Let's notice, what it is enough to check, whether this formula U is identically true in the field, consisting exactly of elements. It follows from this that for formulas of the considered type the following takes place: if the formula U is identically true on some field, it is identically true on any its part.
we will designate the variables accepting values And and L. We will call them variable statements. We will also consider also constant statements. We will also designate them the capital Latin letters somehow noted or it is simple with the additional reservation.
The resolvability problem — this problem is put for formulas of calculation of the predicates deprived of symbols of constant subjects and symbols of individual predicates. In the subsequent statement it is supposed that the considered formulas are that (if it is not made special reservations).
As from our point of view each certain statement represents And or L, expression of F (x) means that from M one of two symbols And or L is delivered to each subject in compliance. In other words, F (x) represents the function defined on a great number of M and accepting only two values And and L. In the same way the uncertain statement about two and more subjects H (x, y), G (x, y, z) etc. predvtavlyat itself function of two, three etc. variables. Thus variables x, y, z run a great number of M, and function accepts values And and L. These uncertain statements, or functions of one or several variables, we will call logical functions or predicates. A predicate with one variables it is possible to express property of a subject, for example "x there is a prime number", "x - a rectangular triangle", etc.
All concepts which we will enter, belong always to some any great number of M which we will call by a field. We will designate elements of this field small Latin letters (we will sometimes supply these letters with indexes). Letters of the end of the Latin alphabet
Let M - any nonempty set, and x represent any subject from this set. Then expression of F (x) designates the statement which becomes certain when x F(b) is replaced with a certain subject from M. F(a)... already represent quite certain statements. For example, if the M natural row, F (x) can designate: "x there is a prime number".
Let R (x) ((x), y..., u) matters And. In that case R ((x), y..., u) matters And for these y..., u and for everyone x. But as function (x) runs all field when x runs a field M, R (x, y..., u) matters And for these y..., u and for all x from. Owing to definition of R (x, y..., u) also accepts I. Obratno's value, if R (x, y..., u) accepts value And, R (x, y..., u) matters And for these y..., u and for everyone x from. In that case expression of R ((x), y..., u) matters And for these y..., u and for everyone x from M as (x) for any x belongs.
We will designate a set of all these elements. let's prove that if a formula U (...) it is true in the field of M, it is true and in the field (as – Yo part of a field M, predicates are defined on). we will deliver to each element x of a field M in compliance the element from belonging to the same class as x. In there is one and only one such element. The element from put in compliance x, we will designate (x). It is possible to tell that we constructed the function defined on a great number of M and accepting values from a set.
The given formula is called normal if it does not contain quantifiers or if at education it from elementary formulas of operation of binding by a quantifier follow all operations of algebra of statements.
If two formulas U and B carried to some field M at all replacements of variable predicates, variable statements and free subject variables respectively with the individual predicates defined on M, individual statements and individual subjects from M accept identical values And or L, we will say that these formulas are equivalent in the field of M.
It is easy to see that, as well as in the previous example, represents the formula formed only by operations of algebra of statements over expressions of P () and Q (), where i = and therefore it can be carried to formulas of algebra of statements at which P () and Q () are elementary variable statements. Whether the formula is identically true?