In the modern world, we increasingly use a variety of machines and gadgets. And not only when it is necessary to use literally inhuman force: move a load, lift it to a height, dig a long and deep trench, etc. Today, cars are assembled by robots, food is prepared by multicookers, and basic arithmetic calculations are performed by calculators. More and more often we hear the expression “Boolean algebra”. Perhaps the time has come to understand the role of man in the creation of robots and the ability of machines to solve not only mathematical, but also
Logics
Translated from Greek, logic is an ordered system of thinking that creates relationships between given conditions and allows one to make conclusions based on premises and assumptions. Quite often we ask each other: “Is this logical?” The received answer confirms our assumptions or criticizes the train of thought. But the process does not stop: we continue to reason.
Sometimes the number of conditions (introductory) is so large, and the relationships between them are so intricate and complex that the human brain is not able to “digest” everything at once. It may take more than one month (week, year) to understand what is happening. But modern life does not give us such time intervals to make decisions. And we resort to the help of computers. And this is where the algebra of logic appears, with its laws and properties. By loading all the initial data, we allow the computer to recognize all the relationships, eliminate contradictions and find a satisfactory solution.
Mathematics and logic
The famous Gottfried Wilhelm Leibniz formulated the concept of “mathematical logic,” the tasks of which were understandable only to a narrow circle of scientists. This direction did not arouse much interest, and until the middle of the 19th century, few people knew about mathematical logic.
A dispute in which the Englishman George Boole announced his intention to create a branch of mathematics that had absolutely no practical application aroused great interest in scientific communities. As we remember from history, at this time industrial production was actively developing, all kinds of auxiliary machines and machine tools were being developed, i.e. all scientific discoveries had a practical orientation.
Looking ahead, let's say that Boolean algebra is the most used part of mathematics in the modern world. So Boule lost his argument.
George Boole
The personality of the author itself deserves special attention. Even taking into account the fact that in the past people grew up before us, it is still impossible not to note that at the age of 16 J. Bull taught at a village school, and by the age of 20 he opened his own school in Lincoln. The mathematician had an excellent command of five foreign languages, and in his free time he read the works of Newton and Lagrange. And all this is about the son of a simple worker!
In 1839, Boole first sent his scientific papers to the Cambridge Mathematical Journal. The scientist turned 24 years old. Boole's work so interested the members of the Royal Scientific Society that in 1844 he received a medal for his contribution to the development. Several more published works that described the elements of mathematical logic allowed the young mathematician to take the post of professor at County Cork College. Let us remember that Buhl himself had no education.
Idea
In principle, Boolean algebra is very simple. There are expressions that, from a mathematical point of view, can be defined only by two words: “true” or “false”. For example, in the spring the trees bloom - true, in the summer it snows - false. The beauty of this mathematics is that there is no strict need to use only numbers. Any statements with an unambiguous meaning are quite suitable for the algebra of judgments.
Thus, the algebra of logic can be used literally everywhere: in scheduling and writing instructions, analyzing conflicting information about events, and determining the sequence of actions. The most important thing is to understand that it does not matter at all how we determine the truth or falsity of a statement. These “how” and “why” need to be abstracted from. What matters is only the statement of fact: truth or falsehood.
Of course, logical algebra functions, which are written with the corresponding signs and symbols, are important for programming. And learning them means mastering a new foreign language. Nothing is impossible.
Basic concepts and definitions
Without going into depth, let's look at the terminology. So, Boolean algebra assumes the presence of:
- statements;
- logical operations;
- functions and laws.
Statements are any affirmative expressions that cannot be interpreted in two meanings. They are written in the form of numbers (5 > 3) or formulated in familiar words (the elephant is the largest mammal). Moreover, the phrase “a giraffe has no neck” also has a right to exist, only Boolean algebra will define it as “false”.
All statements must be unambiguous, but they can be elementary and compound. The latter use logical connectives. That is, in the algebra of judgments, compound statements are formed by adding elementary ones through logical operations.
Boolean algebra operations
We already remember that operations in the algebra of judgments are logical. Just as number algebra uses arithmetic operations to add, subtract, or compare numbers, elements of mathematical logic allow you to construct complex statements, give the negation, or calculate the final result.
For formalization and simplicity, logical operations are written using formulas familiar to us in arithmetic. The properties of Boolean algebra make it possible to write equations and calculate unknowns. usually written using a truth table. Its columns define the elements of calculations and the operation that is performed on them, and the rows show the result of the calculations.
Basic logical actions
The most common operations in Boolean algebra are negation (NOT) and logical AND and OR. This way one can describe almost all actions in the algebra of judgments. Let's study each of the three operations in more detail.
Negation (does not) apply to only one element (operand). Therefore, the negation operation is called unary. To write the concept “not A” the following symbols are used: ¬A, A¯¯¯ or!A. In tabular form it looks like this:
The negation function is characterized by the following statement: if A is true, then B is false. For example, the Moon revolves around the Earth - true; The Earth revolves around the Moon - a lie.
Logical multiplication and addition
Logical AND is called the conjunction operation. What does it mean? Firstly, it can be applied to two operands, i.e. AND is a binary operation. Secondly, only if both operands (A and B) are true is the expression itself true. The proverb “Patience and work will grind everything down” suggests that only both factors will help a person cope with difficulties.
The following symbols are used for recording: A∧B, A⋅B or A&&B.
Conjunction is similar to multiplication in arithmetic. Sometimes they say so - logical multiplication. If we multiply the elements of a table by row, we get a result similar to logical thinking.
A disjunction is a logical OR operation. It accepts the truth value when at least one of (either A or B). It is written like this: A∨B, A+B or A||B. The truth tables for these operations are:
Disjunction is similar to arithmetic addition. The logical addition operation has only one limitation: 1+1=1. But we remember that in digital format mathematical logic is limited to 0 and 1 (where 1 is true, 0 is false). For example, the statement “in a museum you can see a masterpiece or meet an interesting interlocutor” means that you can see works of art, or you can meet an interesting person. At the same time, the possibility of both events happening simultaneously cannot be ruled out.
Functions and laws
So, we already know what logical operations Boolean algebra uses. Functions describe all the properties of mathematical logic elements and allow you to simplify complex compound conditions of problems. The most understandable and simplest property seems to be the property of refusing derivative operations. Derivatives are understood as exclusive OR, implication and equivalence. Since we only got acquainted with the basic operations, we will also consider only those properties.
Associativity means that in statements like “and A, and B, and C,” the sequence of listing the operands does not matter. The formula will be written like this:
(A∧B)∧B=A∧(B∧B)=A∧B∧B,
(A∨B)∨B=A∨(B∨B)=A∨B∨B.
As we see, this is characteristic not only of conjunction, but also of disjunction.
Commutativity states that the result of a conjunction or disjunction does not depend on which element was considered first:
A∧B=B∧A; A∨B=B∨A.
Distributivity allows you to expand parentheses in complex logical expressions. The rules are similar to opening parentheses when multiplying and adding in algebra:
A∧(B∨B)=A∧B∨A∧B; A∨B∧B=(A∨B)∧(A∨B).
Properties of one and zero, which can be one of the operands, are also analogous to algebraic multiplication by zero or one and addition by one:
A∧0=0,A∧1=A; A∨0=A,A∨1=1.
Idempotency tells us that if, relative to two equal operands, the result of the operation turns out to be similar, then we can “throw away” the unnecessary operands that complicate the course of reasoning. Both conjunction and disjunction are idempotent operations.
B∧B=B; B∨B=B.
Absorption also allows us to simplify equations. Absorption states that when another operation on the same element is applied to an expression with one operand, the result is the operand from the absorbing operation.
A∧B∨B=B; (A∨B)∧B=B.
Sequence of operations
The sequence of operations is important. Actually, as for algebra, there is a priority of functions that Boolean algebra uses. Formulas can be simplified only if the significance of the operations is respected. Ranking from the most significant to the least significant, we get the following sequence:
1. Denial.
2. Conjunction.
3. Disjunction, exclusive OR.
4. Implication, equivalence.
As we see, only negation and conjunction do not have equal priorities. And the priority of disjunction and exclusive OR are equal, as well as the priorities of implication and equivalence.
Implication and equivalence functions
As we have already said, in addition to basic logical operations, mathematical logic and the theory of algorithms uses derivatives. The most commonly used are implication and equivalence.
Implication, or logical consequence, is a statement in which one action is a condition, and another is a consequence of its implementation. In other words, this is a sentence with “if... then” prepositions. “If you love to ride, you also love to carry a sled.” That is, to ride, you need to pull the sled up the hill. If you don’t want to slide down the mountain, then you don’t have to carry the sled. It is written like this: A→B or A⇒B.
Equivalence assumes that the resulting action occurs only if both operands are true. For example, night gives way to day when (and only then) when the sun rises over the horizon. In the language of mathematical logic, this statement is written as follows: A≡B, A⇔B, A==B.
Other laws of Boolean algebra
The algebra of judgments is developing, and many interested scientists have formulated new laws. The most famous are the postulates of the Scottish mathematician O. de Morgan. He noticed and defined such properties as close negation, complement and double negation.
Close denial assumes that there is no negation before the parenthesis: not (A or B) = not A or NOT B.
When an operand is negated, regardless of its value, it is said to be addition:
B∧¬B=0; B∨¬B=1.
And finally twice no compensates itself. Those. either the negation disappears before the operand, or only one remains.
How to solve tests
Mathematical logic involves simplifying given equations. Just as in algebra, you must first simplify the condition as much as possible (get rid of complex inputs and operations with them), and then begin to search for the correct answer.
What can be done to simplify? Convert all derivative operations to simple ones. Then open all the brackets (or vice versa, move them out of brackets to shorten this element). The next step should be to apply the properties of Boolean algebra in practice (absorption, properties of zero and one, etc.).
Ultimately, the equation must consist of a minimum number of unknowns, combined by simple operations. The easiest way to find a solution is to achieve a large number of close negations. Then the answer will emerge as if by itself.
Introduction
The topic of the test is “Mathematical Logic”.
BOOL or BUL, also BUUL, George (1815-1864) - English mathematician who is considered the founder of mathematical logic.
Mathematical logic is a branch of mathematics devoted to the analysis of reasoning methods, while the forms of reasoning are primarily studied, and not their content, i.e. formalization of reasoning is studied.
The formalization of reasoning goes back to Aristotle. Aristotelian (formal) logic acquired its modern form in the second half of the 19th century in the work of George Boole “The Laws of Thought”.
Mathematical logic began to develop intensively in the 50s of the 20th century in connection with the rapid development of digital technology.
1. Elements of mathematical logic
The main branches of mathematical logic are propositional calculus and predicate calculus.
A statement is a sentence that can be either true or false.
Propositional calculus is an introductory section of mathematical logic that deals with logical operations on propositions.
A predicate is a logical function of n variables that takes the values true or false.
Predicate calculus is a branch of mathematical logic, the object of which is the further study and generalization of propositional calculus.
The theory of Boolean algebras (Boolean functions) forms the basis of precise methods of analysis and synthesis in the theory of switching circuits in the design of computer systems.
1.1 Basic concepts of logical algebra
Algebra of logic is a branch of mathematical logic that studies logical operations on statements.
In algebra, logicians are only interested in the truth value of statements. Truth values are usually denoted by:
1 (true) 0 (false).
Each logical operation corresponds to a function that takes the values 1 or 0, whose arguments also take the values 1 or 0.
Such functions are called logical or Boolean, or functions of algebra of logic (FAL). In this case, the logical (Boolean) variable x can only take two values:
.Thus,
- a logical function in which logical variables are statements. Then the logical function itself is a complex statement.In this case, the algebra of logic can be defined as a collection of a set of logical functions with all kinds of logical operations specified in it. Such logical operations as conjunction (read AND), disjunction ( OR), implication, equivalence, negation ( NOT), correspond to logical functions for which the notation is accepted
(&, ·), ~, – (), and the truth table holds:| x~y | ||||||
| 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 0 | 1 |
This is a tabular method of specifying FAL. Along with them, specifying functions using formulas in a language containing variables is used x , y , …, z(possibly indexed) and symbols of some specific functions - an analytical way to specify FAL.
The most common language is one containing logical symbols
~, –. The formulas of this language are defined as follows:1) all variables are formulas;
2) if P And Q– formulas, then
P ~ Q, - formulas.For example, the expression
~ - formula. If variable x , y , z assign values from the binary set 0, 1 and carry out calculations in accordance with the operations specified in the formula, then we obtain the value 0 or 1.They say that the formula implements the function. So the formula
~ implements the function h (x , y , z):| x | y | z | h (x, y, z) |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 0 |
Let P And Q– formulas that implement functions f (x 1 , x 2 , …, x n) And g (x 1 , x 2 , …, x n). The formulas are: P = Q, if functions f And g coincide, i.e. their truth tables coincide. An algebra whose main set is the entire set of logical functions, and whose operations are disjunction, conjunction and negation, is called the Boolean algebra of logical functions.
Let us present the laws and identities that define the operations
– and their connection with operations , ~:1. Idempotency of conjunction and disjunction:
.2. Commutativity of conjunction and disjunction:
.3. Associativity of conjunction and disjunction:
.4. Distributivity of conjunction relative to disjunction and disjunction relative to conjunction:
.
5. Double negative:
.6. De Morgan's laws:
=, =.7. Gluing:
.8. Absorption
.9. Actions with constants 0 and 1.
“If all ravens are black, then all non-black objects are not ravens.” This statement is undoubtedly true, and you don't have to be a bird expert to say it. Likewise, one does not need to be an expert in number theory to say that if all perfect numbers are even, then all odd numbers are imperfect. We have given examples of statements that are true regardless of the meaning of the concepts included in them (crows, black, perfect, even) - true by virtue of their very form. The study of statements of this kind is part of the task of logic. More generally, logic is the study of correct ways of reasoning - those ways of reasoning that lead to correct results in cases where the initial premises are true.
The subject of mathematical logic is mainly reasoning. She uses mathematical methods to study them. Let's explain what was said.
Mathematicians build and develop mathematical theories, give definitions, prove theorems, etc. Specialists in mathematical logic, observing this, analyze how mathematicians do this and what comes out of it. Figuratively speaking, the relationship between mathematics and mathematical logic is similar to the relationship between a concert and music theory. We can say that mathematical logic studies the foundations of mathematics, the principles of constructing mathematical theories.
“The book of philosophy is something that is always revealed before our eyes, but since it is written in letters other than the letters of our alphabet, it cannot be read. All the letters of this book are triangles, circles, balls, cones, pyramids and other mathematical figures very suitable for reading it.” G. Galileo
Having established what mathematical logic studies, let us move on to how it does it. We already know that she uses mathematical methods. Let's explain what this means. How are mathematical methods used, for example, in physics? A mathematical model of the physical process under consideration is constructed, reflecting some of its essential properties. Mathematical methods can be used not only in physics, but also in other sciences. For example, the application of mathematical methods in biology consists of constructing mathematical models of biological processes. It is also possible to build mathematical models for the process of development of mathematical theories. This is what mathematical logic does.
How does mathematical theory work? It contains some statements. Some of them are accepted without proof, others can be proven (in this case, the statements are called theorems). The meaning of the words “statement” and “evidence” in everyday practice is very vague. Therefore, if we want to build a mathematical model, then first of all we need to clarify these concepts, i.e. construct their formal analogues in our model. For this purpose, mathematical logicians came up with special formal languages designed to write mathematical statements. Statements written in formal languages are called formulas to distinguish them from sentences in natural languages. Having built a formal language, we get the opportunity to write down some mathematical statements in the form of formulas. This, of course, is not enough. We need to be able to formally write down not only statements, but also evidence. For this purpose, mathematical logicians came up with a formal analogue of the concept of “proof” - the concept of inference (proof written in a formal language). A formal analogue of the concept of “theorem” is the concept of “derivable formula” (i.e. a formula that has a conclusion). The formal language, together with the rules for drawing conclusions, is called a formal system.
What demands are natural to place on a formal system? We want it to be as similar as possible to “live”, informal mathematics. To do this, it is necessary that all the substantive statements that interest us (or at least most of them) can be “translated into formal language,” i.e. written in the form of formulas of this system. In addition, it is necessary that informal proofs can be translated into conclusions of the corresponding formulas.
Currently, quite satisfactory models (formalizations) of most mathematical theories have been constructed. The most important are formal arithmetic and axiomatic set theory. Formal arithmetic is intended to formalize reasoning about natural numbers, and axiomatic set theory is about sets.
The main subject of mathematical logic, therefore, is the construction and study of formal systems. The central result here is the incompleteness theorem proven in 1931 by the Austrian mathematician K. Gödel, which states that for any “reasonable enough” formal system there are sentences that are undecidable in it, i.e. formulas such that neither the formula itself nor its negation has a conclusion. If we identify a formal system with the corresponding field of mathematics, then we can say that in any “reasonable enough” field of mathematics there are statements that can neither be proven nor disproved. We cannot say here exactly what is required of a “sufficiently reasonable” formal system; Let us only note that most formal systems (including formal arithmetic and axiomatic set theory) satisfy these requirements. Using the incompleteness theorem as an example, we see the benefits of building a formal system: we get the opportunity to prove that some statements are unprovable!
The study of formal systems has led to the emergence of many important directions in modern mathematical logic. Let's name some of them. Model theory examines the question of how expressions in formal languages can be given “meaning” and what is achieved by doing so. Proof theory studies the properties of inferences in formal systems. The most important section of logic, which can now be considered as an independent discipline, is the theory of algorithms.
Many signs invented by logicians to construct formal systems gradually came into general use. These include logical connectives (conjunction, “and”), (disjunction, “or”), (implication, “if... then..."), (negation, “it is not true that”) and the so-called quantifiers ( universality, “for all”) and (existence, “exists”). The meaning of logical connectives, in addition to the names indicated in brackets, is explained by so-called truth tables. These tables show whether a complex statement, composed using logical connectives from simple ones, will be true (I) or false (F), depending on the truth of its component parts. Let's bring them.
For example, the fifth column shows that a statement can only be false if true and false. Using these tables, you can create a truth table for more complex statements, for example for the statement 
.
|
|
|
||||
Having compiled it, we will see that this statement (sixth column) is always true, regardless of the truth of statements and (for example, the statement “
which are obtained if we substitute the statement “ – crow”, and instead – “ – black”, or instead – “ – perfect”, and instead – “ – even”.
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
MOSCOW STATE UNIVERSITY
INSTRUMENT ENGINEERING AND INFORMATION SCIENCE
Department: "PHILOSOPHY"
Discipline: "LOGIC"
Topic No. 31: “Mathematical logic: subject, structure and basic principles of operations”
Completed:
1st year student
full-time faculty IT-7
record code 120177IT
Prytkov Yuri Sergeevich
Checked:
Associate Professor, Ph.D.
Blazhko Nikolay Ilyich
Moscow - 2012
Introduction
Mathematical logic
Subject of mathematical logic
Basic principles of operations
Negation
Conjunction
Disjunction
Implication
Equivalence
Quantifier statement
Quantifier with universal quantifier
Quantifier with existential quantifier
Axiomatic method
Conclusion
Introduction
Logic originated in the culture of Ancient Greece. The first work on logic that has come down to us is Aristotle’s “Analytics” (384-322 BC). Formal logic existed without major changes for more than twenty centuries. BOOL or BUL, also BUUL, George (1815-1864) - English mathematician who is considered the founder of mathematical logic.
The development of mathematics revealed the insufficiency of Aristotelian logic and required its further development. Buddhist logic developed independently, but it only recently became a property of European science, so mathematical logic originates from Aristotle’s logic. Mathematical logic is the science of the laws of mathematical thinking. The subject of mathematical logic is mathematical theories in general, which are studied using mathematical languages. At the same time, they are primarily interested in questions of the consistency of mathematical theories, their independence and completeness.
Mathematical logic is distinguished by the fact that it uses the language of mathematical and logical symbols, based on the fact that, in principle, they can completely replace the words of ordinary language and the methods of combining words into sentences accepted in ordinary living languages. The features of mathematical thinking are explained by the features of mathematical abstractions and the variety of their relationships. They are reflected in the logical systematization of mathematics, in the proof of mathematical theorems. In this regard, modern mathematical logic is defined as a branch of mathematics devoted to the study of mathematical proofs and questions of the foundations of mathematics.
Mathematical logic
In the axiomatic construction of a mathematical theory, a certain system of undefined concepts and the relationships between them are preliminarily selected. These concepts and relationships are called basic. Further, the main provisions of the theory under consideration - axioms - are accepted without proof. All further content of the theory is logically derived from the axioms. For the first time, the axiomatic construction of a mathematical theory was undertaken by Euclid in the construction of geometry. Presentation of this theory in Beginnings not flawless. Euclid here tries to define the initial concepts (point, straight line, plane). In the proof of theorems, provisions that are not explicitly formulated are used and are considered obvious. Thus, this construction lacks the necessary logical rigor, although the truth of all the provisions of the theory is beyond doubt.
Let us note that this approach to the axiomatic construction of a theory remained the only one until the 19th century. The works of N. I. Lobachevsky (1792-1856) played a major role in changing this approach. Lobachevsky was the first to explicitly express his belief in the impossibility of proving Euclid’s fifth postulate and reinforced this belief with the creation of a new geometry. Later, the German mathematician F. Klein (1849-1925) proved the consistency of Lobachevsky’s geometry, which actually proved the impossibility of proving Euclid’s fifth postulate. This is how the problems of impossibility of proof and consistency in axiomatic theory arose and were solved in the works of N. I. Lobachevsky and F. Klein for the first time in the history of mathematics. Consistency of an axiomatic theory is one of the main requirements for the system of axioms of this theory. It means that from this system of axioms it is impossible to logically deduce two contradictory statements.
Proof of the consistency of axiomatic theories can be done using various methods. One of them is the MODELING or INTERPRETATION METHOD. Here, elements of a certain set and relations between them are selected as the main concepts and relations, and then it is checked whether the axioms of a given theory will be satisfied for the selected concepts and relations, that is, a model for this theory is built. Thus, analytical geometry is an arithmetic interpretation of Euclid's geometry. It is clear that the modeling method reduces the question of the consistency of one theory to the problem of the consistency of another theory. Most interpretations for mathematical theories (and, in particular, for arithmetic) are based on set theory. However, at the end of the 19th century, contradictions were discovered in set theory (paradoxes of set theory). A striking example of such a paradox is B. Russell's paradox. Let us divide all conceivable sets into two classes. Let's call the set normal , if it does not contain itself as its element and abnormal otherwise. For example, the set of all books - normal multitude, and the multitude of all conceivable things - abnormal a bunch of. Let L be the set of all normal sets. What class does the set L belong to? If L - normal set, then L Î L, i.e. contained in the classroom normal sets, but then it contains itself as its element, and therefore abnormal . If L - abnormal set, then L Ï L, i.e. not included among normal sets, but then L does not contain itself as its element, and therefore it Fine . Thus, the concept normal set leads to a contradiction.
Attempts to eliminate contradictions in set theory led ZERMELO to the need to construct an axiomatic set theory. Subsequent modifications and improvements to this theory led to the creation of modern set theory. However, the means of this axiomatic theory do not allow us to prove its consistency. Other methods of substantiating mathematics were developed by D. GILBERT (1862-1943) and his school. They are based on the construction of mathematical theories as syntactic theories, in which all axioms are written by formulas in a certain alphabet and the rules for deriving some formulas from others are precisely indicated, i.e. The theory includes mathematical logic as an integral part.
Thus, the mathematical theory whose consistency needed to be proven became the subject of another mathematical theory, which Gilbert called METHAMATHEMATICS, or THE THEORY OF PROOF. In this regard, the task of constructing a syntactic one arises, i.e. formalized axiomatic theory of mathematical logic itself. By choosing different systems of axioms and rules for deriving some formulas from others, we obtain different syntactic logical theories. Each of them is called LOGICAL CALCULUS.
Subject of mathematical logic
The main idea of mathematical logic is the formalization of knowledge and reasoning. It is known that the most easily formalized knowledge is mathematical. Thus, mathematical logic, in essence, is the science of mathematics, or metamathematics. The central concept of mathematical logic is ``mathematical proof''. Indeed, “evidential” (in other words, deductive) reasoning is the only type of reasoning recognized in mathematics. Reasoning in mathematical logic is studied from the point of view of form, not meaning. Essentially, reasoning is modeled by a purely ``mechanical'' process of rewriting text (formulas). This process is called inference. They also say that mathematical logic operates only with syntactic concepts. However, it is usually still important how the reasoning relates to reality (or our ideas). Therefore, one must still keep in mind some meaning of the formulas and conclusions. In this case, the term semantics is used (synonymous with the word ``meaning'') and clearly separates syntax and semantics. When people are really only interested in syntax, the term "formal system" is often used. We will use a synonym for this term - ``calculus'' (the terms ``formal theory'' and ``axiomatics'' are also used). The object of formal systems are lines of text (sequences of characters) with which formulas are written.
A formal system is defined if:
An alphabet is specified (a set of symbols used to construct formulas).
There are many formulas called axioms. These are the starting points in the conclusions.
A set of inference rules are specified that allow one to obtain a new formula from a certain formula (or set of formulas).
Basic principles of operations
Negation
Negation of a logical statement is a logical statement that takes the value “true” if the original statement is false, and vice versa. This is a special logical operation. Depending on the location, a distinction is made between external and internal negation, the properties and roles of which differ significantly.
External negation (propositional) serves to form a complex statement from another (not necessarily simple) statement. It asserts the absence of the state of affairs described in the negated statement. Traditionally, a negative statement is considered true if, and only if, the negated statement is false. In natural language, negation is usually expressed by the phrase “it is not true that” followed by the negated statement.
In the languages of formal theories, negation is a special unary propositional connective used to form one formula into another, more complex one. To denote negation, the symbols "\negation", "-" or "-1" are usually used. In classical propositional logic, formula -A is true if and only if formula A is false.
However, in non-classical logic, negation may not have all the properties of classical negation. In this regard, a completely logical question arises about the minimum set of properties that some unary operation must satisfy in order to be considered a negation, as well as about the principles for classifying various negations in non-classical formal theories (see: Dunn J.M. and Hardegree G.M. Algebraic Methods in Philosophical Logic (Oxford, 2001).
In fact, the above traditional understanding of external (propositional) negation can be expressed through a system of the following requirements: (I) If A is true (false), then not-A is false (true); (II) If not-A is true (false), then A is false (true). Formally, requirements (I) and (II) can be expressed through the condition (1) A p-iB => B (= -, A, called “constructive contraposition”. A negation that satisfies condition (1) is usually called a minimal negation. However it turns out that condition (1) can be decomposed into two weaker conditions: (2) A (= B => -, B p-Au (3) A (= - 1 - A, known, respectively, as “contraposition” and "introduction of double negation". As a result, it becomes possible to identify a subminimal negation that satisfies condition (2), but does not satisfy condition (3). It is natural to formulate a condition inverse to (3) and formalizes the principle of "removing double negation": (4) -. - A = A. The minimal negation (i.e., satisfying condition (1) or conditions (2) and (3) together), for which condition (4) is satisfied, is called the de Morgan negation. The minimal negation satisfying the additional property (5 ): If A - B, then for any C it is true that A p C (“the property of absurdity”) - called intuitionistic negation. We can formulate principle (6), which is dual to the principle of absurdity: If B |=Au-S p A, then for any C it is true that C p A. Satisfying this principle of negation. is a type of negation in paraconsistent logic. Finally, the de Morgan negation (properties (2), (3), (4)), for which (5) or (6) holds, is called an ortho-negation. If in the corresponding calculus the distributivity axiom for conjunction and disjunction is accepted, then the ortho-negation negation is called Boole negation, or classical negation.
Internal negation is part of a simple statement. A distinction is made between negation as part of a copula (negative copula) and term negation.
Negation as part of a copula is expressed using the particle “not” standing before the linking verb (if there is one) or before the semantic verb. It serves to express judgments about the absence of some relationships (“Ivan does not know Peter”), or to form a negative predicative connective as part of categorical attributive judgments.
Term negation is used to form negative terms. It is expressed through the prefix “not” or something similar in meaning (“All unripe apples are green”).
Conjunction
The conjunction of two logical statements is a logical statement that is true only when they are simultaneously true (from the Latin conjunctio - union, connection), in a broad sense - a complex statement formed with the help of the conjunction “and”. In principle, one can talk about the conjunction of an infinite number of statements (for example, about the conjunction of all true sentences of mathematics). In logic, a conjunction is a logical connective (operation, function; denoted by: &,); a complex statement formed with its help is true only if its components are equally true. In classical propositional logic, conjunction together with negation constitute a functionally complete system of propositional connectives. This means that any other propositional connective can be defined through them. One of the properties of a conjunction is commutativity (i.e., the equivalence of A & B and B & A). However, sometimes they talk about a non-commutative, i.e. ordered conjunction (an example of a statement with such a conjunction would be: “The coachman whistled and the horses galloped”).
Disjunction
Disjunction of two logical statements - a logical statement that is true only if at least one of them is true
(from Latin disjunctio - disunion, isolation), in a broad sense - a complex statement formed from two or more sentences using the conjunction “or”, expressing alternativeness, or choice.
In symbolic logic, a disjunction is a logical connective (operation, function) that forms from sentences A and B a complex statement, usually denoted as A V B, which is true if at least one of the two disjunctive members is true: <#"justify">Implication
The implication of two logical statements A and B is a logical statement that is false only when B is false and A is true (from the Latin implicatio - intertwining, from implico - closely connecting) - a logical connective corresponding to the grammatical construction “if .., then. ..”, with the help of which a complex statement is formed from two simple statements. In an implicative statement, there is an antecedent (ground) - a statement coming after the word “if”, and a consequent (consequence) - a statement following the word “then”. An implicative statement represents in the language of logic a conditional statement of ordinary language. The latter plays a special role in both everyday and scientific reasoning; its main function is to justify one thing by referring to something else.
The connection between the grounder and the grounded expressed by a conditional statement is difficult to characterize in general terms, and only sometimes its nature is relatively clear. This connection may be, in particular, a connection of logical consequence that takes place between the premises and the conclusion of a correct inference (“If all living multicellular creatures are mortal and the jellyfish is such a creature, then it is mortal”). The connection may be a law of nature (“If a body is subjected to friction, it will begin to heat up”) or a causal relationship (“If the Moon is at the node of its orbit at a new moon, a solar eclipse occurs”). The connection under consideration may also have the nature of a social pattern, rule, tradition, etc. (“If the economy changes, so does politics”, “If a promise is made, it must be kept”).
The connection expressed by a conditional statement presupposes that the consequent “follows” with a certain necessity from the antecedent and that there is some general law, having been able to formulate which, we can logically deduce the consequent from the antecedent. For example, the conditional statement “If bismuth is a metal, it is ductile” presupposes the general law “All metals are ductile,” making the consequent of this statement a logical consequence of its antecedent.
Both in ordinary language and in the language of science, a conditional statement, in addition to the function of justification, can also perform a number of other tasks. It can formulate a condition that is not related to the s.l. an implied general law or rule (“If I want, I’ll cut my cloak”), fix some sequence (“If last summer was dry, then this year it will be rainy”), express disbelief in a peculiar form (“If you solve the problem, I will prove Fermat’s great theorem”), contrast (“If cabbage grows in the garden, then an apple tree grows in the garden”), etc. The multiplicity and heterogeneity of the functions of a conditional statement significantly complicates its analysis.
In logical systems, they abstract from the features of the usual use of a conditional statement, which leads to various implications. The most famous of them are material implication, strict implication and relevant implication.
Material implication is one of the main connections of classical logic. It is defined as follows: the implication is false only if the antecedent is true and the consequent is false, and true in all other cases. The conditional statement "If A, then B" presupposes some real connection between what is said in A and B; the expression “A materially implies B” does not imply such a connection.
Strict implication is defined through the modal concept of (logical) impossibility: “A strictly implies B” means “It is impossible for A to be true and B to be false.”
In relevant logic, implication is understood as a conditional conjunction in its ordinary sense. In the case of relevant implication, it cannot be said that a true statement can be justified by reference to any statement and that any statement can be justified by reference to a false statement.
Equivalence
The equivalence of two logical statements is a logical statement that is true only when they are simultaneously true or false (from Late Latin equivalens - equivalent) - a generic name for all kinds of relations such as equality, i.e. reflexive, symmetrical and transitive binary relations. Examples: equipollence (coincidence in meaning, significance, content, expressive and (or) deductive capabilities between concepts, concepts, scientific theories or formal systems that formalize them) congruence or similarity of geometry, figures; isomorphism; equivalence of sets and other equivalence of any objects means their equality (identity) in some respect
(for example, isomorphic sets are indistinguishable in their “structure”, if by “structure” we mean the totality of those properties with respect to which these sets are isomorphic). Any equivalence relation generates a partition of the set on which it is defined into pairwise disjoint “equivalence classes”; elements of the given set that are equivalent to each other are classified into one class.
Consideration of equivalence classes as new objects is one of the main ways of generating (introducing) abstract concepts in logical-mathematical (and in general natural science) theories. Thus, considering fractions a/b and c/d with integer numerators and denominators equivalent, if ad=bc, rational numbers are introduced into consideration as classes of equivalent fractions; considering sets as equivalent, between which one-to-one correspondence can be established, the concept of cardinality (cardinal number) of a set is introduced (as a class of sets equivalent to each other); considering two pieces of a substance equivalent, which enter into identical chemical reactions under equal conditions, one arrives at the abstract concept of chemical composition, etc.
The term “equivalence” is often used not (only) as a generic one, but as a synonym for some of its particular meanings (“equivalence of theories” instead of “equivalence”, “equivalence of sets” instead of “equivalence”, “equivalence of words” in abstract algebra instead of “identity” " and so on.).
Quantifier statement
Quantifier with a universal quantifier.
A quantifier logical statement with a universal quantifier (“xA(x)”) is a logical statement that is true only if for each object x from a given population the statement A(x) is true.
Quantifier with existence quantifier.
A quantifier logical statement with an existential quantifier ($xA(x)) is a logical statement that is true only if in a given collection there exists an object x such that the statement A(x) is true.
Structure of mathematical logic
The section “mathematical logic” consists of three parts: on the informal axiomatic method, on propositional logic and on predicate logic (first order). The axiomatic construction method is the first step towards formalizing the theory. Most of the problems considered in mathematical logic consist of proving certain statements. Mathematical logic has many ramifications. It uses a tabular construction of propositional logic, uses a special symbol language and propositional logic formulas.
Informal axiomatic method
An axiomatic method that does not fix the rigidly applied language and thereby does not fix the boundaries of the meaningful understanding of the subject, but requires an axiomatic definition of all concepts special to the given subject of study. This term does not have a generally accepted interpretation.
The history of the development of the axiomatic method is characterized by an ever-increasing degree of formalization. The informal axiomatic method is a certain step in this process.
The original axiomatic construction of geometry, given by Euclid, was distinguished by the deductive nature of its presentation, which was based on definitions (explanations) and axioms (obvious statements). From them, based on common sense and evidence, consequences were drawn. At the same time, the conclusion sometimes implicitly used assumptions of geometry and character that were not fixed in the axioms, especially those related to movement in space and the relative position of lines and points. Subsequently, geometry, concepts and axioms regulating their use were identified, implicitly used by Euclid and his followers. At the same time, the question arose: were all the axioms really identified? The guiding principle for resolving this issue was formulated by D. Hilbert: “It should be achieved that we can equally well speak of tables, chairs and beer mugs instead of points, lines and planes.” If the proof does not lose its evidentiary force after such a replacement, then indeed all the special assumptions used in this proof are fixed in the axioms. The degree of formalization achieved with this approach represents the level of formalization characteristic of the informal axiomatic method. The standard here can be the classic work of D. Hilbert “Foundations of Geometry”.
The informal axiomatic method is used not only to give a certain completeness to the axiomatically stated concrete theory. It is an effective tool for mathematical research. Since when studying a system of objects using this method, their specificity, or “nature,” is not used, the proven statements are transferred to any system of objects that satisfies the axioms under consideration. According to the informal axiomatic method, axioms are implicit definitions of original concepts (rather than obvious truths). It does not matter what the objects being studied are. Everything you need to know about them is formulated in axioms. The subject of study of an axiomatic theory is any interpretation of it.
The informal axiomatic method, in addition to the indispensable axiomatic definition of all special concepts, has another characteristic feature. It is the free, uncontrolled, content-based use of ideas and concepts that can be applied to any conceivable interpretation, regardless of its content. In particular, set-theoretic and logical concepts and principles are widely used, as well as concepts related to the idea of counting, etc. The penetration into the axiomatic method of reasoning based on meaningful understanding and common sense, and not on axioms, is not explained by the fixed nature of the language, on which the properties of an axiomatically given system of objects are formulated and proven. Fixing the language leads to the concept of a formal axiomatic system and creates a material basis for identifying and clearly describing admissible logical principles, for the controlled use of set-theoretic and other general or non-specific concepts for the field under study. If a language does not have the means (words) to convey set-theoretic concepts, then all evidence based on the use of such means is eliminated. If a language has means for expressing certain set-theoretic concepts, then their use in proofs can be limited by certain rules or axioms.
By fixing language in different ways, different theories of the main object of consideration are obtained. For example, considering the language of narrow predicate calculus for group theory, one obtains an elementary group theory in which it is impossible to formulate any statements about subgroups. If we move on to the language of predicate calculus of the second stage, then it becomes possible to consider properties in which the concept of a subgroup appears. The formalization of the informal axiomatic method in group theory is the transition to the language of the Zermelo-Frenkel system with its axiomatics.
Axiomatic method
The axiomatic method is a way of constructing a scientific theory, in which it is based on certain initial positions (judgments) - axioms, or postulates, from which all other statements of this theory must be deduced in a purely logical way, through evidence. The construction of science based on the axiomatic method is usually called deductive. All concepts of a deductive theory (except for a fixed number of initial ones) are introduced through definitions that express them through previously introduced concepts. To one degree or another, deductive proofs, characteristic of the axiomatic method, are used in many sciences, but the main area of its application is mathematics, logic, and some branches of physics.
The idea of the axiomatic method was first expressed in connection with the construction of geometry in Ancient Greece (Pythagoras, Plato, Aristotle, Euclid). The modern stage of development of the axiomatic method is characterized by the concept of the formal axiomatic method put forward by Hilbert, which poses the task of accurately describing the logical means of deriving theorems from axioms. Hilbert's main idea is a complete formalization of the language of science, in which its judgments are considered as sequences of signs (formulas) that acquire meaning only with some specific interpretation. To derive theorems from axioms (and in general some formulas from others), special formulas are formulated. inference rules. A proof in such a theory (calculus, or formal system) is a certain sequence of formulas, each of which is either an axiom or is obtained from previous formulas in the sequence according to some rule of inference. In contrast to such formal proofs, the properties of the formal system itself as a whole are studied. by means of metatheory. The main requirements for axiomatic formal systems are consistency, completeness, and independence of axioms. Hilbert's program, which assumed the possibility of proving the consistency and completeness of all classical mathematics, turned out to be generally unfeasible. In 1931, Gödel proved the impossibility of complete axiomatization of sufficiently developed scientific theories (for example, the arithmetic of natural numbers), which indicated the limitations of the axiomatic method. The basic principles of axiomatic methods were criticized by supporters of intuitionism and the constructive direction.
Conclusion
Mathematical logic is the science of the laws of mathematical thinking. The application of mathematics to logic made it possible to present logical theories in a new convenient form and to apply the computing apparatus to solving problems that are inaccessible to human thinking, and this, of course, expanded the field of logical research. The scope of application of mathematical logic is very wide. Every year the deep penetration of ideas and methods of mathematical logic into computer science, computational mathematics, linguistics, and philosophy is growing. A powerful impetus for the development and expansion of the field of application of mathematical logic was the emergence of electronic computers. It turned out that within the framework of mathematical logic there is already a ready-made apparatus for designing computer technology. Methods and concepts of mathematical logic are the basis, the core of intelligent information systems. Mathematical logic tools have become an effective working tool for specialists in many fields of science and technology. All specialists need to know mathematical logic, regardless of what environment they work in (be it an engineer, teacher, lawyer or just a doctor).
Bibliography
mathematical logic statement conjunction
Internet resource: #"justify">1.
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- Mathematical logic, Ershov Yu.L.. The book outlines the basic classical calculus of mathematical logic: propositional calculus and predicate calculus; there is a brief summary of the basic concepts of set theory and theory...