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Philosophy 103: Introduction to Logic
The Traditional Square of Opposition

Abstract:  The group of logical relations forming "the square of opposition" are explained and illustrated.  These relations are called contradictory, contrariety, subcontrariety, and subalternation.
I. First, review the major terms introduced previously:   name, form, quantity, quality and distribution.
II. The Traditional Square of Opposition
A. Logical opposition occurs among standard form categorical propositions if ...
1. they have the same subject and predicate terms and...
2. they differ in quality or quantity or both.
B. Consider the kinds of opposition that can arise with differing quantity and quality:
Case 1: differ the quantity and quality.
Case 2: differ the quality, but not the quantity.
Case 3: differ the quantity, but not the quality.
Case 4: same quantity and quality ( the trivial case).
III. Case 1: The propositions have different quantity and quality.
A. Suppose we have an A proposition: "All philosophers are idlers."
1. What proposition is the denial of this statement? I.e., what proposition differs in both quantity and quality?
2. The particular negative, or the O proposition so differs:

"Some philosophers are not idlers."

3. Consider the following logical geography of the two statements.  Note how one is the denial of the other. In the A the shading indicates no individual present; in the O the "X" indicates at least one individual is present. Put together, they yield:

venn_a.gif (1566 bytes)
venn_o.gif (1466 bytes)
4. This kind of opposition is called contradiction and is defined as follows: Two propositions are contradictories if they cannot both be true and they cannot both be false. In other words, the statements have opposite truth values.
B. Suppose we have an E proposition: "No philosophers are idlers."
1. What proposition is the denial of this statement? I.e., what proposition differs in both quantity and quality?
2. The particular affirmative or an I proposition is so opposed:

"Some philosophers are idlers."

3. Notice what would happen if both of these statements could be true at the same time. The lens area of the diagram would have both shading and an "X."  This state of affairs is, of course, logically impossible.

venn_e.gif (1533 bytes)
venn_i.gif (1452 bytes)
4. This kind of opposition is also called contradiction. Note that there is a kind of symmetry. Hence, we do not have to examine the I and O propositions separately in order to find their contradictories. We already know them.
5. These results can be summarized by the diagram of the Square of Opposition.
IV. Case 2: (first part) the propositions differ in quality but are the same in quantity.
A. Suppose we start again with an A proposition:

"All philosophers are idlers."

What proposition is the same quantity but differs in quality?

B. The universal negative or the E proposition does so:

"No philosophers are idlers."

C. This kind of opposition is called contrariety. A and E are contraries.
D. Two propositions are said to be contraries if they cannot both be true, although they might both be false.
If both the A and the E could be true at the same time, then the subject class would be empty. In the traditional square, we assume that the subject of the proposition refers to something that exists.

venn_a.gif (1566 bytes)
venn_e.gif (1533 bytes)
Case 2 (second part): Suppose, on the other hand, we start with an I proposition:

"Some philosophers are idlers."

What proposition is the same in quantity, but differs in quality?

A. The particular negative or the O proposition does so:

"Some philosophers are not idlers."

B. This logical relation is called subcontrariety. Two propositions are said to be subcontraries if they cannot both be false, although they might both be true. In other words, I and O are subcontraries of each other.
What would happen if both the I and O statements could be false?  The diagram shows that if they were then the subject class would have to be the empty class!

venn_e.gif (1533 bytes)E is a false I
venn_a.gif (1566 bytes)A is a false O
C. We can now summarize our results before moving to Case 3. I and O can both be true but they need not be.  The only thing known is that they cannot both be false.

venn_i.gif (1452 bytes)
venn_o.gif (1466 bytes)
V. Case 3 (part 1): The propositions are the same in quality but differ in quantity.
A. Again, let us start with the A proposition:

"All philosophers are idlers."

What proposition is the same in quality but differs in quantity?

1. The particular affirmative or the I proposition does so:
"Some philosophers are idlers."
2. If the A statement is true, we know that the I statement has to be true, (unless there are no members of the subject class, i.e., it's an empty subject class).
3. Note carefully the following truth relations for this logical relation called subalternation:
If A is true, then I is true. (Otherwise, the subject class is empty.)
If A is false, then I is undetermined in truth value.
If I is true, then A is undetermined in truth value.
If I is false, then A is false. (Otherwise, the subject class is empty.)
B. These are summarized on the Square of Opposition.
Case 3 (part two): our last analysis. Let us look at the E proposition:

"No philosophers are idlers."

A. What proposition is the same in quality but differs in quantity from the E? The O statement does so since it is particular and negative.

"Some philosophers are not idlers."

B. A quick look at the Venn Diagrams yields the following truth values listed below. venn_e.gif (1533 bytes)
venn_o.gif (1466 bytes)
If E is true, then O is true. (Otherwise the subject class is empty.)
If E is false, then O is undetermined in truth value.
If O is true, then E is undetermined in truth value.
If O is false, then E is false. (Otherwise, the subject class is empty.)
C. In sum, then E and O are related by subalternation. The logical relation described above is called subalternation. E is often termed the "superaltern" of O, the "subaltern."
All four logical relations on the Square of Opposition are sketched out in the summary chart:  Try the practice quiz on the Square also.Return to Logic Homepage
 
 
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