Logic is the science of the mind, known since ancient times. It is used by all people, regardless of place of birth, when they think and draw conclusions about something. Logical thinking is one of the few factors that distinguish a person from an animal. But simply drawing conclusions is not enough. Sometimes you need to know certain rules. The de Morgan formula is one such law.
Brief historical background
Augustus, or Augustus de Morgan, lived in the middle of the 19th century in Scotland. He was the first president of the London Mathematical Society, but became famous mainly thanks to his work in the field of logic.
He owns many scientific works. Among them are works on the topic of propositional logic and class logic. And also, of course, the formulation of the world famous de Morgan formula, named in his honor. In addition to all this, August de Morgan wrote many articles and books, including “Logic Is Nothing,” which, unfortunately, has not been translated into Russian.
The essence of logical science
At the very beginning, it is necessary to understand how logical formulas are built and on what basis. Only then can we begin to study one of the most famous postulates. In the simplest formulas, there are two variables, and between them a series of signs. In contrast to what is familiar and familiar to the average person in mathematical and physical problems, in logic, variables most often have a letter rather than a numerical designation and represent some kind of event. For example, the variable “a” can mean “thunder will strike tomorrow” or “the girl is telling a lie”, and by the variable “b” they will mean that “tomorrow will be sunny” or “the guy is telling the truth”.
An example is one of the simplest logical formulas. The variable "a" means that "the girl is telling a lie", and the variable "b" means that "the guy is telling the truth."
And here is the formula itself: a = b. It means that the fact that a girl is telling a lie is tantamount to a guy telling the truth. We can say that she tells a lie only if he tells the truth.
The essence of de Morgan's formulas
In fact, everything is pretty obvious. The formula of de Morgan's law is written like this:
Not (a and b) = (not a) or (not b)
Translating this formula into words, the absence of both “a” and “b” means either the absence of “a” or the absence of “b”. Speaking in a simpler language, if there is no a and b, then there is no a or no b.
The second formula already looks a little different, although the essence in general terms remains the same.
(Not a) or (not b) = Not (a and b)
The denial of conjunction is equal to the disjunction of negations.
A conjunction is an operation that, in the realm of logic, is associated with the union of "and."
A disjunction is an operation that in the realm of logic is associated with the union “or”. For example, "either one, or the second, or both at once."
The simplest examples from life
The following situation can be cited as an example: it cannot be said that the study of mathematics is both meaningless and stupid only if the study of mathematics is not meaningless or it is not stupid.
The following statement can be considered as another example: it cannot be stated that tomorrow will be warm and sunny only if tomorrow is not warm or tomorrow will not be sunny.
This is not to say that a student is familiar with physics and chemistry if he does not know physics or does not know chemistry.
It cannot be stated that a man is telling the truth and a woman is telling a lie only if the man is not telling the truth or if the woman is not telling a lie.
Why seek evidence and formulate laws?
The de Morgan formula in logic ushered in a new era. New options for calculating logical tasks became possible.
Without the de Morgan formula, it has already become impossible to do in such fields of science as physics or chemistry. There is also a type of equipment that specializes in working with electricity. There are also in some cases, scientists use the laws of de Morgan. And in computer science de Morgan's formulas managed to play an important role. The field of mathematics, which is responsible for the relationship with the logical sciences and postulates, is also almost entirely based on these laws.
And finally
Without logic, it is impossible to imagine a human society. Most modern technical sciences are based on it. And de Morgan's formulas are undeniably an integral part of logic.