Chapter 11 Categorical Equivalence

11.1 Theory of Immediate Inference

In this chapter, we will discuss what kinds of inferences can be drawn from categorical statements before combining them into a syllogism. In other words, we will find out how to draw inferences from one statement. This is what is called the “theory of immediate inference.” When we introduced E and I statements, we discussed how the terms can be interchanged and the statements remain equivalent (i.e., if some bagels are toasted, then some toasted things are bagels, and if no markers are made of chalk, then no things made of chalk are markers). This was the relation of conversion. The theory of immediate inference helps us to better establish the context of statements and provides tools for properly translating arguments.

We will look at six relations between categorical statements, which are the product of manipulating the order of the terms, the quantity of the statement (whether it is universal or particular), and the quality of the statement (whether it is affirmative or negative). These relations are conversion, contraposition, obversion, contradiction, contrariety, and subcontrariety. As we will see, many of these relations will only hold based on the added condition that the classes in question are not empty.

11.2 Conversion

The converse of a categorical statement is the product of interchanging the statement’s subject and predicate terms. Looking back at the Venn diagrams, you can see that the diagrams for E and I categoricals are symmetrical with respect to the subject and predicate terms—if you switch the order, the diagram looks exactly the same.

The converse of the E statement “no snakes are poodles” is “no poodles are snakes,” and these are equivalent claims, so an E statement is logically equivalent to its converse. And the same goes for an I statement; you can see that an I statement and its converse are logically equivalent: “Some snakes are pretty things” is equivalent to “Some pretty things are snakes.” Conversion thus is the ground of an immediate inference between E and I statements.

A and O categoricals, however, are not equivalent to their converses. Because their diagrams are not symmetrical, interchanging the subject and predicate terms changes the diagram. It does not follow for the fact that all poodles are mammals that all mammals are poodles! Neither does it follow that if some mammals are not poodles, then some poodles are not mammals.

But remember that sometimes the fact that an entity exists in the unit class is relevant to the argument, and then you may need to opt for a particular categorical. Look back at the example about Kristin the professor parent for help here.

One way to better understand converse relationships is to look at the relationship of distribution. An A statement says something about all members of the class identified by the subject term. This is called distribution.

With the A statement “All dogs are mammals,” for example, a claim is being made about all dogs. But the same is not true for the predicate term: “mammals.” Other mammals, in addition to those identified as dogs, may exist, and so we cannot infer that all mammals are dogs. All A statements are the same in this regard: the subject term is distributed and the predicate term is not distributed.

Another way of understanding this is that an A statement distributes the subject to the predicate, but not the reverse. Distribution is a formal property of a categorical statement. E statements distribute both the subject and the predicate terms. One way of understanding this is that in an E statement, the subject and predicate terms both say something about each other. When we say that “no snakes are mammals,” we can infer that “no mammals are snakes.” We exclude mammals from snakeness, and we exclude snakeness from mammals. It goes both ways.

I statements do not distribute either term. I statements don’t say anything about either class (subject or predicate) because they just assert at least one thing exists in the overlap of classes.

O statements are problematic. Like I statements, O statements do not refer to all members of the subject class. But we could interpret them as saying something about the whole predicate class. Saying the statement “Some students are not engineers” is like saying that all engineers are excluded from the subgroup of students picked out by “some students.” In this view, it follows that the subject of any universal statement is distributed, but the subject of any particular statement is not. I statements do not distribute either term. And it would make sense that the predicate of any negative statement is distributed, but the predicate of any affirmative statement is not. This is nice and symmetrical, but it poses problems for existential import (inferences about what actually exists), as we shall soon see. As a result, we will take the view that neither term in an O statement is distributed.

11.3 Contraposition

We saw previously that the complement to a class is the class of everything not in the original class. For example, the class of non-dogs is the complement of the class of dogs. Equally, the class of dogs is the complement of the class of non-dogs. To obtain the contrapositive of a categorical statement, we first obtain the converse (switching the subject and predicate terms), and then we negate the terms by attaching a “non-” to both the subject and predicate terms.

For example, the contrapositive of “All pickles are green things” is “All non-green things are non-pickles,” which is true: if all the pickles are green things, then all the non-green things are non-pickles. But contraposition does not preserve truth in E statements. If “No pickles are red things” is true, it does not follow that no non-red things are non-pickles! Presumably, there are lots of non-red things that are not pickles. For I statements, if we are asserting that some S are P, then saying that some non-P are non-S cannot be inferred. For example, if we say some apples are red things, does it follow that some non-red things are non-apples? Keep in mind the question here. From the I statement, can it be inferred that some non-red things are non-apples from the very fact that some apples are red things? No, that cannot be inferred. We do not know what the rest of the universe of discourse contains. O statements do turn out to be equivalent to their contrapositive, and it involves some double negation. If we say, “Some philosophers are not artists,” we are saying that some philosophers exist outside of the intersection with the predicate class “artists.” Then in a very roundabout way, we talk about those philosophers as “non-artists” in the contrapositive and say that they are not non-philosophers (which is to say that they are philosophers). Then it is true that some non-artists are not non-philosophers.

11.4 Obversion

Obversion is the product of changing both the quality of a categorical statement (changing it from negative to affirmative or affirmative to negative) and replacing the predicate term with its complement (negating it by attaching a “non-” to the predicate term). Every categorical statement is equivalent to its obverse.

In obversion, we change the quality of the statement by changing the quantifier. “All” switches with “no,” and “some” switches with “some are not.” Then the predicate term is replaced by its complement, which we have seen. For an A statement, if we say that “all snakes are reptiles” it follows that “no snakes are non-reptiles.” They are all reptiles! If we say that “no snakes are mammals,” it follows that “all snakes are non-mammals.”

I and O statements require a bit of thinking. If some S are P—say, “Some artists are philosophers”—does it follow that some philosophers are not non-artists? Yes, because the two negations point us back to the category of artists. An O statement is a bit more straightforward. Consider some dogs are not poodles; it would follow then that some dogs are non-poodles (all the other dogs!).

11.5 Negation

We now turn to a discussion of negation. In traditional categorical logic (subject classes are not empty), there are three kinds of negation that form the basis for logical inference between categoricals: contradiction, contrariety, and subcontrariety. Of these, only contradiction holds as a matter of logic on the modern account (subject classes cannot be assumed to have members). The other two, contrariety and subcontrariety, do not hold universally in the modern form of categorical logic because they depend on the interpretation that the classes are not empty. But because this assumption is almost always the case in practice, it is useful to discuss all three forms of negation for the light that they shed on choices for translation. But it is important to keep firmly in mind that inferences involving contraries and subcontraries can only be made in contexts in which reasoners know that the classes of things under discussion are not empty, and that the inferences depend materially on that knowledge.

11.6 Contradiction

The contradictory of a categorical statement is the explicit denial of the whole statement. A categorical statement and its contradiction accordingly always have opposite truth values. This means that they cannot both be true, and they cannot both be false (at the same time). Another way of saying this is that if one is true, the other must be false.

Here you can see that A and O statements are contradictory (fig. 11.1). The A statement completely rules out the possibility of any S that is not P, whereas the O statement claims that there is indeed an S that is not P. A class cannot both be empty and have members at the same time.

Similarly, an E statement asserts that the SP area is empty—it is an impossibility—whereas the I statement asserts that there is an S that is P (fig. 11.2). Again, a class cannot both be empty and have members at the same time. Thus, E and I contradict each other.

11.7 Contrary and Subcontrary

Later in this text (Chapter 18), we will discuss the fallacy of bifurcation. This is where claims that are merely contrary are treated as contradictory. Recall that in a contradiction, the two claims cannot have the same truth value. In categorical logic, often the contrary is mixed up with contradiction. Previously, we established the contradiction is between asserting something exists in a class and asserting that same class is empty. In the case where subject classes are not empty, A and E statements are contraries and I and O statements are subcontraries.

Our immediate inference here is that if one is true, the other must be false. But if one is false, we do not know the truth value of the other. A contrary is identified by the relation between an A statement and an E statement (fig. 11.3).

Let’s look at an example. “All pickles are green,” and “No pickles are green.” It is pretty clear that these statements cannot both be true (all pickles are both green and not green at the same time), but can they both be false? Can it be false that both all and none are green? If it is false that all are green, then there is at least one pickle of another colour. If it is false that no pickles are green, then there is at least one green pickle (recall that E statements are contradicted by I statements). In both cases, the existence of at least one green pickle and at least one other colour pickle do not contradict each other, thus it can be consistent to have both statements be false. But keep in mind that in both cases, we are assuming that the classes are not empty—we are assuming pickles exist.

Adding to this discussion of contraries is the notion of subcontrary. I and O statements are subcontraries of each other (fig. 11.4).

Can it both be true that there is some S that is P and some S that is not P? If you look at the diagrams, you can imagine them overlapping and it wouldn’t be a big deal. There would be one “x” in the SP lens and one “x” in the S lune. In our previous example, we could say there is at least one green pickle and at least one pickle that is not green. These can both be true.

But what if they were both false? What if there is no “x” in SP and there is no “x” in S. Can that be true? It depends on whether the classes are empty. Let’s return to green pickles. If both the I and O forms are false, then there would be no pickles in the world (regardless of colour). So, the relation of subcontrariety relies on the condition that the classes are not empty.

11.8 Subaltern

Subalternation is the final ground for immediate inference we will discuss in this chapter. If subject classes are not empty, then subalternation holds between A and I categoricals and between E and O categoricals. Subalternation represents the fact that one can infer that “Some S are P” is true from the fact that “All S are P,” and that “Some S are not P” is true from the fact that “No S are P.”

Thus, if we know that there are ducks, and we know that all ducks are birds, we can infer that “some ducks are birds.” Similarly, if we know that there are snakes, we can infer from the fact that “no snakes are mammals” that “some snakes are not mammals.” Below, we introduce the traditional square of opposition, where you can see the subaltern relationship on the sides (fig. 11.5).

11.9 Traditional Square of Opposition

All these relationships of immediate inference are summarized in what is known as the traditional square of opposition. The relationships that it demonstrates were of central importance to the development of logic for over two thousand years.

With the development of modern logic and the mathematics of classes, the theory of immediate inference demonstrated in the square of opposition has declined in importance. Of the relations summarized by it, only contradiction holds for modern categorical logic.

In looking at the square, it is helpful to recall the truth-value relationship. The contradictions outlined demonstrate that they cannot both be true, and they cannot both be false; they must have opposite truth values. This is the case in both modern and traditional logic. On the traditional interpretation, the contraries cannot both be true, but they can both be false. Likewise, subcontraries cannot both be false, but they can both be true. Tracking each relation on the square, we can see that: