First, it is directly verified that all axioms are LZZ formulas.
Therefore, formulate the definition of the formula:
and). If Pn is a predicate letter, and t1, t2, …, tn are terms, then Pn (t1, t2, …, tn) is a formula called elementary. All occurrences of subject variables in the formula Pn (t1, t2, …, tn) are called free.
b). If F1, F2 are formulas, then the expressions (ØF1), (F1ÚF2), (F1ÙF2), (F1®F2) are also formulas. All occurrences of variables free in F1 and F2 are free in all four types of formulas.
in). If F (x) is a formula containing free occurrences of the variable x, then "xF (x) and $ xF (x) are formulas.
In these formulas, all occurrences of the variable x are called related. The occurrences of the remaining variables in F remain free.
d). There are no formulas other than those built according to rules a), b) and c).
Remark. Functional letters and terms are introduced into definitions for the potential needs of various specific applied numbers of predicates. In applied calculus, the subject area M is, as a rule, the carrier of a certain algebraic system, so in many cases it is advisable to have the means to describe the operations and relations given by M.
The net calculus of predicates is built for an arbitrary subject area; the structure of this area and the connections (relationships) between its elements are not taken into account, so it is not necessary to enter functional letters and terms.
3. Axioms of predicate calculus form two groups of axioms.
and). The first group consists of axioms of arbitrary numbering of statements (for example, you can take any of the above two systems A1-A10 or S1-S3). Typically, these axioms are schemes of axioms.
b). The second group includes the so-called predicate axioms:
P1. "xF (x) ®F (y),
P2. F (y) ® $ xF (x).
In these axioms, F (x) is any formula that contains free occurrences of x, and none of them is in the range of the quantifier over y. The formula F (y) is obtained from F (x) by replacing all free occurrences of the variable x by y.
The last remark means that the formula F (x) cannot have, for example, the form $ yA (x, y) or "y (A (x) ®B (y)) and so on.
4. The rules of inference in many predicates are the following rules:
and). The rule of inference (modus ponens) is the same as in many statements. b). Generalization rule ("quantifier input rule"): A® "xB (x) is output from A®B (x). In). Quantifier input rule $: $ BB (x) ®A is output from B (x) ® A.
In both last rules, formula B (x) contains free occurrences of x, and A does not contain them.
The rule of substitution in our number is absent. Thus, of the two possible methods of constructing calculus, the method with axiom schemes is chosen. It is possible to construct the number of predicates with the substitution rule, but it is much more cumbersome due to the need to distinguish between free and bound occurrences of subject variables. Therefore more often in logic use the approach with schemes of axioms.
The concepts of derivation (proof) of a formula, the concept of a theorem, derivation of a formula from a set of hypotheses are denoted in numerous predicates in the same way as it was done in numerous statements. There are also theorems similar to Theorems 5.5 and 5.6 of calculus of statements.
Theorem 5.7. Any derivation formula (theorem) of predicate calculus is identically true (logically universal) formula.
This theorem is proved analogously to Theorem 5.5. First, it is directly verified that all axioms are LZZ formulas. Therefore, it turns out that all derivation rules retain the property of lzz.
Theorem 5.8. Any identically true predicate formula is a derivation (theorem) in many predicates.
The proof of this theorem is rather complicated and will not be given here.
From the latter theorems follows a statement similar to the statement of Theorem 5.1.
Theorem 5.9. Predicate formulas A and B are equivalent if and only if the formula ((A®B) Ù (B®A)) is deducible in the number of predicates, ie is lzz.
As before, to reduce the expression ((A®B) Ù (B®A)) enter the operation ~ and write this expression as (A ~ B). Therefore, the last theorem can be reformulated as follows: formulas A and B are equivalent (A = B) if and only if the formula (A ~ B) is deducible in the number of predicates.
Since, as mentioned above, establishing the equivalence of formulas in the logic of predicates is a much more difficult task than in the logic of statements, the very important value of the latter statement is that this problem can be reduced to finding a formal inference for the formula.
The constructed calculus of predicates is called first-order predicate calculus, or first-order theory. In such a theory, the arguments of functions and predicates, as well as variables bound by quantifiers, can only be subject variables.
In the calculations of the second and higher orders, the arguments of predicates can be predicates, and quantifiers can also connect predicate variables, ie admissible expressions, for example, of the form "P (P (x)). The use of such numbers is much less common, so in mathematical logic they are given less attention.
Logistics service: types and forms. Abstract
The concept of logistics service. Formation of a logistics service system. Quality indicators. Logistics service
In the conditions of the crisis of economy in our country the search of new forms and methods of management for its improvement was conducted and continues to be conducted. These include in the recent past the achievements of cybernetics and computer technology; these currently include the achievements of marketing and logistics.
The word "logistics" comes from the Greek word "logistika" which means the art of calculating, reasoning. The history of the emergence and development of practical logistics is far in the past.
Logistics is considered as a set of actions for integrated management of circulation and information flows in ideas for a narrative short story the field of economics and as an interdisciplinary science.
Experts believe that logistics is undoubtedly a scientific field, and its most radical followers and propagandists consider logistics a new science. Logistics, as a science, they say, plays a leading role in the rationalization and automation of production. This is the science of rational organization of production and distribution, which comprehensively from a systemic standpoint covers the supply of raw materials, fuel, materials, semi-finished products, the organization of sales, distribution and transportation of finished products.
In a "buyer’s market" the seller is forced to build their activities based on consumer demand. In this case, demand is not limited to demand for goods. The buyer dictates his terms also in the field of composition and quality of services provided to him in the process of delivery of this product.
A service, in the general sense of the term, means either an action that benefits, helping another. Work on the provision of services, ie on satisfaction or lack, is called a service.
However, the main principle of modern service is as follows: "Who does, he serves." In other words, who makes the product, he organizes and maintains its service.
But in a competitive market, the service is a subsystem of marketing activities of the enterprise, providing a range of services related to the sale and operation by the consumer of products – machinery and equipment, household appliances, vehicles, etc.
Properly oriented service that accompanies the product throughout its life cycle to the consumer, ensures his constant readiness for normal consumption and efficiency. All this explains the importance of working on the organization of the service.
The concept of logistics service
The nature of logistics involves the possibility of providing the consumer with a material flow of various logistics services.
Logistics service is inextricably linked with the distribution process and is a set of services provided in the process of delivery of goods.
The object of logistics service are different consumers of material flow. The logistics service is provided either by the supplier himself or by a forwarding company specializing in the field of logistics services.
All work in the field of logistics services can be divided into 3 main groups:
pre-sales, ie work on the formation of the logistics service system; work on the provision of logistics services carried out in the process of selling goods; after-sales logistics service.
Prior to the implementation process, the work in the field of logistics service includes, mainly, the definition of the company’s policy in the field of services, as well as their planning.
The pre-sales service includes consulting, which corresponds to the preparation of products, and in the case of transfer of equipment for free trial operation – training of the buyer’s staff (or himself), demonstration of equipment in action, providing the necessary documentation. Upon arrival of the goods at the point of sale, the service staff eliminates the problems that arose during transportation, mounts and adjusts the equipment, ie brings it into working condition. Pre-sale service is always free.
In the process of selling goods can be a variety of logistics services, such as:
availability of inventories in the warehouse; execution of the order, including, selection of assortment, packing, formation of cargo units and other operations; ensuring the reliability of delivery; providing information on the passage of goods.
After-sales services are warranty service, obligations to consider customer claims, exchange, etc. After-sales service is divided into warranty and post-warranty on a purely formal basis: "free" (in the first case) or for a fee (in the second) are provided or service list. The formality here is manifested in the fact that the cost of work, spare parts and materials during the warranty period is included in the sale or price in other (post-warranty) services.
During the warranty period, the manufacturer tries to take on all the work on which depends the long-term trouble-free operation of the product (machinery, equipment, appliances), such as consultations on construction issues, the organization of the chief installation. The manufacturer trains the buyer’s staff, controls the correct operation, service personnel inspect the sold equipment without a special call and carry out all necessary maintenance work, replace broken parts.