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Rules of proof The division of proof into types is a certain conditionality – they can pass into each other.

Rules of proof The division of proof into types is a certain conditionality – they can pass into each other.

With these basic truth tables, you can compile truth tables of complex formulas.

Boolean functions and predicates

Definition. Boolean function f (X1, X2, …, Xn) is an arbitrary n – local function, the arguments and values ​​of which belong to the set {0, 1}.

Generally speaking, there is a clear analogy between logical statements, logical connections, and Boolean functions. If the logical functions can take values ​​or true false, then for the Boolean function analogs of these values ​​will be values ​​of 0 or 1.

For Boolean functions, you can also compile tables of values ​​that correspond to the basic logical operations.

X1

X2

ØX1

X1X2

X1ÚX2

X1ÞX2

X1ÛX2

one

one

0

one

one

one

one

one

0

0

0

one

0

0

0

one

one

0

one

one

0

0

0

one

0

0

one

one

The concept of predicates

Definition. The predicate P (x1, x2, …, xn) is a function whose variables take values ​​from some set M, and the function itself takes two values: I (true) and X (false), ie

The predicate from n arguments is called n – local predicate. Statements are considered zero – local predicates.

You can perform ordinary logical operations on predicates, which result in new predicates.

In addition to the usual logical operations, special operations called quantifiers are also applied to predicates.

Quantifiers are of two types.

1) The quantifier of commonality. Denoted ("x) P (x). The quantifier of commonality is a statement sincere, when P (x) is true for each element x of the set M, and false – otherwise.

2) The quantifier of existence. Denoted by ($ x) P (x). A quantifier of existence is a statement that is sincere when there is an element from the set M for which P (x) is true and false otherwise.

The quantizer binding operation can be applied to predicates from a larger number of variables.

For the formulas of the logic of predicates, the validity of all the rules of equivalent transformations of the logic of statements is preserved.

In addition, the following properties are valid:

1) Transfer of the quantifier through negation.

Ø ("x) A (x) º ($ x) ØA (x); Ø ($ x) A (x) º (" x) ØA (x);

2) Making the quantifier in parentheses.

($ x) (A (x) B) º ($ x) A (x) B; ("x)" (A (x) B) º ("x) A (x) B;

($ x) (A (x) Ú B) º ($ x) A (x) Ú B; ("x) (A) (x) Ú B) º (" x) A (x) Ú B;

3) Permutation of quantifiers of the same name.

("y) (" x) A (x, y) º ("x) (" y) A (x, y); ($ y) ($ x) A (x, y) º ($ x) ($ y) A (x, y);

4) Rename related variables. If we replace the related variable of formula A with another variable that is not included in this https://123helpme.me/write-my-lab-report/ formula, in the quantifier and everywhere in the scope of the quantifier we obtain a formula equivalent to A.

The numbering of predicates is based on the above properties and rules, called axioms.

Whatever the formulas A and B, the following axioms are valid for them:

A Þ (B Þ A); (A Þ (B Þ C)) Þ ((A Þ B) Þ (A Þ C)); (ØB Þ ØA) Þ ((ØB Þ A) Þ B); ("xi) A (xi) Þ A (xj), where formula A (xi) does not contain the variable xi. A (xi) Þ ($ xj) A (xj), where formula A (xi) does not contain the variable xi.

literature

Konversky AS Mathematical logic. – K., 1998. Toftul MG Logic. – K., 1999. Khomenko IV, Aleksyuk IA Fundamentals of logic. – K., 1996.

10/23/2011

Schemes and types of proofs. Proof and refutation. Abstract

The concept of proof, its structure. Types and varieties of evidence (proof). Rules of proof. The concept of refutation, types of refutation

The concept of proof, its structure

Obtaining indirect, derived knowledge is not only in the form of inference. Another form of carrying out this process in thinking is proof (proof). It differs qualitatively in complexity compared to the concept, judgment and inference – and therefore is considered separately from them.

Proof (proof) – a form of thinking that justifies the correctness of judgments, the truth of which is not obvious by turning them into judgments directly obvious. In other words, proof is a form of thinking that reveals the truth of some judgments and the falsity of others.

The linguistic form of expression of proof is more or less complex linguistic constructions, which consists of a set of sentences that are in some way related to each other and express a logical chain of inferences. Proof is based on inference, but is not reduced to it, is not a simple arithmetic sum of inferences. Just as judgment represents itself in the form of a connection of concepts, and inferences in the form of a connection of judgments, so proof represents a connection of inferences (and, accordingly, judgments and concepts).

The structure of the proof includes three components:

1) Thesis – a judgment, the truth of which must be proven. Theses can be a variety of judgments, if they are not obvious and need to be proved. In the sciences, these are various provisions (theorems – in geometry, facts and circumstances – in legal practice), in everyday practice – certain beliefs.

A kind of thesis is a hypothesis (from the Greek hypothesis – justification, assumption, conjecture) – not a true or false judgment, but a more or less probable assumption that can become the subject of proof, and eventually gain the status of a scientific position or theory (if the proof is successful). At one time, M. Lomonosov noted that hypotheses are the only way in which prominent people have reached the discovery of the most important truths of science.

Reflecting on the essence of the hypothesis, one of the functions of proof becomes more accurate – to be a necessary tool in the development of theory or its development. Here we can recall the hypothesis of the atomistic construction of matter of Democritus, Titus Lucretius Kara and others, which later formed the basis of elementary physics; I. Kant’s hypothesis about the origin of the solar system from the proto-nebula, which played a major role in establishing a dialectical view of nature.

A kind of hypothesis in legal practice is the version (from the Latin versio – modification, rotation) – the conjecture or assumption of a lawyer about the presence or absence of events, facts, nature and nature of actions and so on.

2) Arguments – the basic parameters of proof, judgments, which prove the thesis. This is the position from which the truth or falsity of the thesis is derived. The role of arguments in proof is extremely large. In everyday practice, they are, in fact, called evidence. In legal theory, the term "legal basis" is used. There are the following types of arguments: reliable facts (most often), definitions, axioms and postulates.

It is the facts in the evidence that have considerable coercive force and, as a rule, convince the most thoroughly – I. Pavlov called them "the air of a scientist". Under the facts, say, legal, means the circumstances that serve as the basis for the emergence or termination of a specific legal relationship.

In addition to facts, another universal type of argument is definition. For example, in geometry, the definition of "point", "line", "plane", etc. is fundamental to further proofs. A similar role of this type of argument in other sciences, in particular, in the humanities – they reveal the general and specific qualities of the subject of proof.

3) Form of proof (argumentation). The presence of the thesis and arguments does not mean that the proof is available. For example, if we have a bunch of car parts, it does not mean that they are already a finished car. In order for the proof to be complete, it is necessary to establish a logical connection between theses and arguments, which is, in fact, argumentation. That is, you need a consistent chain between the thesis, the system of arguments and the conclusion. For this logical chain to be consistent, a person needs to know and follow the laws of logic.

Types and varieties of evidence (proof). Rules of proof

The division of proof into types is a certain conditionality – they can pass into each other. However, there are the following types of evidence:

direct proof – the truth of the thesis is proved directly from the truth of the arguments. A simple example: "The refrigerator is working, because when connected to the network in its chamber, the air temperature decreases according to the parameters set by the manufacturer"; "Anatoly Karpov is an outstanding chess player in the world because he has won more than 150 tournaments and matches." indirect proof – the truth of the thesis is derived from some other judgments. It differs in that the arguments in it substantiate the truth of thesis indirectly through the substantiation of the falsity of another, opposite thesis.

Indirect proof has two varieties: apogogic proof (from the opposite) and proof by division. The apogogic is that first it is accepted to prove thesis that contradicts the original. Then this thesis is reduced to the absurd, or to a contradiction with certain truths – then from the falsity of such a thesis follows the truth of the original. Indirect proof by the method of division is that of the several possible theses by the method of exclusion is one.

For example, a lawyer knows that this crime was committed by either Ivanov, Petrov, or, say, Bender-Zadunaisky (and no one else). Therefore, the task of the lawyer is to gradually cut off the false theses to establish one person guilty of this crime.

According to the method of argumentation, there are two types of proof of the proposed position: direct and indirect (indirect).

1. Direct is the substantiation of the thesis without recourse to competing assumptions with the thesis. Direct proof can take the form of (1) deductive inferences, inductions, analogies.

(1) Deductive proof is expressed in summing up a partial case under the general rule:

the demonstration takes the form of a conditionally categorical inference. With the truth of the premise-arguments and compliance with the rules of inference, it gives reliable results.

(2) Inductive proof is a logical transition from arguments, information about individual cases of a certain kind, to a thesis that summarizes these cases. If the proof proceeds in the form of complete induction with the truth of the foundations, the conclusion will also be true. In case of incomplete induction, the thesis will be substantiated only to a greater or lesser degree of probability.

(3) Demonstration in the form of analogy is a direct substantiation of the thesis, which formulates a statement about the properties of a single phenomenon.

2. Indirect is the proof of the thesis by establishing the falsity of the antithesis or other competing assumptions with the thesis. Competing with the thesis (T) assumptions can be of two varieties: the antithesis (~ T), which contradicts the thesis, (h) members of the disjunction in the divisional judgment: TvAvB; The distinction in the structure of competing assumptions defines two types of indirect proof: (1) similar and (2) divisive.

(1) The substantiation of a thesis by establishing the falsity of a contradictory assumption is called similar. The argument in this case is divided into three stages:

The first stage – the thesis (T) put forward the antithesis ~ T, conditionally recognized as true and deduce the logical consequences:

The second stage – logically derived from the antithesis of the consequences are compared with the provisions, the truth of which is established (F), and in case of discrepancy, these consequences are abandoned:

The third stage – from the falsity of the consequences logically conclude the falsity of the assumption ~ T. The reasoning takes the form of a negative mode of conditional-categorical inference: