Categorical Claims

Categorical Claims by
Greg Tropea

Categorical claims essentially perform the task of telling how one type of thing or action relates to another. Almost any short sentence in English or any other language can be stated as a categorical claim, although some languages build sentences in ways that make such translations pretty difficult. And, conversely, any categorical claim can be translated into a variety of different-looking sentences that are logically identical. Take this sentence as our starting point:

All flowers are things of beauty. This universal affirmative claim has a number of very close relatives that would be taken to mean the same thing in conventional logical practice. Here are some of the possibilities:

What's most noteworthy in this strange little list is not the clumsiness of some of the sentences, which is evident enough. The thing to pay attention to in the list is the form of each sentence. Whatever is being talked about in sentences that have any of these forms, the same thing is being said if you plug the subject and predicate into any of the other forms. This is an important point, because logic is very much about the forms of arguments, and that begins with the forms of the claims that make up the arguments. Examining this list carefully will usually bring a few surprises to people who haven't studied logic. Discovering some surprises in these simple sentences probably ought to count as a reason to study logic carefully.

Realistically...

It's one of the messy facts of life that exceptions often arise to complicate most claims that we would like to think of as universal. Complexity theory, a comparatively new sub-discipline in mathematics that has grown out of chaos theory, leads us to expect xeceptions to general rules whenever we look closely enough at the details or examine a large enough number of events. This is why a practical approach to categorical logic will include both a knowledge of the deductive techniques that establish what is theoretically true as well as some way to handle exceptions to virtually universal claims without having the whole system break down.

Aristotle's solution to this problem was to allow generalizations to be true for the most part, thus letting apparent exceptions be until their natures were better understood. The sixteenth century philosopher René Descartes attempted to improve on Aristotle by requiring absolute certainty of the truth of the premises of any argument that would meet the standard of science, but this view has not held up well in our scientific age as we have come to realize that in the world of experience, the possibilities of absolute certainty are few and far between.

Good logic requires clearly stated propositions.

The basic building blocks of categorical logic are simple sentences called categorical claims. There are four basic forms of these claims, each of which can be used to translate a variety of natural language sentences into a form that is convenient for the operations of categorical logic. Expressing thoughts in standard-form categorical claims is an important step in revealing the relationship between the things or actions named in the claim. A categorical claim always refers to two classes, which by convention are called S (for the subject) and P (for the predicate) in discussions of the general principles of categorical logic. Categorical claims either affirm or deny that one class (S) is partly or completely included in another class (P). The four basic forms of categorical claims say everything that there is to say in this logic. By requiring the translation of natural language sentences into standard-form categorical claims, the clarifying work of categorical logic begins before any inferences occur.

Standard-form categorical claims funnel the wildly diverse expressions of natural languages (such as English or Japanese) into very simple forms that unambiguously reveal the categorical understandings that will be used in the logical operations. Again by convention, we refer to the four different forms of these simple claims as A-claims, E-claims, I-claims, and O-claims. Using our generic classes S and P, this is what they look like:

A-claim (universal affirmative): All S is P / All S are P

E-claim (universal negative): No S is P / No S are P

I-claim (particular affirmative): Some S is P / Some S are P

O-claim (particular negative): Some S is not P / Some S are not P

The claims show up in versions with both "is" and "are" purely as a matter of convenience. The essential thing to recognize about the verbs in standard-form categorical claims is that the four types all have the same verb connecting subject and predicate, specifically, some form of the verb "to be." Logically, it does not matter whether the form is "are," "is," or simply "be." For purposes of categorical reasoning, they all get the job done equally well. Whichever form of the verb seems to fit best can be used to create a well-formed standard-form claim.

In terms of S and P, a universal affirmative claim says that every member of the first class (S) is also a member of the second class (P). Not surprisingly, a universal negative claim says that every member of the first class (S) is completely excluded from the second class (P).

The particular claims require a bit more interpretation, specifically, with regard to the word "some" that appears as the quantifier in both of them. Categorical logic gives "some" an indefinitely wide range of reference, so in a standard-form categorical claim, "some" can refer to as few as one of a class and as many as all the members of the class. Of course, if we know for certain that we are referring to all members of a class, a universal claim would be the appropriate expression to use. What we're saying here is that there might be times when the things you are referring to in an I-claim or O-claim happen to be all there is of that class, and that if this should occur, it doesn't make the I- or O-claim false.

The word "some" is indefinite, and is customarily regarded as

meaning "at least one." So, in terms of our generic classes S and P, a particular affirmative claim says that at least one member of class S is also a member of class P. By the same token, a particular negative claim says that at least one member of the class designated by the subject term (S) is excluded from the class designated by the predicate term (P).

There are some natural language sentences that look pretty much like standard-form categorical claims. Here are a few that all happen to be true:

All vegetables are plants.

No factories are vegetables.

Some businesses are corporations.

Some corporations are not businesses.

Sentences like these readily translate into standard-form categorical claims, but they are by no means the only kinds of sentences we can work with in the operations of categorical logic. Before we get any further into translating natural language sentences into standard-form categorical claims, a few more points need to be brought into the discussion. First, the subject and predicate terms of a standard-form categorical proposition always designate classes, but those terms may be complicated expressions rather than single words. The important point about this is to recognize that whether a class of things or actions is easily named with a single word or requires a paragraph-like specification, it is still one kind of thing or action that we happen to be talking about. The class may be further divisible, as the class of human beings can be divided in to adults and children, but the class (e.g., human beings) is still composed of human beings, whatever else one may be able to say about them.

Closely related to the notion that the subject and predicate terms of categorical claims designate classes is the requirement that classes are always designated by nouns or noun phrases. What this means is that categories, even when they are actions, are treated as groupings of somehow homogeneous things. As we saw in the example from biology, categories are not required to be perfectly homogeneous; human beings belong in one category, but there are certainly many possible divisions within this category.

English and many other languages allow us to create sentences that look similar on the surface, but are really significantly different. Consider this pair:

The first of these has the structure noun-verb-noun, while the second has the structure noun-verb-adjective. To create a categorical claim that expresses what the second claim says, "fun" has to become part of a noun phrase. The easiest way to do this is to recast the second sentence.

Dogs are things that are fun. In this restated claim, "things that are fun" straightforwardly designates a category. Depending upon whether the ambiguous original was intended as a true universal affirmative or something a liitle less absolute, there will be different translations of the claim into standard form. If one wanted to allow for even one exception, the sentence above would be translated into a particular affirmative claim, the kind of categorical claim that is used to allow for some possible exceptions, while still conveying a general idea. So this sentence now translates readily into one or another standard-form categorical claim:

Names of people may create some confusion when it comes time to translate sentences about individuals into standard-form claims. Consider this sentence:

Barbara is a doctor. Though it will look awkward, this sentence is best translated as an A-claim. In English, names don't accept quantifiers (e.g., all, no, some). So categorical claims about individuals might not look natural or grammatical, but as long as the standard-form claims do the job of unambiguously representing categorical relationships, they are doing all they need to do. So, strange as it appears, we will translate sentences about individuals simply as follows:

All Barbara is doctor. The alien grammar of this sentence can serve as a reminder that although the sentences that are translated into standard-form categorical claims use English words, they aren't really English and don't have to obey the rules of English. The main thing is hat the categories of Barbara and doctor are both present and unambiguously repesented.

If names are difficult to render in standard form, time expressions, usually rendered by adverbs such as today, now, later, etc. are just about impossible to translate recognizably. Consider the following sentence, which presents a good challenge to translation into standard form:

We're leaving today. In this case, the subject is all of us. And all of us are things that are leaving today. If we symbolize all of us (we) with W and the category of things that are leaving today with L, the sentence above in standard form is:

All W are L There are numerous other types of sentences that look quite different from standard-form categorical claims, but can be appropriately translated with a little imagination.

Replacing English category names with letter variables, as in the example above, removes some of the stylistic clumsiness of standard form claims and requires less writing also. Consider this two-premise argument, already in standard form for convenience:

Before you head to the refrigerator, consider how we might simplify the processing load imposed on us by this (invalid) argument. First, define the categories and assign shorthand variables to them:

When there is something at stake, it is almost always good practice to define categories explicity like this so you know exactly what you are talking about. If someone stands to benefit from someone else's confusion, there might be some reason to maximize complexity--and people surely do this at time--but we are now getting into moral reasoning issues that are beyond the scope of our discussion of logical techniques here.

So, back to our task of simplifying things. In a simpler form that is logically identical to the original, the argument above now looks like this:

There's nothing necessary about simplifying things this way, but when it's helpful, go for it. One type of situation in which this sort of simplification is helpful occurs when we have to manipulate categories to get a clearer picture of how things relate to each other. That's one of the purposes of the immediate inferences we'll be looking at next.