Possibility of various interpretations of formulas with free variables indicates existence of various or as speak, various interpretations of free variables in formulas. In general distinguish three possible of free variables as a part of YaKLP formulas.
the statement of B ₀ if only follows if this relation takes place between a set of the formulas G and B representing logical forms of the mentioned statements. The last relation of ⊨ takes place, e. i.e. as a part of there is a final subset of formulas A1..., An (n> = such that (A1 &... & An) ⊨ Century. The last ratio, as well as in logic of, is equivalent to that from a set A1..., An follows In that in turn specifies on noted earlier — in logic of statements — the property of the relation of following consisting that if some statement follows from some set of, it is a consequence also of any expansion of this set.
formulas. Among these expressions there are analogs of narrative offers of a natural language, and a vyskazyvatelny forms — the predicates representing special semantic category which is not allocated — at least, explicitly — in a natural language.
is here, but not connected. We consider here only such terms in which all variables can have only occurrences, and, so are free. The formula and a term which are not containing free variables are called according to the closed formula and to the closed t as m about m (it is obvious that for terms here if close a term, it does not contain variables at all).
Determination of values of all logical terms as which are obviously designated, and implicitly containing in is carried out just by means of rules a of truthconditional values to completely formulas of our language (strictly speaking, we have so-called implicit definition of logical constants here, but they are sufficient for understanding of what sense they give to our statements).
The following relation between formulas A ₀ ⊨ B ₀ takes place. i.e. at any interpretation of descriptive in And both In and at any attributings of values to variables at the validity of the first it is true also, in other words, the first is false or the second is true. in a look thus that, first, if some term is somehow interpreted in And, in the same way it is interpreted and in In (of course, in the presence of it in this formulas, and, secondly, to all free occurrences of the same variable in And and In the same value. From a set of statements of ₀
Let's pay attention that according to definition of a free and connected variable the same variable in the same formula can be free and connected. Such is, for example, variable x ₁ in a formula ∀ x ₁ P ¹ (x ₁) ∨ Q ² (x ₁, x ₂); variable x ₂
Logical constants: ,&,,,, — implication, conjunction, a community quantifier, an existence quantifier, a disjunction and denial. (Often only some of these symbols. From quantifiers only ∀ or ∃, from other, called by sheaves, it is enough: ⊃ and to, or ∨ and to, or & and to. Other constants as, however, and other signs, can sya by definition.)
It is obvious that in the mentioned statements with free variables these variables have conditional interpretation to which we will adhere and further though not possibility of the use of such statements, a in conclusions and proofs with interpretation of generality of their free variables. Strictly speaking, conditional corresponds to concept of logical following. And in a of interpretation of generality at creation of conclusions and, special restrictions are required.
Definition: any subject variable and a subject constant is a term; if there are and f ¸ ⁿ is a n-seater subject functor, f ¸ ⁿ (there is a term; anything, except specified in points and, is not a term.
Let's emphasize once again value of interpretation (set of rules I). In the presence of rules III, that is at the set understanding of the logical constants defining language type, various interpretations generate from the set syntactic system actually various languages of this type (in which every time only some part initial descriptive .
As well as in logic of statements, we say that for A ₀ and B ₀ (expressed now in the described language of logic of a predicate, the relation of logical following of A ₀ ⊨ B ₀ if only if it takes place for formulas A and B1 of the specified statements representing logical forms takes place.