The Rudolf Steiner Archive

13. Formal Logic II

[ 1 ] Of course, it is not possible to cover this topic in as much detail as would be desirable within the few days available. If one wanted to discuss this exhaustively, one would have to teach a course of some sort. Therefore, please consider what has been said here merely as a few rough outlines. Nor is the intention to proceed systematically; rather, I would simply like to present a few of the elementary truths of logic to you, so that you have something that you might find useful right now.

[ 2 ] We have formed a concept of the concept itself; we have learned what a judgment is and how an inference arises—namely, through the combination of judgments. It has been said that there are certain internal laws of the technique of thinking that determine how judgments must be combined if one wishes to arrive at correct inferences. We presented the basic form of the syllogism in the first syllogistic form using the example: All humans are mortal. Cajus is a human. Therefore, Cajus is mortal. — In the major premise—All humans are mortal—we have the first judgment; and in the minor premise—Cajus is a human—we have a second judgment. The task now is to derive a new proposition from the combination of these two propositions through an internal law of logic: Therefore, Cajus is mortal. — We call this last proposition the conclusion. We see what this conclusion is based on: We have two propositions that are given and must be present; we know what they assert. The task now is to omit the middle term from these two given propositions. The subject term of the major premise was “All humans,” and the predicate term was “mortal.” In the minor premise, we had the subject term “Cajus” and the predicate term “human.” In the conclusion, the two terms that were present in both propositions are omitted, namely the term “human.” Our ability to form the conclusion depends on how this middle term, “human,” is positioned in the major and minor premises. Our schema was: \(M = P\); \(S = M\); \(S = P\).

[ 3 ] The fact that we may formulate the conclusion in this way stems from the distribution of terms in the major and minor premises. If this distribution were different, we could not draw the conclusion as in the example given earlier: A photograph is similar to a human being (major premise); a photograph is a mechanical product (minor premise). If we were to omit the middle term, which is contained in both propositions, no valid conclusion could be formed here. This is because in both propositions the middle term is linked to the predicate in the same way, as the subject. The middle term must appear first in one proposition and last in the other; only then may we form a valid conclusion. Logic is a formal art of concept formation. The arrangement of concepts alone reveals how one can arrive at valid inferences. We must master as laws the manner in which concepts must be combined. We could also say that this formal logic encompasses the doctrine of concepts, judgments, and inferences.

[ 4 ] We will now consider judgments in a few remarks. Certain laws can be formulated regarding judgments. The laws of inference become understandable only once the principles concerning concepts and judgments have already been established. Today, therefore, we will first consider the laws of judgments and concepts.

[ 5 ] If we begin with the laws of concepts themselves, we can compare a concept such as “lion” with the concept “mammal.” Both are concepts that we can form. They differ in the following way. Think for a moment about everything that falls under the concept of “mammal.” It encompasses a wide range of individual objects—for example, monkeys, lions, marsupials, and so on—which is much more than what we include under the concept of “lion,” which gives us only a small subset of the “mammal” concept. Thus, all concepts differ from one another in that there are concepts that encompass many things and those that encompass only a small area. We say here: Concepts differ according to their scope; but they also differ in other respects. To define the concept “lion,” many properties are necessary—many characteristics such as head, color, paws, teeth, and so on. All of these elements listed to arrive at the concept of “lion” are called the content of the concept. The concept of “mammal” has significantly fewer characteristics than the concept of “lion.” If you were to subsume animals with a certain coat color under this concept, that would no longer be correct. When you form the concept “mammal,” you must have as few characteristics as possible—a narrow content—such as only the characteristic that it gives birth to live young and nurses them. Thus, “mammal” is a concept with a narrow content and a broad extension, while “lion” is the opposite. There are, therefore, concepts with a broad extension and narrow intension, and concepts with a narrow extension and broad intension. The broader the extension of a concept, the narrower its intension; the broader the intension, the narrower the extension. This is how concepts differ in terms of intension and extension.

[ 6 ] Let us now consider judgments in a similar way. If you make the judgment: “All human beings are mortal,” you have a different judgment than: “The crocodile is not a mammal.” The difference between the two is this: In the first case, something is affirmed; the concepts are brought together in such a way that they are compatible. In the second case, the terms are incompatible; they are mutually exclusive; here we have a negative judgment. Thus, we distinguish between affirmative and negative judgments. There are other differences with regard to judgments as well. “All humans are mortal”—this judgment conveys something entirely different than “Some flowers are red.”—In the first case, the statement of a property applies to the entire scope of the subject term; in the second case, other characteristics may be added. The latter judgment is called a particular judgment, in contrast to the former, which is a universal judgment. We thus have affirmative and negative, universal and particular judgments. One can find other distinguishing features in judgments; for example, a judgment may be formulated according to the pattern: “All humans are mortal,” or it may be expressed as: “When the sun shines, it is bright.” - The first judgment unconditionally unites the subject and predicate terms; the second does not necessarily unite the subject and predicate terms, but is only conditional. It merely states that the predicate term is present if the subject term is also present—nothing more. The first—“All humans are mortal”—is an absolute or unconditional judgment; the second—“When the sun shines, it is bright”—is a hypothetical judgment. There are, therefore, absolute or unconditional judgments and hypothetical or conditional judgments. Many more such characteristics of judgments could be cited; but the aim here is merely to show that certain things depend on knowledge of these distinctions. One must master the technique of concepts in order to draw correct conclusions.

[ 7 ] For example, if you take our conclusion based on the first form of the syllogism: All humans are mortal. Cajus is a human. Therefore, Cajus is mortal—in this case, we have a universal judgment in the major premise and a singular judgment in the minor premise, because it applies only to a single individual, namely Cajus. This is a subtype of the particular judgment. This arrangement of judgments is permissible; it yields a correct inference. But let’s try a different arrangement. Take, for example, the major premise: “Some women have red dresses”—this is a particular judgment. And now let’s say: “Ms. NN is a woman.” — Now I must not conclude: “Therefore, Ms. NN has a red dress.” — I must not, because it is impermissible to draw a conclusion using this form of inference when the major premise contains a particular judgment. Only when the major premise is a universal judgment is this form of inference correct. Thus, certain laws can be established here as well. — We could now also cite other properties of judgments. We have said that a judgment can be affirmative or negative. Let’s take a negative judgment: The crocodile is not a mammal. This animal is a crocodile. - Here, we may conclude: Therefore, this animal is not a mammal. - The major premise may therefore be either affirmative or negative.

[ 8 ] There is, therefore, a specific technique of thinking, a law of thought, which is formal—that is, entirely independent of content. If we follow this formal technique, we think correctly; otherwise, we think incorrectly. We must adhere to this technique of thinking, this law of thought, in order to reach correct conclusions.

[ 9 ] We now have another famous classification, originating with Kant, into analytic and synthetic judgments. Indeed, today this classification is very often encountered, especially by those who engage in philosophy. What, then, is the difference in the Kantian sense? An analytic judgment is one in which the subject concept already contains the predicate concept. In contrast, in a synthetic judgment, the subject concept does not necessarily contain the predicate concept. For example, the sentence “A body is extended” is an analytic judgment, because one cannot conceive of a body without simultaneously conceiving of its extension. “Extended” is merely a characteristic of the concept of “body.” A synthetic judgment, however, is one in which the predicate concept is not yet contained within the subject concept. The subject and predicate are brought together by an external ground. For example: “The body is heavy” is, according to Kant, a synthetic judgment. For he maintains that the concept of heaviness is connected to the concept of the body only through external grounds—namely, the law of gravitation. In a synthetic judgment, therefore, there is a looser connection between the concepts.

[ 10 ] A great deal of nonsense has been made of the concepts of analytic and synthetic judgments in modern philosophy. To me, the most illuminating example has always been the story of what is said to have happened to a student taking an exam at a German university. On the eve of the exam, he went to a friend and asked him to quickly teach him a few concepts in logic. The friend, however, realized the futility of such an endeavor and advised him to go as he was and leave it to chance. The next day, the student was asked: “Do you know what an analytical judgment is?” — The sad reply was: “No.” — To which the professor replied: “That is a very good answer, for I cannot say either. And what is a synthetic judgment?” — The student, having gained some courage, replied again: “I do not know.” - Then the professor said, very pleased: “You have grasped the essence of the matter. I congratulate you; you will receive a good grade!” - In a certain sense, the matter does indeed seem to me to shed light on the issue. For the difference between the two types of judgment is in fact an ambiguous one: it depends on what one has in mind when using the term. For example, one person adds the concept of extension to the body; whoever, on the other hand, adds the concept of weight, incorporates more into the concept from the outset than the other person does. The point now is for us to recognize what kind of truly real substance underlies the combination of concepts into judgments, or rather, what the secret goal of all judging is. Judging is, in fact, purely formal at first.

[ 11 ] However, there is something associated with making judgments that will become clearest to you when you compare the following two judgments side by side. Let’s assume—after all, we’re staying on the physical plane—that we have the judgment: “The lion is yellow.” If you form this judgment, it may be correct. But let’s suppose someone imagines some concept—an animal that is half lion, a quarter whale, and a quarter camel. They could very well imagine it; let’s say they call it “Taxu.” They could now form the judgment: “This animal is beautiful.”—Formally speaking, this judgment is just as valid as the judgment: “The lion is yellow.” - How, then, do I distinguish between valid and invalid judgments? - This brings us to a section where we must find the criterion for the ability to form a judgment in the first place. You can alter the judgment “The lion is yellow” at any time, namely by saying: “A yellow lion”—or: “The yellow lion is.” - But we cannot say: “There is a beautiful taxus.” - This leads to a criterion for the validity of a judgment: One must be able to incorporate the predicate term into the subject term and form an existential judgment from it. The transformation of a formal judgment into an existential judgment by adding the predicate to the subject thus constitutes the criterion for validity. In the first case, [empirical] necessity unites the concept “yellow” with “lion”; in the second case, the formation of the concept presupposes that the subject is taken from an existential judgment, whereas in fact it originated only from a formal judgment.

[ 12 ] This is a criterion for the validity of every judgment. The formal correctness of a judgment depends only on the correct combination of terms, but the validity of a judgment depends on the existential judgment. A formal judgment is transformed into an existential judgment by attributing the predicate to the subject; the subject is enriched. And that is precisely the goal of judging and also of reasoning: the formation of concepts that are valid. Form the judgment: “A yellow lion is —”; in doing so, you have considered not only formal correctness but also validity. Now you see that, while formal logic does indeed offer the possibility of, so to speak, equipping us with correct concepts, the formation of valid judgments is what we must aim for; and valid judgments cannot be derived from formal logic alone. The existential judgment in our example—“The yellow lion is”—was derived from external sensory observation. Formal logic gives us the possibility of arriving at correct concepts; with its help, we can form quite fruitful concepts. But for the validity of judgments, logic will still have to draw upon content-related considerations. People generally do not fully grasp what logic actually is. But once one has learned to grasp the concept correctly, independently of content, this is extremely important.

[ 13 ] The validity and formality of a judgment are two different things. Because people do not fully grasp how these things are actually related, grand theories are spun out that are regarded by some as irrefutable, but which would collapse of their own accord if people were to realize the difference between “formal correctness” and “validity.” You know that there is a modern school of psychology that strictly denies human free will. Every human action, it says, is strictly determined by preceding events. There are certain methods for proving this, and these play a fateful role today in statistics, for example. For instance, someone might investigate how many people in France die by suicide. That’s easy enough; you don’t even have to think about it: you record the figures over a period of about five years, then examine them for another five years, and so on. Then the researcher finds that there is a certain difference between these figures. Now they take larger data sets, compare twenty-year periods, and find that the suicide rates are almost the same; not exactly the same, of course, because circumstances change—let’s say they increase at a certain rate. One thus discovers a numerical pattern according to which one can predict how many suicides will occur within a certain period, how many people are bound to die by suicide during a certain time frame. Now there are people who say: if one can predict how many people would commit suicide, how can one still speak of human freedom? The same applies to estimating future crimes. According to an immutable causality—so they say—a certain number of people are bound to become criminals. - This is not to say that the law is invalid. In a certain sense, it is certainly applicable in practice to certain cases. But the moment this law is applied to investigate and fathom the nature of things or human nature, the gravest misunderstanding will result.

[ 14 ] Let’s consider insurance companies that use probability theory. They arrive at very specific formulas based on empirical findings: a certain number out of every hundred married twenty-year-olds will lose their spouse to death over the course of thirty years. They calculate the percentage over a specific period of time and use it as a basis for setting insurance premiums. It is certainly practical to apply such laws in the insurance industry; these laws hold true; but they do not delve into anything deeper. Things get strange when we take these laws more deeply! Imagine that someone were presented with the data from such an insurance company and discovered: There is still a spouse alive who, strictly speaking, should already have died; yet this person is healthy, and according to their inner nature, it does not even occur to them to die yet. - Nevertheless, the insurance company still gets what it is entitled to, for the formal laws certainly apply in the world, but such laws do not allow us to look into the inner nature of a being. And the same is true of all the laws of nature that are derived solely from the collection of external observations. One gains only an understanding of the external course of events, but cannot infer the inner nature of a thing or a person—for example, whether that person is healthy or sick. Similarly, you can never arrive at an understanding of the nature of light simply by observing its phenomena. This must be kept in mind; otherwise, one arrives at conclusions such as those of Exner in his final address as rector in Vienna.

[ 15 ] External facts are irrelevant to the inner nature of a thing. There is still a great deal of unclear thinking about this among humanity. It is not to be claimed that one can learn to think through logic; that is just as impossible as becoming a musician through the study of harmony. But logic is necessary for correct thinking, just as the study of harmony is necessary for composing for the true musician. One must know how judgments and conclusions are formed. However, we must always remain within the same realm if we wish to form formally correct judgments. For example, consider the syllogism: All humans are mortal. I am a human. Therefore, I am mortal—this does not appear to be a fallacy, because it refers back to the subject. The laws of logic, however, apply only if one remains on the same level. The conclusion “Therefore, I am mortal” refers only to the body. Our “I,” however, belongs to a different level; it is not mortal. The conclusion “Therefore, the ‘I’ is mortal” is therefore false. Such formal errors are frequently found among today’s scholars.