De Morgan’s second law: the negation of disjunction is equivalent to the conjunction of negations.
The law of distributivity is a logical law that allows one logical conjunction to be distributed relative to another.
The law of distributivity of a conjunction with respect to disjunction: in formulas it is possible to distribute a conjunction with respect to disjunction. Scheme of the law: (Al (BvC) (AlB) V (AAC)) (“A and (B or C), if and only if (A and B) or (A and Cj”).
Law of distributivity of disjunction with respect to conjunction: in formulas it is possible to distribute disjunction with respect to conjunction. Scheme of the law: (AV (BAC)
Laws of de morgan
De Morgan’s laws are logical laws that link negation, conjunction, and disjunction. De Morgan’s first law: the negation of a conjunction is equivalent to the disjunction of negations. Scheme of the law: (AAB) (AVB) (“False that A and B if and only if false that A, or false that B”).
De Morgan’s second law: the negation of disjunction is equivalent to the conjunction of negations.
Scheme of the law: (AVB) (AAB) (“False that A or B if and only if false that A and false that B”).
De Morgan’s laws make it possible, using negations, to express the logical connection “conjunction” through the logical connection “disjunction” and vice versa.
Modern logic: features, objects and meanings. Abstract
Features of modern logic. Subjects of logic as a philosophical science. The value of logic
Features of modern logic
Modern logic was formed in the late nineteenth century. – in the early twentieth century. but its founder can be considered Gottfried Leibniz – his works were ahead of his era for several centuries.
Modern logic, the founder of which was G. Leibniz, differs significantly from the traditional, the foundation of which was laid by Aristotle. In the second stage of development of logical knowledge, the interests of logicians are significantly expanded. They begin to turn to the analysis of such types of reasoning, which were previously denied the possibility of logical analysis.
Thus, along with various types of theoretical (scientific) considerations, the main purpose of which is to substantiate knowledge, the subject of study of many logicians are practical considerations, the main purpose of which is to explain human actions. New sections of logical knowledge are emerging, essentially related to the various branches of scientific knowledge and the types of reasoning in them. These are mathematics, law, linguistics, philosophy, psychology, economics, computer science, etc.
Initially, modern logic was completely focused entirely on the analysis of only mathematical considerations. With its help, scientists tried to solve the problem of the foundations of mathematical knowledge after paradoxes were found in set theory. This period in its development is sometimes called “classical”.
At the source of classical logic stood alongside many researchers George Bull (1815 – 1864), Augustus (Augustus) de Morgan (1806 – 1871), Charles Pierce (1839 – 1914), Gottlob Frege (1848 – 1925), David Gilbert (1862 -). 1943) and others. In their works, the idea of transferring to logic those methods that are commonly used in mathematics was gradually realized. The result of this work was the creation of such sections of modern logic as the logic of statements.
Subjects of logic as a philosophical science
The subject of logic are the laws and forms, techniques and operations of thinking, through which a person learns about the world around him.
Cognition as a process of reflection of the objective world by human consciousness is the essence of sensory and rational cognition.
Sensory cognition has 3 main forms:
Sensation – a reflection of certain sensory properties of objects (for example: color, shape, smell, taste, etc.).
Perception – a holistic image of the object, arising from its direct impact on the senses.
Imagination is a sensory image of an object that has remained in the mind. If perception is a direct influence, then imagination is when such influence no longer exists.
Images of the imagination can be arbitrarily combined.
Unlike sensory cognition, thinking reflects the external world (and not only!) In abstractions (distractions). Departing from the concrete in things and phenomena, abstract thinking is able to generalize many homogeneous objects, to single out the most important properties, to reveal essential connections.
The main properties of abstract thinking:
Thinking reflects reality in generalized images. Thinking – the process of indirect reflection of reality. Thinking is inextricably linked to language. Thinking – the process of active reflection of reality (new knowledge). Thinking obeys the logical laws of thinking.
It is necessary to distinguish between the truth of thought and the logical correctness of reasoning. The opinion is true if it is true, and vice versa. (Kolomyia – Asia). Logical correctness of reasoning is a condition of truth of opinions. This is a reasoning in which some opinions (conclusions) necessarily follow from other opinions. The law of thought, or the law of logic, is a necessary, essential connection of thoughts in the process of reasoning. A distinction should be made between formal-logical and dialectical laws (oral explanation).
The main forms of abstract thinking – concepts, judgments and inferences.
By highlighting a certain set of general, essential properties or features, we create the concept of an object (Concept A is a set of features a, b, c, etc., which are related in a certain way). Thus, different objects are reflected in thinking in the same way – as a certain connection of their essential features, ie in the form of a concept.
Judgments show the relationships between objects and their properties. This connection is expressed in the form of an affirmation or denial. Any type of judgment is a scheme S – P, where S (subject; the concept of the subject of the judgment) and P (predicate; the concept of the sign), and the sign “-” the relationship between them.
Thus, judgment is a way of connecting concepts, expressed in the form of assertion or denial.
A inference is a combination of several judgments (called bases) from which a new judgment (conclusion) necessarily follows. (“Fedor is a witness” “Witnesses testify” “Fedor testifies”). So, we highlight something in common that has different inferences in terms of content: the way judgments are connected (a general concept). Thus, common to all forms of thinking is the way of connecting the elements of thought – the signs in the concept, the concepts in the judgment and judgments in the inferences.
The logical form, or form of thinking, is a way of connecting the elements of thought, its construction, through which the meaning exists and reflects reality.
The study of logical forms, regardless of the specific content and is the most important task of the science of logic.
Thus, logic is the science of human thinking. But unlike other sciences that study human thinking, logic studies thinking as a means of cognition, its subject is the forms and laws, techniques and principles of thinking by which man learns about the world around him.
The value of logic
Human thinking obeys logical laws and proceeds in logical forms regardless of the science of logic. People think logically, without even knowing that their thinking is subject to logical laws. But it does not follow that lawyers or other people do not need to study logic.
Knowledge of laws and forms of thinking, their conscious use in the process of cognition increases the professional culture of thinking, develops the ability to think more competently, develops a critical attitude to their own and others’ thoughts. Therefore, the view that the study of logic has no practical significance is wrong.
To think logically means to think accurately and consistently, without contradicting one’s reasoning, to be able to expose logical errors. These qualities of thinking are of great importance in any field of scientific and practical activities.
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Antinomy and paradox. Abstract
The concept of paradoxes and antinomies in logic. Variants of the “Liar” paradox. Language and metalanguage. Other solutions to the paradox
It is known that formulating a problem in most cases is more important and more difficult than solving it. The forms in which the problem situation is detected and realized are very diverse. It does not always manifest itself in the form of a direct question that arose at the very beginning of the study. The world of problems is as complex as their process of cognition. Identifying problems is connected with the very essence of creative thinking.
Paradoxes are the most interesting case of implicit, questionless ways of posing problems. Paradoxes are common in the early stages of development of scientific theories, when the first steps are taken in the still unexplored field and the most general principles of approaches to it are felt.
The concept of paradoxes and antinomies in logic
In a broad sense, a paradox is a position that differs sharply from generally accepted, well-established, orthodox views.
The paradox in the narrower and special sense is two opposite, incompatible statements, for each of which there are seemingly convincing arguments.
The sharpest form of the paradox is antinomy, a reasoning that proves the equivalence of two statements, one of which is a denial of the other.
Paradoxes are especially popular in the most rigorous and exact sciences – mathematics and logic. And this is no accident.
Logic is an abstract science. There are no experiments in it, there are not even facts in the usual sense of the word. Logic ultimately comes from the analysis of real thinking. But the results of this analysis are synthetic, indivisible. They are not statements of any particular processes or events that the theory should explain. Such an analysis cannot, perhaps, be called an observation: there is always a specific phenomenon.
In constructing a new theory, the scientist usually starts from the facts, from what can be observed in experience. No matter how free his creative imagination is, it must be considered with one essential circumstance: the theory makes sense only if it agrees with narrative essay topics for grade 9 the facts relevant to it.