Logic as a science is a subject almost as old as mathematics. In ancient times and the Middle Ages, it was part of the trivium (grammar, rhetoric, logic/dialectic), the basic level of education; the mathematical subjects (arithmetic, geometry, astronomy, and music) comprised the next, more advanced level, called the quadrivium. (From the word “trivium” comes one of the mathematicians’ favorite expressions, “trivial.”) The subjects of the trivium were understood to be the sciences of how to write, speak, and therefore reason correctly, without error.

We will talk about how and why mathematical logic arose, what it studies, what its achievements and modern applications are.

The foundations of the doctrine of correct reasoning were laid by Aristotle. He observed that correct inferences follow certain elementary schemes called syllogisms and listed a number of such schemes. (A classic example of a syllogism: “All men are mortal. Socrates is a human being. Consequently, Socrates is mortal.”) The doctrine of syllogisms in turn relied on an in-depth analysis of concepts and their conjunction into statements.

Aristotle’s syllogism was not without its drawbacks, but on the whole it was an outstanding theory and became the basis for the study of logic throughout antiquity and the Middle Ages. In the works of ancient Stoics and medieval scholastics it was modified and supplemented. In this form Aristotelian logic had existed up to the middle of XIX century, where it had met with revolution connected to penetration into logic of mathematical methods.

The emergence of mathematical logic has completely changed the views of scientists, both about the methods of research of logic, and about what constitutes its very subject of study. Nowadays, the statement that logic is the science of correct reasoning seems as true as the statement “mathematics is the science of correct computation”.

The analogy between reasoning and computation is somewhat deeper than it first appears. The emergence of logic as a mathematical science was associated with the work of British scholars George Bull and August de Morgan, who discovered that logical statements could be operated on as algebraic expressions. For example, if addition is read as the logical conjunction “or”, multiplication as “and”, and equality as “equals”, then for any utterance , the laws as well as many other laws of arithmetic that we are familiar with.

This view of the logic of statements and syllogism proved to be both unexpected and fruitful. Nowadays this view has been developed by the field called algebraic logic, and one of its central concepts is the concept of Boolean algebra, named after its discoverer. This field of research, through the concept of relational algebra, which generalizes the Boolean algebra, led in the 1960s to the theory of relational databases, now the basis of the most common query languages such as SQL.