Аннотация: Essential part of mathematics; deals with Boolean algebra.
Logic is
part of mathematics that deals with
Boolean algebra. This algebra deals with logical variables.
Logical variables may have value TRUE or FALSE. The operations
and and inclusive or are allowed with these variables; if two expressions, say,
a and b always have equal values, it is indicated with special symbol "=", which means equivalence;
one writes a=b. With this symbol, the basic rules of logic can be written as follows:
The grouping operations, id est, parenthesis, are used to specify order of application of these operations.
The logical operations satisfy the simple rules:
(aandb) orc = (aorc) and (borc)
(aorb) andc = (aandc) or (bandc)
With these rules, the complicated logical constructions can be considered, simplified and used. Such rules are called "Boolean Algebra".
They are widely used in the computation. Many other objects are constructed on the base of logical variables.
Set theory
All objects considered in mathematics or treated mathematically are supposed to belong to some certain set.
According to the definition of science, the deal with really ANY object from ANY set (id set, the to deal with the full set) is prohibited.
Such a prohibition is not caused by the personal preference of some mathematicians or some caprice of the writer of this Dictionary.
Such a prohibition follows from the pretty practical reason: consideration of objects from the whole set leads to the internal contradictions.
In such a way the set theory is so fundamental as the logic.
Without logic, one cannot deal with sets, and without to define at least one set of logical variables, one cannot develop the logic.
Historically, in the educational process, the students (usually children) first learn some practical bases of arithmetic,
and only then they begin to deal with logic and the sets (if at all, of course). The fundamental of the sets, as well as the fundamental of logic,
look obvious and in many situations they can be used even without the axiomatic and rigorous deduction.
Counting and arithmetics
On the base of logical variables, more complicated objects are constructed.
In particular, the natural numbers are used to characterize sets non-empty sets that contain more that one element.
The existence of some set with a single element is postulated.
Then, one says, that a set has two elements, if after removing one element from this set, it contains just one element.
One defines the "class of equivalence" that is is called "two" for all such sets.
There is special character 2 for denoting of such equivalence.
In the similar way, one defines "three" (and denotes it with character 3) and so on. In such a way, the natural numbers appear.
For them, the most of algebraic rules hold, instead of operation or, for natural numbers one uses character +, and instead of and one uses character -.
The equivalence denoted with sign = is expected to have logical value, either
TRUE or FALSE. In particular, after establishment of rules of operations with integer numbers, one knows that a value has expression
1+1=2 ; such an expression has value TRUE.
It was very fundamental discovery, that the same rules of counting can be applied to pretty different kinds of objects to characterize their amount.
The human, who was first to discover and formulate this, was great mathematician. in order to deal with such a quantity, the natural numbers are used.
Then, on the base of logical variables and natural numbers,
the integer numbers can be constructed as a certain class of equivalence on the set of pairs of natural numbers,
and operations similar to those for logical variables, are defined.
Then one constructs the rational numbers as class of equivalence of integers,
then real as class of equivalence of the fundamental sequences of rationals, and so on.
Conclusion
The logical variables and the logic as rules of operations with them stays at very beginning of any counting.
Other objects in mathematics are constructed using the logic.
One may try to reduce the number of axioms necessary for the deduction of properties of new objects, but the axioms of logic seem to be unavoidable.
