What are the 3 types of conclusions

The syllogisms of Aristotle

In the last blog post, in the words of According to Aristotle, there are three types of conclusions. The logical conclusion, the dialectical conclusion and the fallacy. Now we have to deal more closely with the logical conclusion.

This conclusion, in which a true conclusion is derived from two true premises, is of course the most important conclusion, because it guarantees a secure "transport of truth" of statements. The good news is that there is such a thing, and it cannot be overestimated. The “bad news” would be that there is no truth in it in any way. It is only passed on because the premises must already be true. The question of how and where one can begin with true statements in practice will occupy us a lot later.

In the logical conclusion one can again distinguish between several types. These differ in the type of premises and the conclusion that results from “adding up” the requirements. “Adding up” means “συν-λογισμός” in Greek; one also speaks of a syllogism; the doctrine of syllogisms is called syllogistics.

First, let's look at an example of a syllogism:

All people are mortal
All Greeks are human
So: all Greeks are mortal

So there are three statements, two antecedents, from which one starts, and a conclusion, in which the antecedents are "added up".

The form of the sentences and their representation

In this example all sentences are of the form “All A are B”. The sentence “All A are B” can also be formulated as follows: “B belongs to all A.” This formulation suggests that B is a predicate of all A: B is assigned to all A (predicted). To be mortal belongs to every human being.

The formulation is also the one that comes closest to the Greek text, so it is the more original form. In scholasticism this was then rewritten as “All A are B”. Every A has the predicate B: So every person is mortal.

The above syllogism then reads in its original form:

To be mortal belongs to every human being
To be human belongs to every Greek,
So: to be mortal belongs to every Greek.

In addition, this formulation also suggests a method that was not discovered until much later to illustrate the relationship between two terms. One makes use of the fact that a term can also be described by its scope of meaning. Then each term applies to a number of entities: “People” can be Greeks, but also Egyptians, Thracians, Asians or Europeans. If the set of people is represented by a circle or some closed curve, the set of Greeks as well, then the circle for the Greeks lies within the circle for the people.

Fig. 1: The set of Greeks (A) is a subset of people (B). To be human (B) belongs to all / every Greek (A).

Such images are called Venn diagrams, after the mathematician John Venn (1834 to 1923), who introduced them following Leonard Euler (1707 to 1783). In fact, Gottfried Wilhelm Leibniz (1646 to 1716) already used it. These Venn diagrams therefore generally illustrate the relationships between two sets, regardless of the nature of the elements. In the context of set theory we would also write: A \ (\ subset \) B, i.e. A is a subset of B.

But this is not the only form of a sentence for a relationship between two terms. Aristotle got an overview of all possible forms of sentences or statements. This resulted in: (Quote from (Höffe, 2009, p. 47)):

  • A proposition is a statement that affirms or denies something about something;
  • Such speech is either general or particular or indefinite.
  • In general I call a statement that says that everyone or nobody should
  • particular is a statement that expresses any coming-of-any, non-coming-of-any or not-coming-to-everyone.

In addition to the general sentences such as “All Greeks are human” or “To be human, every Greek belongs to” there is also the negation (no-coming) and the particular statement (any-coming, but not everyone).

Overall, you get the following types of sentences:

B belongs to every A, (All A are B),
B does not belong to any A, (no A is B),
B comes to some A, (some A are B),
B does not belong to some A (some A are not B).

We can illustrate the other types of statements as follows:

In scholasticism, these forms are abbreviated as (A a B), (A e B), (A i B) and (A o B). The letters “a” and “i” should remind of “affirmo”, “e” and “o” of “nego”.

The structure of the syllogisms

Two such forms of statement then form the prerequisites, one the conclusion. Each conclusion can therefore be characterized by three letters from the set {a, e, i, o} and by the position of the three letters that stand for the terms in the respective sentence.

One of these terms, the so-called middle term, must appear in both prerequisites, it can be in first or second position in each of these, or in one prerequisite first and in the other second. This results in four different shapes or figures. The first figure is the following (see example above)

C - B
B - A
C - A.

Now Aristotle seeks out in all forms that combination of two forms of statement which, as prerequisites, necessarily lead to a conclusion. He simply sorts out the conclusions for which he finds a counterexample.

The valid conclusions can then be represented by the form and a certain combination of the letters a, e, i, o. And in order to be able to memorize such combinations better, you have integrated them into corresponding keywords, e.g. you memorize, for example, the combination a a a with the word "Barbara", and also know that the 1st figure is present here. This conclusion corresponds exactly to the example above.

Another important conclusion, also from the 1st form, is called "Celarent" in this way.

In this way Aristotle demonstrated that one could systematically formulate a system of inferences in which the truth of the presuppositions necessarily follows the truth of the conclusion.

With the systematics of an Aristotle one now has a complete overview of all possible conclusions. Before that, in syllogistics, “the whole trick was to search around aimlessly with a great deal of time and effort”. So should Aristotle in his Sophist Refutations have sued (after Schupp, I 275).

With this Aristotelian logic we already have a system that is reminiscent of the formal predicate logic to be introduced later. As there, one can "quantify" using the predicates, i.e. one can operate with quantities such as "all", "none" and "some" for the predicates.

Most important, however, is that Aristotle has to choose the valid inference rules “by hand” from the set of all possible combinations, simply by discarding those that he recognizes as invalid with the help of an example. This "knowing" is an intuitive one, one with "common sense". One does not doubt the correctness of the final rule, but for a strict science in the modern sense, this kind of knowledge is not enough.

Even if one describes relationships of concepts with the help of set theory today, and thus justifies the rules of inference within the framework of set theory, the knowledge is mathematical. This would only indirectly justify the inference rules, because mathematics only uses, as we know today, the inference rules that are ultimately obtained in modern predicate logic. It is only in this that the inference rules can be strictly justified by deriving them from tautologies. We'll see that in a later blog post. In predicate logic, then, the bottom has only been reached on which one can undisputedly gain a true sentence from true sentences. This incontestability is not based on the fact that the conclusions are "immediately obvious", but on a single, much more fundamental assumption in the predicate logic, which makes it even more "more immediate" to us.

We still have to comment on Aristotle's remark on the dialectical conclusion, i.e. on the conclusion in which the assumptions are only likely to be true or only credible. Then the conclusion could only probably be true. It was not possible to say more here at the time. It was not until the beginning of the 20th century that a probability theory emerged with which one can become more precise in this case. This will be explained in more detail in a later blog post on the subject of “Dealing with uncertain knowledge”.

The syllogisms in an axiomatic-deductive system

Aristotle's logic not only showed us how one can safely pass on certain knowledge. He has also shown that “all inferences can be traced back to general inferences from the first figurecan like him in the Doctrine of the End, Book 1, Chapter 7 writes. (Anon., No date)

So we already have a concrete axiomatic-deductive system. This structure of the statements of a field of knowledge represented the ideal of a science for him. In the 2. Analytics, the doctrine of knowledge, book 1, chapter 3 he writes:

  • I contend, however, that all science must be based on evidence, but that knowledge of the unmediated principles cannot be proven. And it is clear that this has to be the case. For since knowledge of the earlier sentences from which the proof is made is necessary, but one stops once at unmediated sentences, these must necessarily be unprovable.
    This is my view, and I maintain that there are not only sciences, but also supreme principles of the same, through which we learn the concepts of inference. (Anon., No date).

The mathematician Euclid of Alexandria will logically organize the geometrical knowledge of his time around -300 in this way. The organization of a scientific theory as an axiomatic-deductive system represents a model for every science to this day. Attempts have been made in all centuries to imitate this organization of a building of thought, ie the knowledge of the sciences "more geometrico", ie in the manner of Euclidean geometry assign. In The Idea of ​​Science - Your Fate in Physics, Law, and Theology I have described how successful you have been in realizing this idea to date (Honerkamp, ​​2017).

It is unclear who was the first to come up with this idea. The mathematical proof was already known to the Pythagoreans. If there are enough statements secured by evidence in a field, one will probably consider at some point which statements one could consider as axioms. This poses the question: Which statements do I need as a specification in order to be able to derive all the others from them? So: How do I create a logical order? Such specifications will then always be special statements, have special properties.

These statements can be immediately understandable, ie “certain by themselves”, like the syllogisms of the first form or like in general in theories of mathematics, but also highly abstract and far removed from our perception like in theories of physics. In any case, a "logical order" has been established, i.e. the truth of a large number of statements has been reduced to the truth of a few statements in an incontestable manner. That is exactly what constitutes a “strict science”.

Josef Honerkamp was professor for theoretical physics for more than 30 years, first at the University of Bonn, then for many years at the University of Freiburg. He has worked in the fields of quantum field theory, statistical mechanics and stochastic dynamic systems and is the author of several text and non-fiction books. After his retirement in 2006, he would like to devote himself even more to interdisciplinary discussions. He is particularly interested in the respective self-image of a science, its methods as well as its basic starting points and questions and can report on the views a physicist comes to in view of the development of his subject. Overall, he sees himself today as a physicist and "really free writer".