While logicians justifiably have the reputation of working with highly abstract ideas expressed in specialized, formal languages, it is worthwhile to remember that the roots of logic are in the practical challenges of everyday life. This is why, in spite of the the sometimes obscure symbolism connected with logic, it's still perfectly reasonable to expect the practical application of logic to provide useful assistance in keeping our beliefs about ourselves and our world in some coherent order. There are certainly other ways besides logic that people have tried out to bring order to their experience, including unquestioning acceptance of traditions, following charismatic leaders, and giving priority to short-term gratification of desires, to name a few. But of the various ways of bringing order to existence, only careful thinking about the relationships of different kinds of things and actions makes full use of our distinctively human intelligence. This is the realm of categorical logic.

Clear thinking about practical matters requires that things be identified appropriately, priorities set, and conflicts resolved. Categorical logic has been serving just these purposes for over two thousand years with remarkably little modification. As its name suggests, this form of logic is especially concerned with how to categorize things and actions. The practical application of categorical logic requires a few capabilities:

- knowledge of several basic logical relations,

- familiarity with a few patterns of inference, and

- a set of techniques for constructing and analyzing arguments. Conveying these fundamentals is the main work of this chapter. Additionally, effective use of categorical logic often calls for a certain amount of background knowledge and plain old good judgment. You'll have to supply these. The essentials of categorical logic fit into just a few pages, but they can be used to focus attention in some very effective ways and can provide the means to analyze a wide variety of claims, arguments, and situations.
 
 

Have you been here before? As we use computers to perform more and more tasks, we accumulate large numbers of files. It doesn't take long before scrolling through them all in one long list becomes impractical. When this happens, it's almost certainly categorical logic coming to the rescue. When you organize your own work on a computer into folders or directories, assuming you are not just randomly dumping files into any old folder or directory, you are applying techniques of categorical logic to make decisions of what goes where. You might, for example, decide that all graphics images should have a place of their own, apart from letters you write, financial files, etc. As a result of studying categorical logic, you might find that the organization of files in your computer becomes noticeably more efficient.

Categorical logic is useful in a variety of ways.

Categorical reasoning of varying levels of complexity shows up commonly in both academic pursuits and everyday business. In the sciences, the genus-and-species classification system biologists use to keep track of the myriad organisms of the world rests on a foundation of categorical logic. Human beings, for example, are a species within the larger category of primates, what all belong to the still larger category of mammals, which in turn belong to the yet larger category of organisms with a backbone, etc. As those who are familiar with the field of taxonomy know well, classification decisions in biology can depend upon some very subtle argumentation. Most of the logical work of taxonomy occurs in the form of deductions that are essentially like those we will be working with in this chapter. The extended debates about classification that have occurred among biologists over the past two centuries are testimony not only to the complexity of categorical problems, but also to the durability of the logic that structures these discussions.

For your consideration.. One of the attractions of an evolutionary explanation of how the earth's current population of species came to be is that it shows why most species can be straightforwardly ordered in a hierarchy of categories with the more general branching out into the more specific. Notice that the mere fact of categorical orderliness does not explain anything. It is interesting, nonetheless, that Darwin's theory provides an explanation that accords with many long-held intuitions about what sorts of things are related to other things.  While there are gaps in the fossil record that leave many missing links, this is to be expected, since only a very small percentage of living things become fossilized after they die. The consensus among most scientists is that the organisms with backbones can trace their ancestry back some common progenitor, though, of course, backbones could have evolved separately in more than one ancient population. So here are two categorical claims: All creatures with backbones have two eyes. Some creatures with two eyes do not have backbones. Do they suggest anything by themselves? How about if you add some ideas from evolutionary theory? (We hope that when you thought about these claims "by themselves," you didn't prematurely smuggle in any evolutionary assumptions!)

Categorical logic is not just concerned with navigating scientific technicalities, though. It also lies behind the organization of the thousands of items in a supermarket into groupings that somehow seem to belong together. Interestingly, the organization of supermarkets varies little from country to country, suggesting that there is some naturalness to the ways things are grouped (although it seems to be the case that only in the United States and Canada will you find the peanut butter and the jellies and preserves next to each other on a shelf). Imagine how differently the supermarket would be laid out if some other method of organization were used, say, product size or color or shipment date or the amount of a bribe paid to a manager. In a very real sense, the order that you perceive in the layout of well-planned supermarket is a direct application of one of the key insights of ancient Greek philosophy.

In the realm of information technology, modern database design would be impossible without highly disciplined categorical reasoning. It turns out that keeping track of business facts in a database is greatly facilitated by breaking these facts up into simpler conceptual units that can subsequently be used to identify all things or events of a similar type. So, if you owned a record store, for example, you might at first imagine simply dividing it up into CDs, vinyl, and tapes. Eventually, though, you'd want a richer set of categories than just these. Within each of these categories, you might find it useful to distinguish among different types of music or spoken word, products from different distributors, and time in your bins. You might also want such categories as new and used, latest releases and regular inventory, full-price and discounted, etc. You'd need to be able to relate all of these categories (and probably more) to each other, with the reward being that you would know a lot about what was working for your business and what wasn't.

Categories on the Internet A few years after people began using the Worldwide Web to convey information, people bumped up against several limitations of the simple browser display technology that was in use then. One of these limitations was the difficulty in sharing and maintaining complex information. The solution to the problem was a classic application of categorical logic: XML (eXtensible Markup Language), a system of hierarchical inclusion in which a large category, such as Recipes, could contain smaller categories, such as Soups, Breads, Pastas, etc., which could contain still smaller categories, such as Split Pea Soup, Hot and Sour Soup, Tomato Soup, etc. The system is set up so that anybody can define systems of categories that will work for their purposes, as long as the categories are logically laid out.

Categorical logic deals not only with things, but also actions. This is an important point which we shall return to when we discuss how to translate natural language into what we will call standard form categorical claims. In a rough and ready sense, you probably already have a grasp of how categorical logic works in the classification of actions. If, for example, you have any sense of how the "rules of the road" apply when you drive, you can thank your existing understanding of categorical logic. In fact, most legal proceedings depend heavily upon categorical logic, as do those parts of business contract negotiations in which people are attempting to be reasonable.

Categorical reasoning is mostly deductive.

While some instances of categorical reasoning proceed by inductive techniques such as analogy, most instances of categorical logic involve quick deductive arguments with a couple of premises. Analogical arguments are common, and sometimes they are the best form of reasoning that is available to us, but trying to determine the status of a thing or action inductively, for example, by accumulating evidence or by constructing an analogy that calculates similarities and differences, is best left to the borderline cases. When a clear-cut categorization can be made deductively, it is preferable, mainly because it is simpler.

Take the question of whether a chemical compound should be classed as a poison, for example. We think of something as poisonous when it either immediately or over time causes tissue damage or death after it is ingested or otherwise absorbed into the body. There has never been much question that copper sulfate fits this definition. But for a long time, there was a question of whether hexachlorophene fit the definition of a poison. A few decades ago, it was a featured ingredient in a well advertised toothpaste. As evidence piled up about hexachlorophene, it became identified as a probable carcinogen with an ever-inreasing level of confidence. Both of these compunds now belong unquestionably in the category of poisons, as does methyl alcohol, which tastes pretty close to ethyl alcohol, which is not classed as a poison. But what about ethyl alcohol, the alcohol which is found in beer, wine, and distilled spirits? In comparatively small quantities, wine seems to promote health, while in large quantities, it clearly destroys health. So is wine a poison or isn't it? How about Vitamin A, which behaves the same way? Maybe the definition of poison needs to be changed somehow so that it better captures what is dangerous, but this will be a difficult task.

For your consideration... How might the concept of "poison" be further refined so that it would be clear what belonged in the category and what did not? Would you want to include a clause stating that no substances that are necessary to life are poisons? Would you include some specification of quantity in the definition? If so, what would the language look like? Is the category of poison so complicated that we should abandon all hope of ever defining it precisely?

There are many cases similar to the poison example above. Some things clearly belong in a category because they obviously possess the category's defining characteristics, some things clearly do not belong in that category, and other things are borderline.

For example, there continues to be much wrangling about what one is paying for when one "buys" computer software. The license agreements that come with just about every software package state that one does not buy the software, but rather a license to use it, usually on just one computer. For purposes of establishing the rights of software producers and the rights of their customers, some determinations about the nature of software need to be made. Interestingly, the analogy of the book was used by both sides to support their respective positions. Software producers argued that software was like a book in that it could only be in one place at a time, that people who purchased a book were not owners of the creative work that went into the book, but only of the physical medium that contained a copy of it, that there was no guarantee of satisfaction after reading the book, no right to make copies of the book, and so on. All of these ideas show up regularly in software licensing agreements, along with others that are rather unbooklike, such as that the software may communicate back to the company over the internet without the customer's knowledge, that the company may disable the software by remote control under certain conditions, that the software may expire and become unusable (thus imprisoning the customer's files until they are unlocked), that the software may be re-installed a limited number of times, even if the customer is simply trying to recover from mechanical problems, and more. Consumer advocates have argued that purchasers of books could read them wherever they wanted to, read them together with someone else, access them without special codes (registration and copy protection), lend them to friends, sell them, expect that the book would be free of errors that would render it unusable, and more. In this case, the prevailing categorical argument, which will probably be analogical in form, will have implications that run into the billions of dollars.

When categorical determinations can be made with certainty using the more straightforward techniques of deduction or immediate inference, which we'll be exploring shortly, are the right tools for the job. A deductive argument, we recall, is one that is intended to be valid, that is, one whose premises, if true, are supposed to provide airtight support for the truth of its conclusion. This, of course, requires us to put a lot of thought into whether or not the premises of a deductive argument are, in fact, true. Insufficient care in making sure that the premises in a deduction say exactly what we believe to be true is the most common cause of disappointment with the outcome of this sort of reasoning. This is a special concern in categorical logic, since universal claims (ones that refer in a positive or negative way to all of a class) are common in this way of thinking. We expect, for example, that a law will apply to everybody.

Of course, not every argument that is intended to be deductively valid actually makes the grade. So, any attempt at a deductive argument is going to be either valid or invalid. As we may recall, the usual practice in logic is not to accept an argument as valid unless somebody explicitly proves it is so. In the traditional arrangement of categorical reasoning into simple, two-premise deductive arguments called syllogisms, there are only fifteen valid argument forms. One would think that with such a smallnumber of valid forms--and simple ones, at that--keeping our categorical reasoning in perfect order would be incredibly easy. It doesn't work out that way, though, mainly because these forms are inconvenient for most people to memorize and the methodical evaluation of reasoning that allows us to prove an argument valid or invalid is not the widespread skill it might be. Consequentially, it's rather common to encounter someone who will affirm the validity of any argument that has an agreeable conclusion. Some people will actually go so far as to claim that if the conclusion of a valid argument is true, the premises must also be true. Judgments like these do not have anything to do with proper application of logic, though people who traffic in such lazy thinking will often try to wrap their prejudices and preferences in assertions that they are logical or self-evident. We need to remember that bits and pieces of truth are not evidence that anybody has accomplished a logically valid inference; the conclusions of invalid arguments, even invalid arguments with false premises, may still happen to be correct. And, we emphasize, simply having a correct conclusion does not make an argument valid. In our categorical deductions, we want it all: true premises and valid logic. We are so interested in valid arguments with true premises (that is, sound arguments) because these provide us with the best reliability possible when we are wondering about the truth of a claim.

In working with categorical arguments, it is inevitable that we will encounter some invalid arguments with conclusions that look plausible, given the premises. While it is possible for an invalid argument to provide some evidence to support a conclusion, invalid arguments present problems of reliability that require techniques beyond those of categorical deductions for adequate evaluation of the strength of the argument when there is anything important at stake. The practical techniques for determining what confidence, if any, we may have in an invalid argument present complex difficulties of judgment unlike any found in categorical logic or other types of deduction, so if we accept the principle that we should try to solve problems in the simplest and most economical way, we will find ourselves making as much use of valid deductions as we can. If we discover through methodical application of categorical logic that we can't know essentially for sure about such questions in a case under consideration, then we might decide either to recast the problem or start applying the far more "expensive" techniques of induction.
 
 

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 exceptions 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 Rene 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 categoricallogic. 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 register 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:

Dogs are mammals.

Dogs are fun.

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 universal affirmative or particular affirmative claim, this sentence now translates readily into a standard-form categorical claim:

All dogs are things that are fun. (A-claim)

Some dogs are things that are fun. (I-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) that would make the categorical claims 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 that 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.