Categorical Logic

© 2011 By Paul Herrick

A Survey of the Main Branches of Logic

Categorical logic seeks the universal standards that govern categorical arguments. What’s that? A categorical argument is one like this:

  1. All cats are mammals.
  2. All mammals have lungs.
  3. So, all cats have lungs.

And like this

  1. No cats are birds.
  2. Some cats are pets.
  3. So, some pets are not birds.

Aristotle was the first to discover and systematize universal standards of categorical reasoning. To do so he began by breaking categorical reasoning down into its essential parts. After this, he studied how the parts fit together to compose the many different forms that categorical reasoning can take. By formulating precise definitions for the type of sentences that make up a categorical argument, and a precise definition of a categorical argument (or “syllogism”), he discovered that, once defined properly, there are only four basic forms that a categorical sentence can take, and it followed precisely from this that there are exactly 256 basic forms of categorical reasoning, exactly 24 of which are unconditionally valid patterns of reasoning.

On this basis, Aristotle formulated a set of general algorithms that could be used to determine, with precision, in every possible case, whether a syllogism is valid or invalid. (An algorithm is a very precise rule for accomplishing a task that is guaranteed to give a definite outcome after a finite number of steps.) He also developed a precise axiom system, similar in form to the axiom system of geometry, allowing him to prove, with mathematical precision, the validity or invalidity of every possible form of categorical reasoning.

Thus, using precise algorithms, or his axiom system, logicians could now determine exactly which abstract patterns or forms of categorical reasoning are templates for valid arguments, and which are not.

The system of standards for categorical reasoning, that Aristotle produced in the 4th century B.C., was so accurate, and worked so well, because he identified and defined every moving part and then assembled the parts like a master engineer  constructing a great cathedral.

I want to draw attention at this point to one of Aristotle’s most important discoveries. He found that effective logical thinking requires precise definitions for all key terms; trying to reason with loosely defined words is like trying to play catch in the wind with cotton balls or puffs of smoke. For this reason, logical theory, like mathematics, has always been concerned with formulating detailed, exact definitions for all of its important terms.

It is also commonsense, of course, that if we do not adequately define our words, we may not really know what we are talking about. It is also commonsense that hen we are reasoning with others, we may not all agree on what it is we are talking about, if we have not first defined our terms. This is why most logic textbooks contain a chapter on the art of defining words: Knowing how to define your terms accurately is an important logical skill.

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Just a Taste of Categorical Logic

For those who would like to take a brief look at some of the details of Aristotle’s system, here are just some of the basics. A categorical argument is defined as one composed of categorical statements.  A categorical statement is a statement expressing a relation between two categories or groups of things by stating either that all, none, or some of one category, belong to, or do not belong to, a second category of things. There are thus four possible forms that the sentences making up a categorical argument can take. Letting A and B stand for categories of things, the four possible forms are:

  1. All A are B.
  2. No A are B.
  3. Some A are B.
  4. Some A are not B.

For example:

  1. All cats are mammals.
  2. No cats are reptiles.
  3. Some cats are pets.
  4. Some cats are not pets.

Each categorical sentence can be broken down into four elements: a quantifier, a subject term, a predicate term, and a copula. These can be illustrated with the first sentence: “All cats are mammals.”

"Cats" is the subject term.

"mammals" is the predicate term.

"All" is the quantifier.

"are" is the copula.

The subject term identifies the subject category—the category the sentence is about. The predicate term identifies the predicate category. The quantifier states how many members of the subject category (“cats”) belong to, or do not belong to, the predicate category (“mammals”). The copula joins subject and predicate and states whether the quantity refers to things that belong to the predicate category or that do not belong to it. Sentence 1 is thus to be interpreted as saying that all the members of the subject category
("cats") belong to the predicate category ("mammals").   

In contrast, sentence 2 says that none of the members of the subject category belong to the predicate category. Sentence 3 says that some (meaning one or more) of the members of the subject category belong to the predicate category, while sentence 4 says that some of the members of the subject category do not belong to the predicate category. A categorical sentence in standard form consists of one quantifier, followed by a subject term, a copula, and a predicate term, in one of the four forms stated above.

If the sentence states a claim about all the members of the subject category, it is called a universal sentence. If it makes a claim only about some of the subject category, the sentence is a particular sentence. If the sentence asserts that the members of the first category belong to the second category, the sentence is called affirmative; if it states that the members of the first category do not belong to the second category, the sentence is a negative sentence. The four forms of categorical sentence can now be characterized as follows:

  1. Universal affirmative
  2. Universal negative
  3. Particular affirmative
  4. Particular negative.

Beginning in the Middle Ages, these four forms were labeled with the Latin vowels A, E, I, and O:

A: Universal affirmative. All A are B.

E: Universal negative. No A are B.

I: Particular affirmative. Some A are B.

O: Particular negative. Some A are not B.

Finally, Aristotle defined a categorical syllogism as an argument having all of the following properties:

  1. It is composed of two premises and one conclusion, each a categorical sentence.
  2. Only three different terms appear in the argument.
  3. Each term appears exactly twice.

The predicate term of the conclusion is called the major term, the subject term of the conclusion is called the minor term, the premise containing the major term is the major premise and the premise containing the minor term is the minor premise. The term appearing in both premises is called the middle term. A categorical syllogism is said to be in standard form if the major premise appears first and the conclusion appears last.

Finally, the figure of a categorical syllogism is determined by the arrangement of its middle term.  There are four possible placements, and thus four possible figures:

Figure 1: The middle term appears in premise 1 as a subject term and in premise 2 as a predicate term.

Figure 2: The middle term appears in premise 1 as a predicate term and in premise 2 as a predicate term.

Figure 3: The middle term appears in premise 1 as a subject term and in premise 2 as a subject term.

Figure 4: The middle term appears in premise 1 as a predicate term and in premise 2 as a subject term.

All of these definitions lead ultimately to the definition of logical form for a categorical syllogism. Where the mood of a syllogism is a listing of the letters for its sentences, in order when the syllogism is in standard form, the logical form of any given categorical syllogism is then specified by stating its mood and figure. For example, the logical form named in Medieval times “Barbara” looks like this:

  1. All B are C.
  2. All A are B.
  3. Therefore, All A are C.

B is the middle term since it appears in both premises. Since the middle term appears in the first premise in the subject slot and in the second premise in the predicate slot, the syllogism is in figure 1. Since each sentence is an A sentence, the mood is AAA. The logical form of Barbara is thus AAA-1.

An interesting logical experiment, if you have a spare evening, would be to develop your own catalog of all 256 possible forms of the categorical syllogism. Once this is complete, develop universal algorithms determining exactly which of the 256 forms is a template for valid reasoning and which is not. To get you started, here are the unconditionally valid forms in the first figure, identified by the Latin names assigned by European logicians during the Middle Ages:

Latin name Mood Figure Structure
Barbara AAA 1 All M are P; all S are M. So, all S are P
Celarent EAE 1 No M are P; all S are M. So, no S are P
Darii AII 1 All M are P; some S are M. So, some S are P
Ferio EIO 1 No M are P; some S are M. So, some S are not P

where M stands for the middle term, S for the subject term of the conclusion, and P stands for the predicate term of the conclusion.

(Did you notice the relation between the letters in the form and the vowels in the corresponding Latin name? There was a reason behind this “coincidence”: students in the Middle Ages chanted the names in a rhyme song in order to memorize the forms.)

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Categorical Logic and the Birth of the Computer

It is probably no accident that the first person in history to envision and design a mechanical computer was an Aristotelian logician, that is, a logician working in the tradition started by Aristotle. As we have seen, Aristotle formulated the first algorithms in the history of logic—precise rules that could be used to decide if any categorical syllogism is valid or invalid. It seems logical to suppose that a machine can be designed to carry out the instructions contained in any set of sufficiently precise algorithms, but nobody had ever thought of this and done it, until the 13th century, when Raymond Lull (1232-1315), a medieval logician who was also a Catholic priest, designed a computing machine consisting of two rotating disks, each inscribed with symbols for categorical propositions. The disks were aligned in such a way that one could turn a dial and see which statements validly follow from a given statement. And what were the rules Lull used to “program” his device? The rules of Aristotelian categorical logic. For the first time in history, someone had conceived of a machine that takes inputs of a certain sort and then, on the basis of rules of logic, computes an exact answer which is then read off some other part of the device. Some historians of computer science have called Lull the “founder of computer science.”

We usually associate computing with mathematics; perhaps this is why it is interesting that the first designs in history for computing machines were designs for devices driven by the laws not of mathematics but of logic. Of Aristotle’s logic, to be exact. Categorical logic, and its long history, is a fascinating subject!

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