Whether a categorical sentence is true (or false) depends entirely on the relation between the extensions of its subject and predicate. Obviously, a sentence of the form “All S are P” (SaP) will be true if and only if the extension of S (the class of the things that are S) is a subset of the extension of P (the class of the things that are P). For example, “All cats are intelligent” will be true if and only if the class of cats is a subset of the class of intelligent beings (the notion of subset allows the classes to be the same class – the case when all cats are intelligent and nothing else is).
Let us look at some elementary relations between two sets and hence between the extensions of two terms:
X and Y are different classes only if they have different members – if their members are the same, they are the same class. We say that a class X is a subset of a class Y (in symbols, X ⊆ Y) if and only if all members of X are members of Y. This definition entails that X will be a subset of Y also when the elements of X and Y are the same and therefore they are the same class – each class is a subset of itself (in symbols, X ⊆ X). If X is a subset of Y and Y is a subset of X, their members will be the same and they will be the same class. If X is a subset of Y but Y is not a subset of X, in other words if all members of X are members of Y but Y has members that are not in X, we say that X is a proper subset of Y (in symbols, X ⊂ Y).
Two classes are disjoint, or mutually exclusive, if and only if they have no common elements. For example, the class of all cats and the class of all parrots are disjoint since there is no a thing that is both a cat and a parrot. On the other hand, the class of opera singers and the class of drivers are not disjoint since there is at least one opera singer who is a driver.
Two classes are intersecting if and only if they have at least one common element. It follows that X and Y are intersecting exactly when they are not disjoint.
The following diagrams show all possible relation between the extensions of two terms S and P:^{1}
1) corresponds to the case when the extension of S is a proper subset of the extension of P. In this case all S are P, but not all P are S. 2) corresponds to the case when the extension of S is the same class as the extension of P. In this case all S are P, and all P are S. 3) represents the case in which the extensions of S and P are intersecting but neither is a subset of the other. In this case some S are P and some P are S but it is not true that all S are P nor is it true that all P are S. 4) represents the case when the extensions of S and P are disjoint (do not have common elements). In this case no S are P and no P are S. Finally, 5) corresponds to the case in which the extension of P is a proper subset of the extension of S. In this case all P are S, but not all S are P.
As mentioned above, a universal affirmative statement “All S are P” will be true if and only if the extension of S is a subset of the extension of P. Moreover, the extension of S does not have to be a proper subset of the extension of P – they might be the same class. For example, the sentence “All squares are rectangles with equal sides” is true because the class of all squares and the class all rectangles with equal sides are the same class. Therefore, an A-sentence is true in cases 1) and 2) and is false in the remaining three cases – in all of them there are S that are not P. On the other hand, a particular negative sentence (“Some S are not P”) is true in cases 3), 4), and 5), as then there are S that are not P, and is false in cases 1) and 2), as then there are no S that are not P. So, an O-sentence is true in the cases in which an A-sentence is false and is false in the cases in which an A-sentence is true. This means that an universal affirmative (A) and a particular negative (O) sentences that have the same subject and predicate are exact negations of one another – they always have the opposite truth values. For example, if it is true that all cats are intelligent, it will be false that some cats are not intelligent, and if it is false that all cats are intelligent, it will be true that some cats are not intelligent. So, we have the following two equivalences:
SaP ⇔ ¬SoP |
¬SaP ⇔ SoP |
To say “All cats are intelligent” is the same as to say “It is not true that some cats are not intelligent”, and to say “It is not true that all cats are intelligent” is the same as to say “Some cats are not intelligent”. This, by the way, gives us another alternative way of asserting universal affirmative and particular negative sentences – by negating the corresponding particular negative or universal affirmative sentence, respectively.
The above diagram also shows that the same relation exists between universal negative (E) and particular affirmative (I) sentences with the same subject and predicate. An E-sentence (“No S are P”) is true only in case 4), when the extensions of S and P are disjoint. In this case the I-statement (“Some S are P”) is false. In all remaining cases, the E-statement is false and the I-statement is true. Therefore, they always have the opposite truth value. For example, if it is true that no cats are intelligent, it will be false that some cates are intelligent, and if it is false that no cats are intelligent, it will be true that some cats are intelligent. E- and I-sentences with the same subject and predicate are exact negations of one another:
SeP ⇔ ¬SiP |
¬SeP ⇔ SiP |
To say “No cat is intelligent” is the same as to say “It is not true that some cats are intelligent”, and to say “It is not true that no cat is intelligent” is the same as to say “Some cats are intelligent”. We have another alternative way of asserting universal negative and particular affirmative sentences – by negating the corresponding particular affirmative or universal negative sentence, respectively.
Тo illustrate the logical relations between the four categorical sentences when they have the same subject and predicate, a diagram with the shape of a square was invented in the Middle Ages called the square of opposition. In it, the four categorical sentences are located at the four vertices of a square as follows:
The universal sentences are on the top, the particular are on the bottom, the affirmative are on the left and the negative are on the right. In addition, the diagonals of the square are drawn. In this way for each pair of sentences there is a line corresponding to the logical relation between them.
We already discussed the relations on the diagonals of the square – between A and O sentences, on the one hand, and Е and I, on the other. We saw that they are negations of one another and always have the opposite truth value. Such categorical sentence, are called contradictories. Universal affirmative and particular negative sentences (with the same subject and predicate) are contradictories, and so are universal negative and particular affirmative sentences. This is the most important relation in the square of opposition.
Of two contradictories, the one is true and the other is false – they cannot be both true or both false. |
If we know the truth value of an arbitrary categorical sentence, we can infer the truth value of its contradictory – it is the opposite one (false if the first sentence is true and true if it is false).
Let us now consider the relation between a universal affirmative (A) and a universal negative (E) sentence with the same subject and predicate, to which the upper side of the square corresponds. Clearly, if “All S are P” is true, then “No S are P” cannot be true and vice versa – if no S are P, it is not possible for all S to be P^{2}. So, the two universal sentences (with the same subject and predicate) cannot be both true. However, they can be both false if some S are P and some are not. For example, if some cats are intelligent but others are not, the sentences “All cats are intelligent” and “No cats are intelligent” will be both false. Such sentences that cannot be both true but can be both false are called contraries. The universal affirmative (A) and the universal negative (E) sentences are contraries.
Two contraries cannot both be true but they can both be false. (Also, one of them can be false and the other true.) |
Therefore, if one of the universal sentences is true, the other universal sentence is necessarily false. In other words, from the truth of a universal sentence (affirmative or negative) we can infer the falsehood of the other. However, the situation is not the same in the other direction (from falsehood to truth). If “All S are P” is false, it is possible that no S are P (in which case “No S are P” will be true) but it is also possible that some S are P and other S are not P (in which case “No S are P” will be false). For example, if it is not true that all cats are intelligent (if the sentence “All cats are intelligent” is false), it is logically possible that some cats are intelligent and some are not (in which case “No cats are intelligent” will be false). However, it is also logically possible that there are no intelligent cats (in which case “No cats are intelligent” will be true). Therefore, if we know that an A-sentence is false, we cannot say solely on logical grounds what the truth value of the corresponding E-sentence is – it may be true or it may be false. In this case we say that the E-sentence is undetermined, meaning that its truth value is not determined by the truth value of the contrary.
Similarly, if an E-sentence is false, the corresponding A-sentence is undetermined. If it is not true that no cats are intelligent (if the sentence “No cats are intelligent” is false), then it is possible that all cats are intelligent but it is also possible that some cats are intelligent and some are not. Under the first possibility the sentence “All cats are intelligent” will be true and under the second it will be false.
Particular sentences (affirmative or negative) are called subaltern to the corresponding universal sentences. The particular affirmative “Some S are P” (SiP) is subaltern to the universal affirmative “All S are P” (SaP) and so is the particular negative “Some S are not P” (SoP) to the universal negative “No S are P” (SeP). Thus, the left and the right sides of the square of opposition represent the relation of subalternation between categorical sentences with the same subject and predicate.
According to traditional logic, each general sentence logically entails its subaltern, i.e. particular affirmative sentences are logical consequences of the general affirmative sentences with the same subject and predicate and particular negative sentences are logical consequences of the corresponding universal negative sentences At first glance, this seems obvious. For example, if all cats are intelligent, it seems trivially true that at least one of them is intelligent, and similarly if no cats are intelligent, at least one of them has to be unintelligent (because all of them are such). However, an essential requirement to be entitled to draw such conclusions is the existence of at least one thing that is S. The reason is as follows. If there are no S, the particular affirmative and the particular negative sentences are both false since they can be rephrased as “There is an S that is a P” and “There is an S that is not a P”, and since there is no S whatsoever, they are both false. However, in that case the universal affirmative and the universal negative sentences are true, since we can rephrase them respectively as “If anything is an S, it is a P” and “If anything is an S, it is not a P”. Those sentences are not committed to there being an S and therefore they are not false when there are no S. Thus, if the class of all S is empty, the universal sentences are true and the particular are false, which means that the latter cannot be logical consequences of the former. Traditional logic implicitly assumes that whenever we are asserting general sentences (whether affirmative or negative) the extension of their subjects (S) are not empty, which has as a consequence that, for that logic, the universal sentences entail their subalterns:
SaP ⇒ SiP |
SeP ⇒ SoP |
Subalternation is the only relation in the square of opposition that represents logical inference, i.e. whereby the truth of a sentence may be validly inferred form the truth of another. The other relations enable us to infer the truth of a sentence from the falsehood of another or vice versa.
If a universal statement (A or E) is false, its subaltern (I or O, respectively) is undetermined (it might be true as well as false). For example, if all we know about cat intelligence is that it is false that all cats are intelligent, then there are two possibilities – either no cats are intelligent, in which case the subaltern (“Some cats are intelligent”) will be false, or some cats are intelligent and others are not, in which case the subaltern will be true. Similarly for E and O-sentences.
Particular affirmative (I) and particular negative (O) sentences with the same subject and predicate are called subcontraries. Characteristic of their relation, represented by the bottom side of the square, is that they can both be true but cannot both be false. Imagine that some cats are intelligent and some are not. Then the I-sentence (“Some cats are intelligent”) and the O-sentence (“Some cats are not intelligent”) will both be true. On the other hand, if it is false that some cats are intelligent, then no cats are intelligent and therefore (trivially) some cats will be not intelligent. So, if “Some cats are intelligent” is false, the subcontrary “Some cats are not intelligent” will be true. Similarly, in the other direction, if a particular negative (O) sentence is false, the subcontrary (I) sentence will be true. Compared to contraries, subcontraries behave in the opposite way – they can both be true but cannot both be false, while contraries can both be false but cannot both be true.
Based on the relations in the square of opposition, if we know the truth value of a categorical sentence, we are able to infer the truth values of (some of) the other three types of sentences having the same subject and predicate. For this, we do not actually need all the relations in the square. For example, using only the relations on the diagonals (between contradictories) and on the upper side (between contraries) will suffice for any such inference. Let us see why.
Suppose that we know that some particular negative (O) sentence is true and we want to know the truth values of the other three sentences with the same subject and predicate. On the diagonals, contradictories always have opposite truth values, so the A-sentence is false. Then, taking the relation on the upper side of the square, we know that if a universal sentence is false, the contrary is undetermined. As the A-sentence is false, the E-sentence is undetermined. Next, using the relation on the diagonals, we may infer that the contradictory I-sentence is also undetermined. The reason for this is that contradictory sentences are like communicating vessels – they always have the opposite truth values. Therefore, if we were able to say what the truth value of the O-sentence is, the contradictory A-sentence would have not been undetermined. So, given the truth value of an O-sentence, we were able to determine (as far as possible) the truth values of the other three sentences with same subject and predicate using only the relations between contradictories and contraries. We saw that if an O-sentence is true, the A-sentence is false and the E and the I-sentences are undetermined.
Now let us see what the other three sentences are if an O-sentence is false. Across the diagonal, the contradictory A-sentence (the negation of O) is true. By the relation on the upper side of the square, if a universal sentence is true, the contradictory is false. Therefore, the E-sentence is false. From the falsehood of the E-sentence, we conclude that the contradictory I-sentence on the other side of the diagonal is true. Again using only the relations on the diagonals and the upper side of the square, we got that if a particular negative (O) sentence is false, the corresponding A and I-sentences are true and the E-sentence is false.
In a completely analogous way, if we know the truth value of an A, E, or I-sentence, we can use the relations between contradictories and contraries to determine the truth values of the other three sentences with the same subject and predicate. Generally, we will always have one of the following two results. Either we will be able to determine only the truth value of the contradictory sentence (the other two remaining undetermined, as was the case with the O-sentence being true above), or we will be able determine the truth value of all three sentences (as was the case with the O-sentence being false).
This exhausts the content of the square of opposition. Two are the most important things in it, which we will continue to use. 1) The universal affirmative (A) and the universal negative (E) sentences are exact negations of the particular negative (O) and the particular affirmative (I) sentences respectively, which is why they always have the opposite truth value. 2) (According to traditional logic) the particular affirmative (I) and the particular negative (O) sentences follow logically from the universal affirmative (A) and universal negative (E) sentences respectively, i.e. a particular sentence is implied by the general sentence with the same quality.
The table below summarizes the relations between sentences in the squire of oppositions.
If a sentence is | true | false |
its contradictory is | false | true |
its contrary is | false | undetermined |
its subaltern is | true | undetermined |
its subcontrary is | undetermined | true |
(1) If the truth value of a categorical sentence is as follows, what are the truth values of the other three types of sentences having the same subject and predicate? |
1) | A universal affirmative sentence is true. |
2) | A universal affirmative sentence is false. |
3) | A universal negative sentence is true. |
4) | A universal negative sentence is false. |
5) | A particular affirmative sentence is true. |
6) | A particular affirmative sentence is false. |
(2) Assuming that the following sentences are true, what may we infer about the truth values of the other three types of sentences with the same subject and predicate? |
1) | Some successful businesspersons are intelligent people. |
2) | No animals with horns are predators. |
3) | All uranium isotopes are highly unstable substances. |
4) | Some good writers are not entertaining lecturers. |