How is traditional logic possible from the modern logic point of view?

Anatoliy Konversky,academician of National Academy

of Science of Ukraine,Dean of Philosophy Faculty

Taras Shevchenko National University of Kyiv

Dear colleagues, participants of the conference!

Logic is one of the oldest areas of human knowledge. It has a special place in the spiritual culture of man. Its role in the modern world of science is extremely important and multifaceted. Of course, during the time changes the orientation of logical researches, logical methods begin to improve, appear new trends, which are relevant to today's needs of scientific and technological progress.

It should be stressed that after the death of the ancient civilization the first thing, which was recovered from the ancient science - this is the logic of Aristotle. We also know the negative attitude of the Middle Ages to the Antique science, but its fundamental recognition started from the first seven chapters of the Aristotelian "Analitics".

During the Renaissance Period, logical

methods, which were opened in ancient times, were reinstated and actively used. From this begins the philosophy of Rene Descartes and other thinkers, from time begins the whole science of modern times.

Aristotle created the logic as a way of defending the truth and expose the sophistry. This logic quality remains indispensable for more than two millenniums.

During the Middle Ages period scholastics continued to develop different problems of logic. They entered the Latin terminology into the logic. In modern times, Francis Bacon explored the basis of inductive inferences. The teachings of the great German philosopher and mathematician Leibniz initiated the second phase of logic - modern or symbolic logic (which appeared in the middle of the nineteenth century).

When we want to write an apprentice of traditional logic appears rather complex problem, which is connected with striking successes of modern logic. Because of these successes, especially in the theory of inference and logical semantics, appeared the idea about the uselessness of traditional logic.

Nevertheless, people as two thousand years ago, continue to argue, improve, refuse and use the natural language during these actions. And in these circumstances, the traditional logic unit is the most effective means.

So, appeared the next question: how to reconcile the teaching of traditional logic with the results of modern logic? In modern logic, many problems of traditional logic have a new look, for example, a problem with the relationship judgments, complex judgments, laws of logic, etc. Modern logic has opened new forms of reasoning, new kinds of logical relationships. Therefore nowadays it is impossible to expound the traditional logic without the achievements of modern logic.

But, how to combine the material of traditional and modern logic in a single course? Clearly, that these are two different logical systems, two different languages, two different stages of the same science. But we are talking about the traditional logic as an academic discipline, so this association is necessary.

So, the main difficulty of this situation is a fundamental difference between traditional and modern logic in the approach of the analysis of reasoning.

Traditional logic analyzes thinking, in particular, such its shape as concept, judgment, reasoning, and modern logic explores language and its substantial sense. Therefore its major category is not a "form of thought" but "statement" and those language expressions which have independent value as part of the utterance. And it is very difficult to combine these two approaches.

Misunderstanding of the sense of this approach to the presentation of the course of traditional logic leads to the extremes that have emerged over the last 50-60 years in modern and foreign scientific literature.

In some textbooks of traditional logic authors include the material from modern logic. But it looks like a foreign body in the content of textbooks.

Sometimes a textbook of traditional logic is called a textbook of modern logic, because the summary of the main topics of the traditional logic course is diluted by information of modern logic without any explanation.

We can select some textbooks, where the material of traditional logic is presented entirely without the achievements of modern logic. For example, it takes place in the textbook "Logic for Lawyers“ by Vladimir Kurbatov (Publishing House" Nauka -Press", Moscow, 2006). However, this textbook includes chapters "Classical logic" and "non-classical logic". But they are not connected with that part of the textbooks, which refers to the traditional logic.

Thus, we can see a clear desire to embellish and to retouch the traditional logic.

What the consequences of this approach do we have today? We can say about reducing of logic courses for non-specialist departments, or worse, complete extraction of logic from course curriculum. A consequence of this is the absence of prospects for the formation of theoretical thinking skills in students of the humanities and natural faculties.

The way out is to strictly distinguish the difference between the goals and objectives of traditional logic (we must not reduce it to Aristotle logic) and modern. Each of these logics has its goals and objectives.

And if traditional logic explores how the thinking and knowledge "live", primarily in natural language, the modern study of logic explores the same thinking and knowledge, which have a theory as a unit of organization, in the language of science.

Therefore, where it is appropriate, where it is effective to apply the achievements of modern logic in the traditional logic course - it is essential. But if it is not appropriate to apply the achievements of modern logic in the traditional logic course then it is unnecessarily to do it.

It should be axiomatic for those who today are writing a tutorial of traditional logic or who are teaching such logic course.

The foregoing can be reformulated in words of the great Stephan Kleene, which he expressed about a strict discrimination between object-language and meta-language: "We must always remember the distinction between the logic which we study (subject) logic and logic as a means of such study (the logic of the researcher). Those who are not ready for it, must immediately close this book and find something else to do (for example, drawing charades or beekeeping)".

But those who insist on lecturing the logic course to students of non-special faculties (students of historical, geographical, physical, philological, legal departments , etc. ) without setting out the ways of thinking, it is necessary to build a logic course in accordance of nature and objectives of modern logic.

In this case, logic textbook or logic course should include sections of propositional logic and predicate logic, each of them is represented by algebra and calculus.

Then, even if the tittle of the textbook or course will not be marked, as a textbook of modern logic, then from the table of contents you can understand that the term "logic" does not refer to traditional logic, and refers to the modern logic.

This version of the logic textbook lets the corresponding faculty orders not only a "Logic" course, and the course of "Classical logic" or "non-classical logic". And in the standard curriculum must be the requirement for the author of the course. According to this requirement, the author should not give only a variety of information from the logical science, but strictly presents the material from the course of "classical logic" and "non-classical logics". The author must not read a lecture which includes modern logic elements and fragments of traditional logic“.

Returning to the version of the "Logic" textbook for students of faculties where there is no specialization in logic, I want to pay your attention to the fact that its basic maintenance must be followed after the special "Prolegomena“.

Such "Prolegomena" must necessarily include following sections:

Section I-st. Thought and language; (in this section should be emphasized that in the process of thinking we operate mental content that does not directly coincide with the objective reality from which it is abstracted. Only in a language this maintenance really exists as something ideal. Therefore, the language is a reality and logic deals with it.

Section II-nd. Levels of semiotic analysis of language; (here it is specially necessary to point out the distinction between empirical and pure semiotics. The task of empirical semiotic - to study historically arisen languages. The task of pure semiotic – to analyze artificial language systems, especially those, which were created in logic and for logic needs).

Thus, the logic uses a specific language, which is not spoken language, but it is a tool, a method of researching of the subject of logic.

Section III- rd. Formalization as a method of logic; (this section should be focused on the fact that the purpose of the use of artificial languages in the logic is not replacing of natural language words for some special characters in the process of description of logical rules and procedures, but reproduction of logical deductions).

Section IV- th Semantic analysis of natural language expressions; (the goal of semantic analysis of natural language expressions is to identify those idioms which are carriers of logical forms, their properties and relations).

Section V- th. Elements of the theory of names; (the objective of this section is to show that the logic of the study is interested in the following points:

a) how to relate the name and concept (or more precisely the meaning of the name and content of the concept).

b) how a logical value of statement depends on the names, which it contains;

c) what kind of logical means can provide the invariability of statements in their interaction, in the process of reasoning).

Section VI-th. Functional analysis in logic; (in this section we should show that we use the functional approach in logic because the language does not directly express the forms of thought. Therefore, only when we analyze the method of operation (use) of the corresponding segment in the structure of language statement, we can clearly say what kind of logical form it is.

So "Prolegomena", which was represented, allows the most efficiently and with maximum benefit to apply modern logic device when we present the material of standard course of "Logic" for non-special faculties.

This approach we will illustrate in specific themes of this course. Let’s take the theme "Concept". It is known that the main difference between the concept and the name shows that the concept generalizes objects, and the name calls the objects. So, arises the question: what is the object of thought, which is displayed in the concept and how it can be represented by means of modern logic?

In traditional logic with a subject-predicate structure of judgment, the object of thought is presented by a logical subject (S). In modern logic the subject is a member of class, the carrier of its own name.

In other words, the difference between the positions of traditional logic and modern logics how that the object of thought in these logics is represented by different semantic categories. In traditional logic this is a predictor, and in modern logic this is a term.

Using the capabilities of modern logic we can more strictly (excluding psychological insight, intuition, common sense) reveal the process of generalizing of objects of thought in the concept:

a is Q - true b is Q - true c is Q - true .......... - ........... .......... - ........... .......... - ...........

n is Q – true_____________________

x is Q - true

The scheme shows that pre-condition of generalization is a presence of totality of true statements about each individual (a, b, c, ... n). In other words , any member of the set of objects a, b, c, ... n (we can denote it by x) also has the feature of Q, it means «X is Q» or Q(x). But Q(x) is a uniform way to represent a trait and, therefore, is nothing but a logical predicate or predicate.

We want to remind you, that if in traditional logic the subject and the predicate of judgment belong to the same semantic category – to a predictor, and in modern logic the "object of thought" is represented" by the term".

Due to this difference, the predicate is a form of propositional functions. Understanding the predicate as a propositional function allows strictly represent the process of formation of concepts. Expression Q (x) is the closest to the meaning of the expression x Q. So next statement will be fair:

x (xQ) = Q(x)

The predicate is a propositional function, so the range is the set (great number) of true singular statements:

1) (Q (a), Q (b), Q (c) ..... Q (n)); 2) (Q (a, b), Q (a, c), Q (b, c) ...),

a range of definition is the individual items or their characteristics.

Second function of logic (first function is propositional) is conceptual:

хQ (x).

In this function, the value is separate objects or characteristics of objects, and the arguments are single utterances (P (a), P (b), P (c) ... P (n)).

When we compare the predicate and conceptual function we can see that the predicate as a propositional function is the value of (Q (a), Q (b), Q (c) ... Q (n)) for concepts function is an argument.

After these remarks, with help of the modern logic we can strictly represent the structure concept as a form of thinking:

x Q (x) - the object of thought in the concept

Q (x) – the content of the concept Wx Q (x) –the scope of the concept

Based on this syntax of concept we can make a conclusion that if the scope of the concept “A” is included in a scope of the concept “B”, then each element of the concept “A” is also an element of the concept “B”. So we have the equality:

1. Wx A(x)WxB(x) = Vx (xWx A(x)) xWx B (x)),but it is known:

2. XWxA(x) = A(x),then the equality1. becomes:

3. WxA(x)WxB(x) = Vx(A(x)B(x))

Equalitcy 3. is a formula of the inverse relation between the volume and content of the concept.

But now the formulation of the law of the inverse relation between the volume and content of the concept, which is expounded in traditional logic on a descriptive level, acquires strict theoretical interpretation: "If the scope of the concept A is included in the scope of the concept B, then from the content of concept A follows the content of concept B. This means that the extensional relation of "including" between the volumes of concepts corresponds to the intensional relation of the following between their contents.

Moreover, it becomes possible to understand why the logical term "implication" can not show full value of the relation of "logical consequence“.

In the theme "Judgment" when we use the material from the section "Semantic analysis of natural language expressions" we can to introduce the structure attributive judgments, judgments, attitudes and judgments of existence by single and standard way. Such standardization is provided by application of language of predicate logic during the recording of the structure of the adopted judgments.

If in the traditional logic the structure of attributive judgment was shown by formula

«S is/is not P», then in modern logic, because of the using of means of predicate logic, it becomes possible to allocate the following nuances:

If as an object of values of variable domain we select multiple items, which are fixed by predictors in a position of logical subject, then the formula which is a translation of the attributive judgments from the language of traditional logic to the language of modern logic will contain a simple predicate:xP(x) илиxP(x);

If we change the area of value of the subject variable and consider it as a variety of arbitrary objects, then formula of predicate logic, which we use for the transferring of attributive judgments, will include a complex predicate:

x(S(x)P(x) or x(S(x)&P(x))

When we write attribute judgments, attitude judgments and judgments of existence by language of predicate logic, as we noted above, it allows by single and standard way to fix their logical structure, and thus we theoretically reproduce the principles of logical deduction during the constructing of reasoning.

In the presentation of deductive reasoning in the theme "Inference" appears the problem of lighting of inference logic judgment. If the findings from categorical judgments in traditional logic were outlined more or less in a systematic manner, it means, that the presentation of the conclusions of logic judgment was based on common sense, sense-certainty, psychologisms.

We can do a systematic review of theconclusions of logic judgment because of theusing the apparatus of propositional logic.

Thanks to the language of propositional logic we can write the scheme of inference of logic judgment in this way:

A1, А2, А3, …Аn ╞ В

It is considered that the scheme is acceptable, and the conclusion is true if and only if:

╞ A1& А2& А3, … &Аn В

When we review conclusions of logic judgment this approach will allow us not to go from the specific examples of such findings but, from the rules, that allow you to do these findings.

Systematization of rules of logic judgment by means of language of propositional logic has made it possible to use the substitution method, the method of equivalent transformation method of analytical tables to verify the correctness of any conclusion of logic judgment. It means, that we can use the logical tool that is effective in propositional logic.

These cases of using of modern logic in the main sections of traditional logic show that where the application of the apparatus of modern logic is relevant and rigorously thoughtful, only in those cases it will contribute to a didactic presentation of the material of traditional logic and will show the presentation of traditional logic not only as a simple set of empirically heterogeneous information about logical methods, rules and laws, but as a special theory of reasoning.

I would like to pay your attention at one more moment, at the nature of logical science.

The adjective "formal“ is often credited to logic, and it means primarily traditional or Aristotelian logic. But we know that Aristotle did not use the term “Logic", and the adjective "formal" had a different connotation in his works.

Two famous German philosophers Kant and Hegel called logic "formal". But, as always happens, the contemporaries of the great discoveries are not always aware their nature and importance. And many things, which Kant and Hegel called figuratively and transmitted aphoristic, commentators of their teachings often interpreted literally.

Kant opposed his transcendental logic to formal logic and Hegel opposed his dialectical logic to formal logic. But the "thing", which Kant and Hegel call the logic is not actually logic, this is their philosophical system.

Sometimes harsh stance against formal or Aristotelian logic does not mean that Kant and Hegel crossed out genius Stagirite.

Kant and Hegel realized value of Aristotle's logic teaching, they were critical of those dogmatic, unscientific complications, which were introduced into the logic more than two thousand years ago.

In the 40-th years of the XX century in the journal "Problems of Philosophy" from the side of new interpreters of Kant and Hegel started a heated debate about the nature of "formal logic". "Formal logic" was compared with the so-called "dialectical logic". For comparison, it was taken the analogy of arithmetic and higher mathematics. Formal logic (read "traditional") is announced the elementary (in the sense of the primitive, outmoded), and dialectical logic is the highest level of logic. For confirming of this position it was used the works of Marx, Engels, and Lenin.

But the thing is that the logic (traditional logic, modern logic) is really a formal logic. This is because the method of logic is formalization. And if you can call logic formal logic it means that you can called salt salty.

Another thing is that in the history of philosophy and logic an adjective "formal" does not always have the same meaning. In the tradition of classical German philosophy this adjective meant stiff, frozen, unmoving, metaphysical concepts (in this sense) education.

For Aristotle, "form" is an active, creative principle which merges with an entity. But Aristotle distinguished a form as opposed to matter (the study of philosophy) and a form as the iconic model of content of thought and knowledge.

Great Frege uses the example with Greeks, who defeated the Persians at Plataea, announces the distinction between grammatical and logical structure of the sentence. At the same time Jan Lukasiewicz loudly declares that logic deals with the thinking not more than with mathematics or any other science. But, it does not mean that Frege and Lukasiewicz did not see the contribution of Aristotle in the formation of logic as a science. Frege and Lukasiewicz, in a such acute and easily accessible form, wanted to dissociate the true Aristotle from Aristotle, who was represented by not always talented his followers and commentators.

Lukasiewicz shows in some detail in his fundamental work "The Aristotelian syllogistic from the standpoint of modern logic" that Aristotle introduces variables in the logic and uses the formalization method when he presented his theory of inference, etc.

Aristotle became a great philosopher because his logical doctrine became the basis for many future directions of logical science.

In conclusion I want to emphasize once again, that logic is the science, which has been developing all time and its value does not consist in the getting knowledge (it is an aim of concrete sciences), but in preserving the cleanliness, orderliness, perfection, perfection of knowledge, by extracting from it all new heuristic calls.

Thank you for attention!