and predicates, to bring out more clearly the formal aspects of the propositions.

For example, we could avoid considering the specific subject matter by using expressions such as “All S is P” or “Some S is not P.”

All S is P.All S is P. No S is P. No S is P.

AA EE

II OOSome S is P.Some S is P. Some S Some S

is not P.is not P.