1. Formal Logic

Formal logic, which was established by Aristotle, is more the study of the forms and laws of thinking, and it does not deal so much with the content of thinking. According to Kant, “That logic has already, from the earliest times, proceeded upon this sure path is evidenced by the fact that since Aristotle it has not required to retrace a single step…. It is remarkable also that to the present day this logic has not been able to advance a single step, and is thus to all appearance a closed and completed body of doctrine.” ¹ Formal logic has existed, almost without change, for two thousand years since Aristotle. This is because formal logic contains considerable truth, in so far as it is concerned with thinking. Let me now introduce the main points of formal logic, and point out which parts are valid, and which are insufficient.

1.1. The Laws of Thought

Formal logic enumerates the following four laws as the laws of thought.

(1) The Law of Identity
(2) The Law of Contradiction
(3) The Law of the Excluded Middle
(4) The Law of Sufficient Reason

The law of identity can be expressed by the form “A is A,” as in the statement, “A flower is a flower.” This implies that, in spite of changes in phenomena, the substance of the flower remains unchanging. This also implies identity in thinking itself. That is to say, the concept of “flower” has one and the same meaning in every case. Furthermore, this principle can also imply that two concepts are in agreement, as in the statement, “A bird is an animal.”

The law of contradiction can be expressed by the form “A is not not-A.” This can be regarded as the principle of identity stated in reverse. In saying that “a flower is not a non-flower,” one is actually saying that “a flower is a flower.” Likewise, in saying that “a bird is not a non-animal,” one is actually saying that “a bird is an animal.” One is an affirmative way of expression, and the other is a negative way of expression, but the content remains the same.

The law of the excluded middle can be expressed as, “Everything is either A or not-A.” This means that there can be no third or middle judgment.

The law of sufficient reason was first advocated by Leibniz. Its meaning is that every act of thinking comes into being due to necessary reasons. Expressed in a more general way, it becomes the law of cause and effect, which states that everything has a sufficient reason for its existence. Reason here has two meanings: namely, basis and cause. Basis is the opposite concept to conclusion, and cause is the opposite concept to result. Therefore, this law means that thought always has its basis, and that existence always has its cause.

There are many other laws, but all of them are derived from these four fundamental ones. Formal logic also consists of three fundamental elements, that is, three elements of thought: concept, judgment, and inference. I will explain each of these next.

1.2. Concept

A concept is a general representation (or idea) through which the essential characteristics of a thing are grasped. A concept has two aspects, namely, intension and extension. Intension refers to the qualities, or properties, common to a certain concept, and extension refers to a set of beings to which the concept is applied. To explain these, let me take living beings as an example.

Living beings can be classified into concepts on various levels, such as animals, vertebrates, mammals, primates, and human beings. Living beings are those beings that have life. Animals, in addition to life, have sense organs. Vertebrates have a backbone. Mammals have the nature of suckling their young.

Primates have the ability to grasp things. Human beings have reason. In this way, the living beings of each level, represented by a certain concept, possess a certain common nature. The qualities, or properties, common to a certain concept are called the intension of that concept.

Among living beings, there are animals and plants, and among animals there are mollusks, arthropods, vertebrates, etc. Among vertebrates, there are reptiles, birds, mammals, etc. Among mammals, there are primates, carnivores, etc. Finally, among primates there are the various kinds of apes and human beings. A set of beings to which a certain concept is applied is called the extension of that concept (see table. 10.1).

When we compare any two concepts, that concept whose intension is broader and extension narrower is called a “specific concept” (or subordinate concept), and that concept whose intension is narrower and extension broader is called a “generic concept” (or superordinate concept). For example, when we compare the concept of vertebrate with the concepts of reptile, bird, or mammal, the former is a generic concept in relationship to the latter; and the latter are specific concepts in relationship to the former. Also, when we compare the concept of animal with the concepts of mollusks, arthropods, or vertebrates, the former is a generic concept, and the latter are specific concepts. Further, when we compare the concept of living beings with the concepts of plants or animals, the former is a generic concept, and the latter are specific concepts. If we repeat this operation over and over again, we will eventually reach the highest generic concept, beyond which no other concept can be traced. Such concepts are called “categories” (see fig. 10.2).

In addition, the pure concepts that reason possesses by nature (rather than through experiences) are also called categories. These categories vary from philosopher to philosopher. The reason for this is that the most important and fundamental concepts in each thought system are considered categories. Accordingly, the definition of categories varies from philosopher to philosopher.

Aristotle was the first philosopher to establish categories. He set up the following ten categories, taking clues from grammar:

In the modern age, Kant established twelve categories, which were mentioned in “Epistemology,” based on the twelve forms of judgment.

1.3. Judgment

a) What is a Judgment?

An assertion of something about a certain object is called a “judgment.” Logically, a judgment is an affirmation or denial of a relation among certain concepts. When expressed in language, a judgment is called a proposition.

A judgment consists of the three elements of subject, predicate, and copula. The object to which thinking is directed is the subject; the predicate describes its content; and the copula connects the two. Generally, the subject is expressed as ‘S,’ predicate as ‘P,’ and copula as ‘―’. A judgment is formulated as “S―P.”

b) Kinds of Judgment

As for the kinds of judgment, the twelve forms of judgment proposed by Kant are still employed in formal logic today. The Kantian twelve forms of judgment refer to the four main headings of quantity, quality, relation and modality, each of which is divided into three subdivisions. They are as follows:

Quality: Affirmative Judgment: S is P. Negative Judgment: S is not P. Infinite Judgment: S is not-P.

Relation: Categorical Judgment: S is P. Hypothetical Judgment: If A is B, C is D. Disjunctive Judgment: A is either B or C.

Modality: Problematic Judgment: S may be P. Assertive Judgment: S is in fact P. Apodictic Judgment: S must be P.

As explained above, Kant established three forms of judgment in each of four headings of quantity, quality, relation, and modality. In our daily life, we face various incidents and situations, and in order to cope with them, we think in various ways. Needless to say, the content of thinking is different from person to person. However, as far as judgment is concerned, it is in accordance with the above-mentioned forms of judgment.

That is, a judgment is either a judgment of quantity (much or little, many or few), a judgment of quality (is or is not), a judgment of relation (among concepts), or a judgment of modality (How is it certain?).

c) Basic Forms of Judgment

Of the above forms of judgment, the most basic is the categorical judgment. If the universal and particular forms of judgment concerning quantity, and the affirmative and negative forms of judgment concerning quality are combined with the categorical judgment, the following four kinds of judgment can be obtained:

The twelve forms of judgment, with the exceptions of disjunctive and hypothetical judgments, can be treated as categorical judgments. Then, if we arrange these categorical judgments in terms of quantity (a singular judgment can be treated as a universal judgment) and quality (an infinite judgment is included in the affirmative judgment), we arrive at the four basic forms of judgment, A, E, I, and O. The code letters A, E, I, and O derive from the first two vowels of the Latin words affirmo (‘I affirm’―A, I) and nego (‘I negate’―E, O).

d) Distributed and Undistributed Terms

In order not to fall into error in making a categorical judgment, one must examine the relationship between the extension of the subject and that of the predicate. In one case, a term (subject or predicate) in a judgment applies to an entire extension, but in other cases, it does not. When a term in a judgment applies to an entire extension, that term is said to be “distributed.” When a term applies to only a part of its extension, that term is said to be “undistributed.”

Distribution and undistribution of subject and of predicate are important concepts in a judgment. In a judgment, there is a case where both subject and predict are distributed, but there is a case also where subject and predicate can not both be distributed, and there is yet another case where only one of either subject or predicate can be distributed. For example, in the universal affirmative judgment “every man (S) is an animal (P)” (judgment A), the subject is distributed while the predicate is undistributed (see fig. 10.3). In other words, the term ‘man’ applies to the proposition “every man is an animal,” throughout its entire extension, but the same is not true about the term ‘animal’.

In the universal negative judgment “every bird (S) is a non-mammal (P),” subject and predicate are both distributed (see fig.10.4).

In the particular affirmative judgment “some flowers (S) are red (P),” both subject and predicate are undistributed (see fig. 10.5).

In the particular negative judgment “some birds (S) are non-carnivorous animals (P),” the subject is undistributed, since some S does not belong to P, while the predicate is distributed (see fig. 10.6).

In the above judgments A, E, I, and O, the distribution of terms is a rule of judgment. If one violates the rule, one’s judgment will fall into error. If, for example, one draws the conclusion “every lover of mountains is a hermit” from the judgment “every hermit is a lover of mountains,” one will fall into undue distribution; thus, the judgment is a fallacy. In a universal affirmative judgment, S should be distributed, whereas P should be undistributed. In this example, however, both S and P are regarded as distributed.

1.4. Inference

Inference refers to the process of reasoning whereby a conclusion is derived from one or more propositions. In other words, a conclusion “therefore, S―P” is derived from already known judgments, which are called premises. When there is only one proposition as the premise, the inference is called a “direct inference.” When there are two or more propositions as premises, it is called an “indirect inference.” Indirect inference includes syllogism, induction, and analogy. Let me briefly explain each of these.

a) Deduction (Deductive Method)

Deduction refers to an inference wherein a particular conclusion is drawn from more than one universal, general premise. The representative deduction is the syllogism, as indirect inference, which draws a conclusion from two premises. The first premise in the syllogism is called the major premise, and the second premise is called the minor premise. In the categorical syllogism, the major premise contains the major term (P) and the middle term (M), and the minor premise contains the minor term (S) and the middle term (M). The conclusion contains the minor term (S) and the major term (P). The following is an example of the categorical syllogism.

The above can be expressed with signs as:

M is P.
S is M.
Therefore, S is P.

In this syllogism, the extension of the major term (P) is larger than that of the middle term (M), which is larger than that of the minor term (S), as illustrated in figure 10.7.

b) Induction

The method by which one attempts to reach a general assertion from a number of observed particular facts is called inductive inference, or induction. It is regarded as an application of the syllogism. The following is an example of induction:

Horses, dogs, chickens, and cows are mortal.
Horses, dogs, chickens, and cows are animals.
Therefore, all animals are mortal.

Is the conclusion “therefore, all animals are mortal” correct? This conclusion is a universal affirmative judgment. The term “animal,” therefore, has to be distributed. In this inference, however, it is undistributed, since horses, dogs, chickens and cows are a part of animals. The conclusion is stated in the form of a universal affirmative judgment as shown in fig. 10.3. However, this conclusion is, in fact, a particular affirmative judgment as shown in fig. 10.5.

Thus, strictly speaking, this inference is erroneous. However, such an inductive inference is possible in natural science because of the application of the “principle of uniformity in nature” and the “law of causality.” The former means that all phenomena in the natural world have the same form, and the latter means that the same effect is always brought about by the same cause. Accordingly, from our experiences the induction is considered to be correct.

c) Analogy

Another important mode of inference is analogy. Suppose there are two objects of observation, A and B, and it is known, through our observations, that A and B both have common natures (a), (b), (c), and (d). Furthermore, suppose that A has another nature (e), and it is difficult to observe whether B has the nature (e). In this situation, one may conclude that B also has the nature (e), which A does. This is an analogy.

For example, through observations of the earth and Mars, it is known that the two planets have the following common natures:

(a) Both are planets, revolving around the sun while rotating on their axes.
(b) They have air.
(c) They have almost the same temperature.
(d) They have the changes of four seasons, and they have water.

Then, based on these facts, one may conclude that there are living beings on Mars, since such beings exist on the earth.

Analogy is often used in our daily lives. For example, present-day advanced scientific knowledge has been acquired through analogy, especially in the early stages of the development of science. Also, analogy plays an important role in our family life, group life, school life, business life, and creative activities. Therefore, the accuracy of analogy becomes an important issue. The requisites for the accuracy of analogy are:

(a) There should be as many similarities as possible in the objects to be compared.
(b) Those similarities should not be accidental, but rather essential.
(c) There should be no incompatible qualities in these similarities.

In formal logic, there are several other kinds of inferences to be dealt with, such as direct inference, hypothetical syllogism, disjunctive syllogism, the theory of fallacy, and so forth, but I will conclude here, since my intention was only to introduce the main points of formal logic.