CHAPTER 43. THE LOGIC OF CLASSES.

1. Subsumptive or Nominal.

2. Classes.

3. Classes of Classes.

 

1. Subsumptive or Nominal.

We have to distinguish between the subsumptive use of a word, and its (say) ‘nominal’ use. In the former case, the word has an only incidental role, serving to direct our minds to the objects we label by it; in the latter case, the word itself is the object of our attention, while the things it refers to are incidental. Thus, for example, when we speak of dogs, we think of our tail-wagging and barking friends; but when we speak of “dogs”, we mean the word enclosed by inverted commas itself.

This should not be confused with the distinction between denotative and connotative terms, which we made in discussing permutation. We can take denotative terms subsumptively or nominally (as we did with above, comparing dogs and “dogs”), and we can take connotative terms likewise subsumptively or nominally (for example, caninity and “caninity”). That is not what is at issue here. What is at issue is, whether our focus is purely objective (as in, dogs and caninity), or we are focusing on the instrument (as in, “dogs” and “caninity”).

Thus, in subsumptive intent, we mean what the word refers to, and the instrument is transparent; whereas in nominal intent, the instrument itself is what we mean, and what the word refers to specifically is of lesser moment. Normally, our intent is subsumptive (let us symbolize this as an X); but sometimes, especially in epistemological discussions, our intent is nominal (let us symbolize this as “X”).

Nomenclatural propositions have the primary forms: “X” is the name of all X; or: all X are the referents of “X”.

We may extend the distinction between subsumptive and nominal intent to other aspects of our instruments of thought, not only to the verbal. An ‘idea’ or a ‘class’, or any similar construct, may like a word be considered ‘nominally’, in contrast to subsumptively. This does not mean to imply that words, ideas, classes, and such, are all one and the same thing, but only that they have in common the property we mentioned. There is no doubt that they are significantly different concepts, yet also somehow related.

The precise relation between these various concepts is not the topic of this chapter. Rather, we shall specifically explore some of the mechanics of classification, in an effort to better understand the logical relations between things and our concepts of them. This research into the way knowledge is organized, has been of great interest to logicians of this century, under the heading of ‘the logic of classes’.

 

2. Classes.

We think of a class and its members, as having a similar relation to that of a receptacle and the things it contains. The container, an elastic and permeable wrapping, is a figment of our imagination; yet, its shape and size are determined by the contents. This visual analogy is not perfect, but is a starting point.

The subsumptive outlook is directed at the contents, labeling each member as X; this is the only kind of classificatory relationship we have dealt with so far: it is the concern of Aristotelean logic. The nominal outlook is directed at the container, labeling the class as “X”; this gives rise to a new field of logic.

a. Definitions.

For any thing X, we can invent a corresponding thing “X”, such that:

· whatever is X, is a member of “X”; and whatever is not X, is not a member of “X”.

Conversely, we say:

· “X” is the class of (all) X; and “X” is a class of anything which is X, and not a class of anything which is not X.

These two sets of statements mean the same thing, they are just two sides of the same coin, they commute each to the other; we call the whole relationship ‘class-membership’.

The above begin to formally define the difference between what we mean by X and “X”, relating them through a new pair of copulas, which are different from the copula ‘is’. In one direction, the copula is labeled ‘is a member of’, and has a subsumptive as subject and a nominal as predicate; in the other direction, the copula is labeled ‘is a class of’, and has a nominal as subject and a subsumptive as predicate.

Since in speech, unlike writing, we have no way to display inverted commas, we merge the two and say: this thing is a member of the class of X; the latter expression, class of X, is equivalent to “X”, in relation to X. We understand “X”, or the class of X, as a mental construct of some sort, which we intend to bear a certain relation to the things we have labeled as X. We assign the virtually same label to the construct as we did to the original things, except for a small distinguishing mark (“” or the class of) to keep their distinction in mind.

The plain name is subsumptive, referring directly to the things concerned, the marked name is nominal, referring rather to the invented correspondent of the things. Note that, although the member to class relation has some similarity to the relation of an individual to a group, they are not identical. The subsumptive versus nominal distinction, should not be confused with the dispensive versus collective (or even collectional) distinction, which we made earlier (in the discussion of quantity).

Thus far, what we have done is to point to a set of phenomena, which we commonly encounter in our current ways of thinking, and sorted them out somewhat, and named the various factors. But all we have achieved is at best a technical definition; a fuller definition requires some further understanding of the distinctive properties of these factors. That is what we will look into now.

Consider the following example, which accords with our normal manner of speaking. Dogs are dogs, and are members of “dogs” (or the class of dogs). But, dogs are not “dogs”, only members of “dogs”; and “dogs” is not a dog, and not a member of “dogs”. Note that there is no self-contradiction in saying that dogs are not “dogs”, or that “dogs” is not a dog, even though the statement that dogs are not dogs is of course absurd.

Such examples suggest the following features and processes. (Note that I concentrate mainly on the properties of ‘is a member of’; those of ‘is a class of’ follow obviously, I do not highlight them, to avoid repetitions.)

b. Features.

Whereas in a proposition of the form ‘this thing is X’, the subject and predicate are both subsumptive — in a proposition of the form ‘this thing is a member of “X” (or the class of X)’, the predicate is nominal. This principle is necessary, because the whole concept of membership was built with the intent to study that special kind of term we call nominal. Membership by definition relates any kind of thing to one kind of thing specifically: mental constructs.

With regard to the subject of membership, the above definition concerns only subsumptive subjects, but we shall presently consider nominal ones.

What is X, is not “X”, but only a member of “X”. The copula ‘is a member of’ positively relates two things, X and “X”, which the copula ‘is’ negatively relates, at least in examples like ours (dogs are not “dogs”, but only members of “dogs”).

“X” is not an X, nor a member of “X”. A class is not a member of itself: the relation of membership is not reversible, at least not in examples like ours (“dogs” is not a member of “dogs”, since it is not a dog).

With regard to the latter two principles, the examples only prove that they hold in some instances; however, we may generalize from such cases, if we find no examples to the contrary.

c. Immediate inferences.

Obviously, by definition, since all X are X, all X are members of “X”; and since no nonX are X, no nonX are members of “X”. The class of X includes all things which are X, and excludes all things which are not X. Similar eductions apply for the class of nonX, or “nonX”. It follows that membership in “X” and membership in “nonX” are exact contradictories.

More broadly, we can infer from the above definition of membership that: if any X is Y, that X is a member of “Y”; and if any X is not Y, that X is not a member of “Y”. Any thing which is X and also Y, is an X which is a member of “Y”; any thing which is X but not Y, is an X which is not a member of “Y”. That is, “X” is the class of all X, but not the only class for any X; there are normally other classes like “Y”, of which we can say that it is a class of some or all X. For examples, retrievers are members of the class of dogs, but not members of the class of cats.

It follows that: if all X are Y, then all X are members of “Y”; if only some X are Y, then only some X are members of “Y”; if no X is Y, no X is a member of “Y”. (Note in passing, in the middle case, we regard the membership of some X in “Y”, as ‘accidental’, or ‘incidental’ to their being X, since not all X fall in this category; or we say that these X are Y, but not ‘qua’ X or not as X ‘per se‘, not by virtue of being X.)

These statements are reversible: if all X are members of “Y”, then all X are Y; if some X are members of “Y”, then some X are Y; if some X are not members of “Y”, then some X are not Y; if no X are members of “Y”, then no X are Y.

d. Deductive arguments.

We thus can construct the following syllogisms for the copulas ‘is (or is not) a member of’, on the basis of Aristotelean syllogisms for the copulas ‘is (or is not)’.

Figure 1.

All Y are members of “Z”,

all/this/some X is/are member(s) of “Y”,

so, all/this/some X is/are member(s) of “Z”.

Likewise, with negative major and conclusion.

Figure 2.

No Z are members of “Y”,

all/this/some X is/are member(s) of “Y”,

so, all/this/some X is/are not member(s) of “Z”.

Likewise, with positive major and negative minor.

Figure 3.

All/this/some Y are members of “Z”,

some/this/all Y is/are member(s) of “X”,

so, some X are members of “Z”.

Likewise, with negative major and conclusion.

Figure 4.

No Z are members of “Y”,

some Y are member(s) of “X”,

so, some X is/are not members of “Z”.

Such deductions are easily validated, by translating them into their customary forms. Note that a term may be subsumptive in one proposition and nominal in another, according to its position by virtue of the figure.

e. Modal class-logic.

The definition of class-membership is easily modalized, if we wish to work out a more modal class logic.

Thus, for natural modalities: if something can be X, then it can be a member of “X”; and if something cannot be X, it cannot be a member of “X”; if something must be X, then it must be a member of “X”; and if something can not-be X, it can not-be a member of “X”. Similarly for temporal modalities. The quantification of these singular forms represents extensional modality.

Note that these definitions are in the form of extensional conditionals. The logical properties of their consequent forms are easily derived from the modal logic of their antecedent forms, which are ordinary categoricals. That includes: oppositions, eductions, and deductions.

 

3. Classes of Classes.

In the previous section, we defined and analyzed the membership of a non-class (subsumptive) in a class; now, we need to look into what we mean when we say of a class (nominal) that is a member of another class.

a. Definitions.

We propose that, for any X and Y:

· if all X are Y, then “X” (or the class of X) is a class of Y, and therefore is a member of “classes of Y”, (or the class of classes of Y).

Conversely:

· if less than all or no X are Y, then “X” (or the class of X) is not a class of Y, and therefore not a member of “classes of Y” (or the class of classes of Y).

Note the variety in wording; we also often abbreviate ‘class(es) of Y’ to ‘Y-class(es)‘.

This definition of so-called classes of classes reflects our common practise. For examples, since all dogs are animals, “dogs” is an animal-class, or a member of “animal-classes”; but since some dogs are not black animals, “dogs” is not a class of black animals, or a member of “classes of black animals”.

Now, this is an artifice. The reason why we construct this new concept is that we want to be able to talk about classes in the same way as we talk about things. We build up a parallel domain, in which classes bear relations to each other, somewhat similar to the relations between their ultimate referents. Thus far, the stratification of things had no equivalent in the realm of classes, since nominal terms were defined as predicates of the ‘is a member of’ copula. In order to place classes as subjects of similar propositions, we introduce appropriate special predicates: classes of classes. A class of classes is a subsumptive whose referents are specifically nominal.

Note that an ordinary class (that is, one which is not a class of classes) stands as subject of membership when the predicate is a class of classes; there are no grounds for assuming that an ordinary class can ever be a member of another ordinary class. We cannot, for instance, say “dogs” is a member of “animals”, but only, dogs are members of “animals”, or “dogs” is a member of “animal-classes”.

This was already suggested in the previous section, in the claim that “X” is not a member of “X”; now, we can generalize further, and say that “X” cannot be a member of any “Y”, granting that these terms are not classes of classes of anything. Other than the above defined case, there are no known conditions regarding any X and Y, under which we could conclude that “X” is a member of “Y”.

Similarly, there are no known conditions under which propositions of the form: “X-classes” is a member of “Y”, may arise. However, as we shall presently see, propositions of the form: all/some X-classes are (are not) members of “Y-classes”, do indeed arise, directly out of the definition of classes of classes. However, note that the subject is subsumptive here, not nominal.

Let us now investigate how successful our above definition of classes of classes is, some of the logical properties it implies.

b. Features.

“X” is an X-class, and a member of “X-classes” (or the class of X-classes), since all X are X, and even though “X” is not an X, nor a member of “X”. This principle proceeds deductively from the definition, by substituting X where we find Y. It means that every class is a member of the class of classes bearing its name. It does not mean that it is a member of itself, however; we should not confuse a class with a class of classes; thus far, we have no cause to doubt the earlier postulate that classes cannot be members of themselves. For example, “dogs” is a dog-class, and a member of “dog-classes”.

However, no X is an X-class, nor a member of “X-classes”, even though all X are members of “X”, and “X” is a member of “X-classes”. The definition of a class of classes refers to a nominal “X” as its subject, not a subsumptive X. The relationship of membership is not passed on all the way down the chain to the individuals subsumed by X; the only individuals subsumed by a class like “X-classes” are classes like “X”. For example, dogs are members of the class of dogs, but not of the class of dog-classes.

Similarly, no X is a Y-class, nor a member of “Y-classes”, even if all X are Y, and therefore members of “Y”. Contrast those statements to saying that “X” is a Y-class (or a class of Y), and therefore a member of “Y-classes” (or the class of Y-classes, or the class of classes of Y). Keep the distinctions clear.

We might strengthen these insights by calling ordinary classes, classes ‘of the first order’, and classes of classes, classes ‘of the second order’; then we can say: members of a class of the first order cannot be members of a class of the second order; at best, they might be said to be members of a member of a class of the second order. This may be referred to as the principle of intransmissibility of membership across orders of classification.

c. Immediate Inferences.

Obviously, by definition, if “X” is a Y-class, then all X are Y; and if “X” is not a Y-class, at least some X are not Y. Likewise, with any of the alternative wordings.

The following theorems are important, because they construct propositions in which a class of classes is the subject, a novelty; thus far, classes of classes only appeared as predicates.

If all X are Y, then all X-classes (including “X” itself) are Y-classes, or members of “Y-classes”, the class of Y-classes. Proof is by exposition: consider any class “W” which fits the definition of an X-class, so that all W are X, then (since all X are Y) all W are Y, and it will follow that “W” is a Y-class; this can be repeated for any “W”, and even “X” fits in (since all X are X). For example, all dog-classes (such as “retrievers”) are animal-classes.

A corollary is: if “X” is a Y-class, then all (other) X-classes are (also) Y-classes; the conclusion follows, since the premise implies that all X are Y.

If some X are Y, then some X-classes are Y-classes. Proof: those things which are both X and Y can be said to be XY, and self-evidently all XY are X and all XY are Y; thus we have, in the case of “XY” at least, an X-class which is a Y-class.

If some X are not Y, then some X-classes are not Y-classes. Proof: those things which are X but not Y can be said to be XnonY, and self-evidently all XnonY are X and no XnonY are Y; thus we have, in the case of “XnonY” at least, an X-class which is not a Y-class.

If no X is Y, then no X-classes are Y-classes. For if, say, “W” is an X-class, then all W are X; and since no X is Y, it follows that no W is Y, which means that “W” is not a Y-class.

Thus, note well, if some X are Y, it follows only that some X-classes are Y-classes, for we may find a class “W” (other than “X”) for which all W are X and yet no W is Y. Likewise, if some X are not Y, it follows only that some X-classes are not Y-classes, for we may find a class “W” (other than “X”) for which all W are X and also all W are Y.

Conversely, if all X-classes are Y-classes, then all X are Y; if some X-classes are Y-classes, then some X are Y; if some X-classes are not Y-classes, then some X are not Y; and if no X-classes are Y-classes, no X is Y.

d. Deductive arguments.

It is important to note that syllogistic reasoning with the copula ‘is a member of’ depends for its validity on the manner of reference of its terms.

We saw that, if any X is a member of “Y”, and “Y” is a member of “Z”, it follows that that X is a member of “Z”. The proof being, since that X is Y, and all Y are Z, then that X is Z.

However, if even all X are members of “Y”, and “Y” is a member of “Z-classes”, it does not follow that any X is a member of “Z-classes”. For, even though it be implied that all X are Z, this only signifies, as already pointed out, that “X” is a member of “Z-classes”, not that any X is a Z-class.

Thus, we have the same arrangement of premises, with the copula ‘is a member of’ in both cases, yet the conclusions are of fundamentally different form. In the former case, subsumptives are members of an ordinary class; in the latter case, a nominal is member of a class of classes. This of course illustrates the earlier mentioned principle of intransmissibility of membership.

The following arguments may be validated with reference to the indicated Aristotelean syllogisms.

Figure 1 (from 1/AAA).

“Y” is a member of “Z-classes”,

and “X” is a member of “Y-classes”,

therefore, “X” is a member of “Z-classes”.

Figure 2 (from 2/AOO).

“Z” is a member of “Y-classes”,

and “X” is not a member of “Y-classes”,

therefore, “X” is not a member of “Z-classes”.

Figure 3 (from 3/OAO).

“Y” is not a member of “Z-classes”,

and “Y” is a member of “X-classes”,

therefore, “X” is not a member of “Z-classes”.

However, no other arguments of that sort are possible. In the first figure, a negative major premise, “Y” is not a member of “Z-classes”, would only imply that some Y are not Z, from which no conclusion can be drawn; and as for 1/AII, it has no equivalent here, since “X” is a member of “Y-classes” would require that all X be Y. In the second figure, likewise with regard to a negative major premise; and as for 2/AEE, it has no equivalent here, since “X” is not a member of “Y-classes” only implies that some X are not Y. We can similarly write off the remaining moods of the third figure. The fourth figure has no equivalent here, since the minor premise of 4/EIO is not enough to imply membership of a class in a class of classes.

Thus, we only have three valid moods for propositions of this kind; no other moods are valid. The first is used for including a class in a class of classes, the other two for purposes of exclusion. These can be restated as follows, in accordance with the theorems of immediate inference:

Figure 1 (1/AAA)

“Y” is a Z-class (or, all Y-classes are Z-classes),

“X” is a Y-class (or, all X-classes are Y-classes),

so, “X” is a Z-class (or, all X-classes are Z-classes).

Figure 2 (2/AOO).

“Z” is a Y-class (or, all Z-classes are Y-classes),

“X” is not a Y-class (or, some X-classes are not Y-classes),

so, “X” is not a Z-class (or, some X-classes are not Z-classes).

Figure 3 (3/OAO)

“Y” is not a Z-class (or, some Y-classes are not Z-classes),

“Y” is an X-class (or, all Y-classes are X-classes),

so, “X” is not a Z-class (or, some X-classes are not Z-classes).

For examples. (i) The class of retrievers is a class of dogs, and the class of dogs is a class of animals, therefore “retrievers” is an animals-class. (ii) “Roses” is a class of plants, but “dogs” is not a class of plants, therefore “dogs” is not a member of “classes of roses”. (iii) “roses” is not a class of animals, but “roses” is a class of plants, therefore “plants” is not a member of the class of classes of animals.

Although the subsumptive relation between classes and classes of classes allows for only these three valid moods, it is clear that the subsumptive relation of classes of classes with each other allows for a fuller range of syllogistic argument. The three arguments indicated in brackets are obviously not all the valid moods for such terms, but any valid Aristotelean syllogism might be applied here. For example: some X-classes are Y-classes, no Y-classes are Z-classes, therefore some X-classes are not Z-classes. The explanation is simply that first order classes are effectively singular, whereas second order class subsume many such singulars.

e. Modal class-of-classes logic.

To modalize the concept of classes of classes, we would have to appeal to a collectional proposition, of the form ‘all X can be Y. This, you may recall, signifies, not only that for each X there are some circumstances in which it is Y, but also that there is at least one set of circumstances in which all X at the same time are Y.

The modal definitions are: for any X and Y, if all X simultaneously can be Y, then “X” can be a class of Y; but if some X cannot be Y, or all X can be Y, but not all at once, then “X” cannot be a class of Y; and if all X must be Y, then “X” must be a class of Y; but if some X can not-be Y, then “X” can not-be a class of Y.

From these definitions, the entire modal logic of classes of classes is easily derived, with reference to the logic of ordinary modal categoricals and collectionals. Note that the defining propositions are all intended as extensional conditionals, but two of them are special in that they contain a collectional antecedent.