The Opposition of Propositions

philosopherIntroduction to Logic, Part 6

By Jonathan Dolhenty, Ph.D.


We have learned something about the general and special types of propositions in a previous essay. It’s time now to investigate certain properties of these propositions as they are compared to one another. This brings us to the matter of logical opposition. Propositions are said to be logically opposed to each other when they have the same subject and predicate but with a change in quality or quantity or both.

The Nature of Logical Opposition

We have already learned that all truth is based on the three laws of thought known as the Principle of Identity, the Principle of Contradiction, and the Principle of the Excluded Middle. These three principles are the foundation for all human knowledge. They are self-evident and need no proofs or demonstrations. In fact, they cannot be “proved” in the ordinary sense of the term. If we reject them, however, we are at the end of rational discussion since we need to accept them as true in order to initiate and continue any rational discussion.

The Three Laws of Thought

There are three important laws of thought that every critical thinker needs to know. Without them, we would find it very difficult to reason correctly.

The Principle of Identity

The Principle of Identity states that “A is A.” Other ways of saying the same thing are “What is, is,” “Everything is what it is,” and “A thing is identical with itself.” Can we seriously challenge such a principle?

The Principle of Contradiction

The Principle of Contradiction states that “A cannot be A and not A at the same time in the same respect.” It can also be stated as “Whatever is, cannot at the same time not be under the same circumstances,” or “It is impossible for the same thing both to be and not to be at the same time from the same point of view.” From the standpoint of logic, the Principle of Contradiction can be read as “The same attribute cannot at one and the same time be both affirmed and denied of the same thing in the same respect.”

The Principle of Excluded Middle

The Principle of Excluded Middle can be stated in different ways: “A thing either is or is not,” “Everything must either be or not be,” and “Any attribute must be either affirmed or denied of any given subject.” For the purpose of the study of logic, the principle can be stated: “If we make an affirmation, we thereby deny its contradictory; if we make a denial, we thereby affirm its contradictory.”

Propositional Constructions

The logical opposition of propositions is the relation which exists between propositions having the same subject and the same predicate, but differing in quality, or in quantity, or in both.

There are four possible way in which a proposition having the same subject and the same predicate may appear:

1. as a universal affirmative (A)

  • “All men are content.”

2. as a universal negative (E)

  • “No men are content.”

3. as a particular affirmative (I)

  • “Some men are content.”

4. as a particular negative (O)

  • “Some men are not content.”

These four propositions represent the four types of opposition. They can be diagramed, together with their mutual relations as opposites, in what is called a Square of Opposition.

The Traditional Square of Opposition

Here are explanations of the four types of opposition and the four relations resulting from the opposition:

Subalternation

Subalternation: the opposition existing between a universal and particular affirmative (A and I), and between a universal and particular negative (E and O). Both propositions, the universal and the particular, are called subalterns. The universal is the subalternant (A and E). The particular is the subalternate (I and O).

Contradiction

Contradiction: the opposition existing between a universal affirmative (A) and a particular negative (O), and between a universal negative (E) and a particular affirmative (I).

Contrariety

Contrariety: the opposition existing between a universal affirmative (A) and a universal negative (E).

Subcontrariety

Subcontrariety: the opposition existing between a particular affirmative (I) and a particular negative (O).

Here is a diagram showing the logical opposition of propositions (with examples provided):

The Laws of Logical Opposition

We can now formulate certain laws of truth and falsity regarding propositions which contain these various relations.

The Law of Subalternation

The Law of Subalternation: A–I and E–O. This law has two phases, depending on whether we begin with the truth or the falsity of one of the subaltern propositions.

The First Rule
 

Beginning with the truth of one of the subaltern propositions (A–I, E–O), the first rule states: The truth of the universal involves the truth of the particular (A to I, E to O); but the truth of the particular does not involve the truth of the universal (I to A, O to E).

In other words:

  • If A is true, I must also be true.
  • If E is true, O must also be true.
  • If I is true, A need not be true, but is doubtful.
  • If O is true, E need not be true, but is doubtful.

There are, therefore, two sections to this first rule.

The First Section of the First Rule
 

The first section states: “It is always logical to conclude from the truth of the universal to the truth of the particular.” After all, what is true of all individuals of a class must also be true of some of these individuals. What is true of the whole must be true of every part of the whole.

Examples: If “All men are mortal,” then surely “Some men are mortal.” If “No men are dogs,” then “Some men are not dogs, either.”

The Second Section of the First Rule
 

The second section states: “The truth of the particular does not involve the truth of the universal; the truth of the universal will always be doubtful.” What is true of some need not be true of all. What is true of a part of a class need not be true of the whole of the class.

Examples: If it is true that “Some men are content,” we cannot conclude, on the basis of the proposition alone, that “All men are content.” If it is true that “Some men are not content,” we cannot conclude that “No men are content.”

We can see from the examples that the truth of the particular propositions (I and O) does not involve the truth of the universals (A and E). Although the particular propositions I and O are true, their respective universals A and E are false.

It could happen, of course, that what is true of some is also true of all and what is true of a part is also true of the whole. In this case, both the particular propositions (I and O) are true, and their respective universals (A and E) are also true. But we are never permitted to conclude from the truth of the particular to the truth of the universal. It may be so, but it need not be so. We cannot validly argue from some to all and from the part to the whole.

The Second Rule
 

The second rule of the Law of Subalternation states: The falsity of the particular involves the falsity of the universal; but the falsity of the universal does not involve the falsity of the particular. Here we begin with the falsity of one of the subaltern propositions (I to A, O to E). The rule states:

  • If I is false, A is also false.
  • If O is false, E is also false.
  • If A is false, I need not be false.
  • If E is false, O need not be false.

There are also two sections to this second rule.

The First Section of the Second Rule
 

The first section states: “We can always validly conclude from the falsity of a particular proposition to the falsity of the universal.” This makes sense. For something to be true of all, it must be true of every individual that belongs to the all. For something to be true of the whole, it must be true of every part contained in the whole.

Example: The particular proposition I, “Some men are dogs,” is false. Actually it would be true to say that “Some men are not dogs.” In order, however, for the statement to be true that “All men are dogs,” it could not be true to say that “Some men are not dogs,” because all must include some, and the whole must include every part.

Another example: The particular proposition O, “Some men are not mortal,” is false. We should say “Some men are are mortal.” But if proposition E, “No men are mortal,” is true, it would follow that the same some are and are not mortal at the same time.

If it is false that “Some men are dogs,” it is all the more false to state that “All men are dogs.” If it false to say that “Some men are not mortal,” it is also false to say that “No men are mortal.”

From the falsity of the particular proposition (I or O), we must conclude to the falsity of the respective universal proposition (A or E).

The Second Section of the Second Rule
 

The second section of this rule states: “If A is false, I need not be false; if E is false, O need not be false.” In order that a universal be true, every individual of the class and every part of the whole must be included in the truth of the universal. The universal, therefore, will be false if not every individual of the universal and not every part of the whole is included in the truth of the universal statement.

This means that if a universal proposition is false, some of its individuals must also be false, but some of the others may be true. But if some may be true, even if the universal is false, it is obvious we cannot validly conclude from the falsity of the universal to the falsity of the particular.

We can now see the truth of the Law of Subalternation. The truth of the universal involves the truth of the particular, but the truth of the particular does not involve the truth of the universal. The falsity of the particular involves the falsity of the universal, but the falsity of the universal does not involve the falsity of the particular.

The Law of Contradiction

The Law of Contradiction: A–O and E–I. This law has two phases.

The First Rule
 

The first rule states: “Contradictories cannot be true together.”

  • If A is true, O is false.
  • If O is true, A is false.
  • If E is true, I is false.
  • If I is true, E is false.

In an affirmative universal (A) proposition, it is asserted that the predicate is affirmed of each and every individual belonging to the subject, as in, for example “All men are mortal.” If this is true, then it must be false to deny this statement of some of the individuals. Therefore, the statement that “Some men are not mortal” (O) cannot be true.

In a negative universal (E) proposition, it is asserted that the predicate must be denied of each and every individual belonging to the subject, as in, for example “No men are dogs.” If this statement is true, then it must be false to say that “Some men are dogs” (I).

What is true of all, must be true of every one of the class. To state at the same time that all are and some are not, and that none are and some are, would violate the Principle of Contradiction.

The Second Rule
 

The second rule states: “Contradictories cannot be false together.”

  • If A is false, O is true.
  • If E is false, I is true.
  • If O is false, A is true.
  • If I is false, E is true.

Example: If it is false that “All men are content,” it must be true that “Some men are not content” (A–O). If it is false that “No men are content,” it must be true that “Some men are content” (E–I).

Another Example: If it is false that “Some men are not mortal,” it must be true that “All men are mortal.” If it is false that “Some men are dogs,” it must true that “No men are dogs.”

We now can state the following conclusions:

  • From the falsity of the affirmative universal (A) follows the truth of the particular negative (O).
  • From the falsity of the universal negative (E) follows the truth of the particular affirmative (I).
  • From the falsity of the particular negative (O) follows the truth of the universal affirmative (A).
  • From the falsity of the particular affirmative (I) follows the truth of the universal negative (E).

The Law of Contrariety

The Law of Contrariety: A–E. There are two rules to be considered.

The First Rule
 

The first rule states that “Contraries cannot be true together.” If A is true, E is false and if E is true, A is false. If one of the contraries is true, the other contrary must be false.

Example: If “All men are mortal” (a universal affirmative proposition–A) is true, then “No men are mortal” (a universal negative proposition–E) must be false. If A is true, E is false.

Another example: If “No men are dogs” (a universal negative proposition–E), then “Some men are dogs” (a particular affirmative proposition–I) must be false. The universal affirmative proposition–A) “All men are dogs,” must also be false.

The Second Rule
 

The second rule states: “Contraries may be false together.” If one contrary is false, the other contrary may also be false, although it need not be false, and may be true.

Example: Consider the proposition “All men are content.” This is a universal affirmative proposition (A) and let’s consider this false. Since this statement is false, its contradictory, a particular negative proposition (O) “Some men are not content,” must be true. But the Law of Subalternation states that the truth of the particular proposition does not involve the truth of the universal. Therefore, although it is true that “Some men are not content,” we cannot validly conclude from this that its universal (“No men are content”) is also true. “No men are content” (E) may be true or false. Therefore, both contraries may be false.

The second rule is established. From the truth of one contrary we can conclude to the falsity of the other; but from the falsity of one contrary we cannot conclude to the truth of the other.

The Law of Subcontrariety

The Law of Subcontrariety: I–O. There are two rules to this law.

The First Rule
 

The first rule states: “Both subcontraries cannot be false together.”

The first rule says:

  • If I is false, O is true.
  • If O is false, I is true.

Example: Consider the statement “Some men are dogs.” Let’s say that this particular affirmative (I) proposition is false. Since this is false, its contradictory (a universal negative–E) must be true, that is, “No men are dogs.” If a universal proposition is true, its particular proposition is also true (the Law of Subalternation). So, since E is true, O must also be true and must state that “Some men are not dogs.” If I is false, O is true.

Another example: Let’s say that O is false, that is, “Some men are not mortal.” Its contradictory A, that is, “All men are mortal,” must be true (the Law of Contradiction). But if A (“All men are mortal”) is true, then I (“Some men are mortal”), must also be true (the Law of Subalternation). If O is false, I must be true.

We can see now the truth of the first rule regarding subcontrary propositions (I and O). Subcontraries cannot be false together, at least one of the two must be true.

The Second Rule
 

The second rule of subcontraries (I and O) states: “Both subcontraries may be true together.”

  • If I is true, O may be true.
  • If O is true, I may be true.

Example: Let’s suppose it’s true that “Some men are content,” (a particular affirmative proposition–I). The contradictory of this proposition, “No men are content” (a universal negative proposition–E), must be false. We know, however, that the falsity of the universal does not involve the falsity of the particular (the Law of Subalternation). Therefore, even though E (“No men are content”) is false, we cannot conclude that O (“Some men are not content”) is false. This proposition may be true.

Another example: Let’s suppose that “Some men are not content” (a particular negative–O) is true. It’s contradictory, “All men are content” (a universal affirmative–A), is false. We cannot conclude, however, from the falsity of the universal to the falsity of its particular (the Law of Subalternation), so it does not follow that I (“Some men are content”) is also false. The statement “Some men are content” may be true.

The two rules regarding subcontrary propositions (I and O) have now been established. Both subcontraries cannot be false together, but both subcontraries may be true together.

Summary of the Laws

We can now state the following conclusions:

  • If A is true: then I is true, E is false, O is false.
  • If A is false: then O is true, E is doubtful, I is doubtful.
  • If E is true: then O is true, A is false, I is false.
  • If E is false: then I is true, A is doubtful, O is doubtful.
  • If I is true: then E is false, A is doubtful, O is doubtful.
  • If I is false: then O is true, A is false, E is true.
  • If O is true: then A is false, E is doubtful, I is doubtful.
  • If O is false: then I is true, E is false, A is true.

The summary diagrammed:

A
E
I
O
If A is true
False
True
False
If A is false
Undetermined
Undetermined
True
If E is true
False
False
True
If E is false
Undetermined
True
Undetermined
If I is true
Undetermined
False
Undetermined
If I is false
False
True
True
If O is true
False
Undetermined
Undetermined
If O is false
True
False
True

Logical Opposition of Modals

The Square of Opposition refers solely to categorical propositions which do not contain a mode affecting the copula. The way we treat modal propositions is similar to ordinary categorical propositions, but the logical opposition affects the mode itself. (Remember there are four modes we learned about.)

  • The necessary mode resembles the A proposition. “Man must be moral.”
  • The impossible mode resembles the E proposition. “Man cannot be moral.”
  • The possible mode resembles the I proposition. “Man can be moral.”
  • The contingent mode resembles the O proposition. “Man need not be moral.”

Obviously, the subject of the modal proposition may be either a universal term or a particular term (all, no, or some).

The opposition becomes more complicated thereby, but the general scheme must be carried out according to the logical opposition intended.

If the logical opposition intended affects both the quantity and the mode of the propositions, the Square of Opposition would be as it appears here. Thus,

 
On the other hand, if the logical opposition affects only the mode, but not the quantity, of the propositions, the quantity (all or some) will remain the same and only the mode will change.

 
Or,

 
From these relations of opposites it will be clear that we are often entitled to conclude from the truth or falsity of one proposition to the truth or falsity of another. This method of concluding from the truth or falisty of one statement to the truth or falsity of another is called immediate inference; it is called immediate because we can pass directly from the one to the other, without the necessity of adducing any other idea or judgment as proof.

The Square of Opposition and the Three Laws of Thought are sufficient to make their truth or falsity evident, provided we know beforehand that one of these opposites is true or false.

The Square of Opposition (with its relations of subalternation, contradiction, contrariety, subcontrariety) will act as a powerful aid toward correct thinking.

Equivalent Sentences

Besides the immediate inference of logical opposition, we have the immediate inference of eduction.

Eduction is a mental process whereby, from any proposition taken as true, we derive another proposition implied in it, though differing from the first proposition in subject or predicate or both. There are three main forms of eduction: obversion, conversion, and contraposition.

The purpose of this technique is to transform certain sentences into other sentences which are equivalent in meaning, but may have a different logical form. The advantage of this is that an argument which may not be in strict syllogistic form can be transformed into a syllogism.

Consider the following argument:

  • No unwise people are trustworthy,
  • All wise people are unaggressive,
  • Therefore, no trustworthy people are aggressive.

This argument appears to be valid but we can’t test it by the rules already discussed because it contains more than three terms. It appears, in fact, to have five terms: unwise people, trustworthy people, wise people, unaggressive people, and aggressive people.

But look at the second premise. It means the same thing as “All aggressive people are unwise.” So, if we substitute this latter sentence for the original sentence, we get the following argument:

  • No unwise people are trustworthy,
  • All aggressive are unwise,
  • Therefore, no trustworthy people are aggressive.

The argument now contains only three terms (unwise, trustworthy, aggressive) and is a syllogism. It can now be tested using the standard rules for testing the validity of syllogisms. We can then see that the argument is valid.

Main Forms of Eduction

Obversion

Obversion is an eduction in which the inferred judgment, while retaining the original subject, has for its predicate the contradictory of the original predicate.

The original proposition is called the obvertend and the inferred proposition is called the obverse.

In obverting a given sentence we do two things:

  • We change the quality of the sentence. If it’s negative, we make it affirmative. If it’s affirmative, we make it negative.
  • We then negate the predicate.

Example: Consider the sentence “All men are mortal.” First, we change the quality, and the sentence becomes “No men are mortal.” Then we negate the predicate and the sentence becomes “No men are non-mortal.” The sentence “No men are non-mortal” is equivalent to the sentence “All men are mortal.”

Every A, E, I, and O sentence can be obverted. Study the following diagram:

Type of
Sentence
Original
Obverse
A
All men are mortal. No men are non-mortal.
E
No men are mortal. All men are immortal.
I
Some men are mortal. Some men are not immortal.
O
Some men are not mortal. Some men are immortal.

Care must be exercised in obverting sentences in ordinary language since some negative English terms can be confusing and some terms, which may appear to negate, do not negate at all. For instance, “large” is not the negation of “small.” Certain prefixes (“im,” “un,” “in”) do not always express negation. Logicians prefer, therefore, to use the prefix “non” in order to negate the predicate. Therefore, the negation of “rich” is not “poor,” but “non-rich.”

When obverting, there must be a change in the quality of the sentence only. Do not change the quantity. Then a universal sentence remains a universal sentence and a particular sentence remains a particular sentence.

Conversion

Conversion is an eduction in which the inferred judgment takes the subject of the original proposition for its predicate, and the predicate of the original proposition for its subject. In other words, when we convert we merely interchange subject and predicate.

The original proposition we call the convertend and the inferred proposition we call the converse.

Example: Consider the sentence “No dogs are horses.” This sentence is equivalent to the sentence “No horses are dogs.”

Conversion is unlike obversion because not every standard sentence has an equivalent converse. Only the E and the I sentences can be converted.

Example: The O sentence cannot be converted. From “Some woman are not nuns,” we cannot infer “Some nuns are not woman.”

Example: The A sentence cannot be converted simply. From “All dogs are animals,” we cannot infer that “All animals are dogs.” It is possible, however, to partially convert the A sentence, using a technique logicians call Conversion by Limitation. When we convert a true A sentence, we can transform it into a true I sentence. The sentence “All dogs are animals” can be partially converted into “Some animals are dogs.” Partial conversion, however, does not give us a sentence which is exactly equivalent in meaning to the original. This is because the quantity of the original sentence is changed.

Study this diagram of permissible conversions:

Type of
Sentence
Original
Converse
E
No men are mortal. No mortals are men.
I
Some men are mortal. Some mortals are men.
A
All men are mortal. Some mortals are men. (Partial Converse)

Contraposition

Contraposition is an eduction in which the subject of the inferred proposition is the contradictory of the predicate of the original proposition. It is (read carefully!) the obverse of a converted obverse.

In order to obtain the contraposition of a sentence, three operations must be performed:

  • First we obvert.
  • Then we convert.
  • Then we obvert once again.

Example: Consider the sentence “All dogs are animals.”

  • Step 1: Obvert – “All dogs are non-animals.”
  • Step 2: Convert – “No non-animals are dogs.”
  • Step 3: Obvert – “All non-animals are non-dogs.”

Contraposition cannot be applied to all four standard sentences. The A sentence and the O sentence have contrapositives. The I sentence has no contrapositive and the E sentence has only a partial contrapositive. Usually, contraposition is only applied to A sentences.

You should now be able to transform large parts of ordinary language into arguments of a syllogistic form, making it possible to test the validity of the argument.

The chart below will help you see how to form equivalent sentences.

Type
Form
Obverse
Converse
Contrapositive
A
All S is P No S is non-P Some P is S All non-P is non-S
E
No S is P All S is non-P No P is S Some non-P is not non-S
I
Some S is P Some S is not non-P Some P is S None
O
Some S is not P Some S is non-P None Some non-P is not non-S

The Moral Liberal recommends: Great Books of the Western World


The late Dr. Jonathan Dolhenty was the Founder and President of The Center for Applied Philosophy and the Radical Academy, and is Honorary Philosophy Editor at The Moral Liberal. The Moral Liberal has adopted these projects beginning with a republishing and preserving of all of Dr. Dolhenty’s work.


Copyright ©1992 -2011 The Radical Academy. Copyright renewed in 2011 -2014 © The Radical Academy (a project of The Moral Liberal).


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