Judgments, conclusions, syllogisms ... or the achievements of ancient logic in one post

When I was in school, we studied logic, but now for some reason they don’t even teach it in my beloved lyceum. Moreover, I learned that most of my acquaintances (even those who successfully graduated from higher educational institutions) do not know about the logical square, nor about the various modes. In this small topic, I will try to briefly tell you about everything. At once I will say that the gurus of discrete mathematics are unlikely to learn anything new, but the rest should be at least interesting, but as useful as possible.

Judgments

Let's start with the basics. Those. with judgments. If you do not introduce strict definitions, then in the judgments just something is approved or denied. In the overwhelming majority of languages, judgments are constructed according to the following form S is P, while S is called the subject of the judgment, and P is the predicate.
Judgments can be divided according to a variety of criteria. For example, on simple (judgments without logical connectives) and complex. Simple judgments can be divided into attributive (state or deny the presence of the attribute), existential (state or deny the existence of something) and judgments with relationships. Another way to classify judgments is a classification by quality: statements are divided into affirmative and negative. The most interesting classification for us is the classification by quantity. We will take a closer look at it.
• The general affirmative statements of the form “all S are P” are called type A judgments.
• Partly affirmative statements like “some S are P” are called type I judgments.
• Generally negative judgments of the form “no S is P” are called judgments of type E.
• Private negative judgments of the form “some S is not P” are called O-type judgments.
The tradition to denote judgments by the letters AIEO originated in the Middle Ages. These vowels are taken from the Latin words affirmo (affirm) and nego (deny).

Logical square

The classification of judgments by number is important, because on its basis was built the famous logical square.

The corners of the square show the types of judgments, and on the sides and diagonals of the square indicate the relationship between the respective judgments. These relationships require some explanation.
If between judgments acts of submission , then their truth can say the following. If the general judgment is true, the subordinate judgment is also true. If the general judgment is false, then nothing definite can be said about private judgment. If the private judgment is true, then nothing definite can be said about the general. If the private judgment is false, then the general judgment is also false.
Contradictions can be false at the same time, but they cannot be true at the same time.
Subcontracted judgments can be true at the same time, but they cannot be false at the same time.
Contradictory judgments cannot be both true and false. Those. if one of the counter-judgments is true, then the second is necessarily false and vice versa.

Inference

Having dealt with the judgments, you can go to the rules for obtaining new judgments, i.e. conclusions. Let's start with the most simple immediate inferences.

Simple conclusions

Transformation

New judgment is obtained by changing the quality of the package. For this, it is necessary to insert two negations into the initial judgment: before the bundle and before the predicate. In fact, this is a very simple inference, in fact it comes down to the following transformations of AE, EA, IO, OI.
Those. the judgment “some people are talented” after transformation becomes the judgment “some people are not talented” (OI).

Appeal

In the appeal, the new inference is obtained after rearranging the subject and the predicate in some places. Those. the judgment “S is P” turns into a judgment “P is S”. Unfortunately, this operation cannot be applied to any statements, otherwise from the statement “cats are mammals,” we would get “mammals are cats.”
In order for the appeal to be correct, the following rules must be observed:
• Generally affirmative judgments turn into privately affirmative
• Generally negative judgments turn into generally negative judgments.
• Private affirmations turn to private affirmative statements.
• Private negative judgments are not addressed at all.

Opposition to predicate

This is the most complex operation, which is essentially a combination of transformation and circulation. In practice, it looks like this: “S is P” turns into “not P is not S”. I will not specifically mention here the restrictions imposed on the opposition to the predicate, so that you yourself can think a little.

Syllogism

And while we begin to consider syllogisms. Syllogisms are the most popular type of judgments, it includes three judgments (two premises and a conclusion) and three terms.
The lesser term (S) is the subject of the judgment that came out as a conclusion. The larger term (P) is an output predicate. The middle term (M) is included in both premises, but is missing in the output.
For a syllogism to be correct, it must obey three groups of rules: the rules of terms, the rules of premises, the rules of figures.

Terms of Terms

• There should be exactly 3 terms in the syllogism.
• The average term should be taken in full in at least one of the premises.
• If the term is not taken in full in the parcel, then it cannot be taken in full and in output.
To understand the importance of these rules, I’ll give just one example: “Some living creatures are poisonous. Seals are living creatures. Seals are poisonous. " Which rule is violated, try to determine for yourself.

Shipping Rules

• Of the two negative premises does not follow any conclusion.
• If one premise is negative, then the output must be negative.
• Of the two private parcels does not follow any conclusion.

Rules of shapes

Do you remember that there are three propositions and three terms in syllogism? By the mutual arrangement of terms in judgments, syllogisms can be divided into 4 classes (figures):

Rules of shapes:
• In the first figure, the first premise must be a general proposition, and the smaller one an affirmative
• In the second figure, the big premise must be a general judgment, and the smaller premise and conclusion are negative.
• In the third figure, the smaller premise must be an affirmative judgment, and the conclusion is private.
• The fourth premise is less common, it has two rules: 1. if the total premise is an affirmative judgment, then the smaller premise must be a general judgment; 2. If one of the parcels is negative, then the big parcel should be common.
Again, breaking the rules of the figures leads to quite funny logical errors: “All the seals drink water. I drink water. I'm a cat.
In fact, syllogisms can be divided not only by the mutual arrangement of S, P, M, but also by the types of judgments (A, I, O, E) that are included in syllogism. It is not difficult to notice that in total 64 different syllogisms are possible, these syllogisms are called modi. If we apply all the restrictions and rules that we talked about to the modes, it turns out that there are only 19 logically correct modes and they are distributed among the figures as follows:

If you know these rules and use them, then you can, first of all, not allow stupid mistakes yourself, and secondly, notice these mistakes of your opponents in disputes.
In fact, what is described in the post - only a small part of the achievements of ancient thinkers. We didn’t talk at all about enthymema, epichauram, nor categorical syllogism, nor about ... Yes, we almost didn’t talk about anything, but I still hope that you were interested.