Symbolic logic: concept, language of logic, traditional and modern logic

Symbolic logic is a branch of science that studies the correct forms of reasoning. It plays a fundamental role in philosophy, mathematics and computer science. Like philosophy and mathematics, logic has ancient roots. The earliest treatises on the nature of correct reasoning were written more than 2000 years ago. Some of the most famous philosophers of ancient Greece wrote about the nature of retention more than 2,300 years ago. The thinkers of ancient China wrote about logical paradoxes around the same time. Although its roots go back in time, logic still remains a vibrant area of ​​research.

Mathematical Symbolic Logic

One must also be able to understand and reason, which is why logical conclusions were given special attention when there was no special equipment for analysis and diagnostics of the most diverse areas of life. Modern symbolic logic arose on the basis of the works of Aristotle (384-322 BC), the great Greek philosopher and one of the most influential thinkers of all time. Further successes were made by the Greek stoic philosopher Chrysippos, who developed the foundations of what we now call propositional logic.

Active development was received only in the 19th century by mathematical or symbolic logic. The works of Buhl, de Morgan, Schroeder appeared, in which scientists carried out the algebraization of the teachings of Aristotle, thereby forming the basis of the calculus of propositions. This was followed by the work of Frege and Pris, in which the concepts of variables and quantifiers were introduced, which began to be applied in logic. So the calculation of predicates - statements about the subject was formed.

Logic implied the proof of indisputable facts, when there was no direct confirmation of the truth. Logical expressions were supposed to convince the interlocutor of truthfulness.

Logical formulas were built on the principle of mathematical proof. So they convinced the interlocutors of accuracy and reliability.

However, all forms of argument were written in words. There were no formal mechanisms that would create a logical calculus of deduction. People began to doubt whether the scientist was hiding behind mathematical calculations, hiding behind them the absurdity of his guesses, because everyone can present their arguments in another favor.

The birth of meaningfulness: solid logic in mathematics as proof of truth

Towards the end of the 18th century, mathematical or symbolic logic appeared in the form of science, which implied the process of studying the correctness of conclusions. They should have a logical end and a connection. But how was it to prove or justify these studies?

The great German philosopher and mathematician Gottfried Leibniz was one of the first to recognize the need to formalize logical arguments. It was Leibniz's dream: to create a universal formal language of science, which would reduce all philosophical disputes to a simple calculation, reworking the reasoning in such discussions in this language. Mathematical or symbolic logic appeared in the form of formulas that facilitated problems and solutions in philosophical matters. And this area of ​​science has become more significant, because senseless philosophical chatter then became the bottom on which mathematics itself relies!

Nowadays, traditional logic is symbolic Aristotelian, which is simple and unpretentious. In the 19th century, science was faced with a paradox of sets that gave rise to inconsistencies of those very famous solutions of Aristotle's logical sequences. This problem had to be solved, because in science there cannot be even superficial errors.

Lewis Carroll Formality - Symbolic Logic and the Stages of its Transformation

Formal logic is now a subject that is part of the curriculum. However, it owes its appearance to a symbolic one, the very one that was originally created. Symbolic logic is a method of representing logical expressions using symbols and variables, rather than a common language. This eliminates the ambiguity that accompanies common languages, such as Russian, and makes it easier to work.

There are many systems of symbolic logic, such as:

• Classic propositional.
• First order logic.
• Modal.

The symbolic logic in the understanding of Lewis Carroll would have to indicate a true and false statement in the question asked. Each may have separate characters or exclude the use of certain characters. Here are some examples of statements that complete the logical chain of conclusions:

1. All people who are identical to me are creatures that exist.
2. All heroes that are identical to Batman are creatures that exist.
3. Therefore (since Batman and I were never seen in the same place), all people identical to me are heroes identical to Batman.

This is an invalid form syllogism, but it is the same structure as the following:

• All dogs are mammals.
• All cats are mammals.
• Therefore, all dogs are cats.

It should be obvious that the symbolic form given above is not valid in logic. However, in logic, justice is defined by this expression: if the premises were true, then the conclusion would be true. This is clearly not the case. The same will be for an example with a hero who has the same shape. Validity applies only to deductive arguments, which are designed to prove their conclusion with certainty, since a deductive argument cannot be valid. These “corrections” are also used in statistics when there is a result of a data error, and modern symbolic logic as a formality of simplified data helps in many of these issues.

Induction in modern logic

The inductive argument is intended only to demonstrate its conclusion with high probability or rebuttability. Inductive arguments are either strong or weak.

As an inductive argument, the Batman superhero example is simply weak. It is doubtful that Batman exists, so one of the statements is already incorrect with a high probability. Although you have never seen him in the same place as someone else, it is ridiculous to consider this expression as evidence. To understand the essence of logic, imagine:

1. You have never been seen in the same place as a native of Guinea.
2. It is not plausible that you and the person from Guinea are the same person.
3. Now imagine that you and the African never saw one place. It is not plausible that you and the African are one and the same person. But the Guinean and the African intersected, so you cannot be with both at the same time. The evidence that you are African or Guinean has declined significantly.

From this point of view, the very idea of ​​symbolic logic does not imply an a priori relation to mathematics. All that is required to recognize logic as a symbol is the widespread use of symbols to represent logical operations.

Caroll's Logical Theory: Entanglement or Minimalism in Mathematical Philosophy

Carroll studied some unusual ways that made him solve the rather complicated problems that his colleagues faced. This prevented him from achieving significant success due to the complexity of the logical notation and systems that he received as a result of his work. The meaning of Carroll's symbolic logic is a fix. How to find a conclusion that should be drawn from a set of premises regarding the relationship between given terms? Eliminating "average terms."

To solve this central problem, mid-nineteenth-century logic invented symbolic, diagrammatic, even mechanical devices. However, Carroll’s methods for processing such “logical sequences” (as he called them) did not always provide the right solution. Later, the philosopher published two documents on hypotheses that are reflected in the journal Reason: The Logical Paradox (1894) and What the Turtle Sayed to Achilles (1895).

These articles were widely discussed by nineteenth and twentieth-century logicians (Pierce, Russell, Ryle, Prior, Quine, etc.). The first article is often referred to as a good illustration of the paradoxes of material implication, and the second leads to what is known as the inference paradox.

Simplicity of characters in logic

The symbolic language of logic is a replacement for long ambiguous sentences. It is convenient, since in Russian one can say the same thing about different circumstances, which will make it possible to get confused, and in mathematics, symbols will replace the identity of each meaning.

1. First, brevity is important for efficiency. Symbolic logic cannot do without signs and designations, otherwise it would remain only philosophical, without the right to true meaning.
2. Secondly, symbols facilitate viewing and formulate logical truths. Paragraphs 1 and 2 encourage "algebraic" manipulation of logical formulas.
3. Third, when logic expresses logical truths, a symbolic formulation encourages the study of the structure of logic. This is due to the previous paragraph. Thus, symbolic logic lends itself to the mathematical study of logic, which is a branch of the subject of mathematical logic.
4. Fourth, when repeating the answer, the use of characters is an aid in preventing the ambiguity (e.g., multiple meanings) of a common language. It also helps ensure uniqueness of meaning.

Finally, the symbolic language of logic allows predicate calculus introduced by Frege. Over the years, the symbolic designation of the predicate calculus itself has been refined and become more effective, since good notation is important in mathematics and logic.

Ontology of Antiquity of Aristotle

Scientists became interested in the work of the thinker when they began to use Slinin Ya. A. methods in their interpretations. “Symbolic Logic” is a textbook written by a scientist and published in St. Petersburg. The book presents theories of classical and modal logic. An important part of the concept was the reduction to CNF in the symbolic logic of the formula of the logic of utterance. Abbreviation means conjunction or disjunction of variables.

Slinin, Ya. A., suggested that complex negatives, which require repeated reduction of formulas, should turn into a subformula. Thus, he converted some values ​​to more minimal ones and solved problems in an abbreviated form. Work with negatives came down to de Morgan's formulas . The laws that bear the name de Morgan are a couple of theorems connected with each other that make it possible to turn statements and formulas into alternative and often more convenient ones. The laws are as follows:

1. The denial (or inconsistency) of a disjunction is equal to the union of the negation of alternatives - p or q is not p and not q, or symbolically ~ (p ⊦ q) ≡ ~ p · ~ q.
2. The negation of the conjunction is equal to the disjunction of the negation of the original conjuncts, i.e., not (p and q) is not equal to p or not q, or symbolically ~ (p · q) ≡ ~ p ⊦ ~ q.

Thanks to these initial data, many mathematicians began to apply formulas to solve complex logical problems. Many people know that there is a lecture course where the area of ​​intersection of functions is studied. And matrix interpretation is also based on logical formulas. What is the essence of logic in algebraic connection? This is a level linear function, when the science of numbers and philosophy can be put on one side as a “soulless” and unprofitable sphere of reasoning. Although Kant E. believed differently, being a mathematician and philosopher. He noted that philosophy is nothing until proven otherwise. And the evidence must be scientifically sound. And it so happened that philosophy began to have significance due to the combination with the real nature of numbers and calculations.

The application of logic in science and the material world of reality

Philosophers usually do not apply the science of logical thinking only in some ambitious post-degree project (usually with a high degree of specialization, for example, by adding to social science, psychology or ethical categorization). It is paradoxical that philosophical science “gave birth” to the method of calculating truth and lies, but philosophers themselves do not use it. So for whom are such clear-cut mathematical syllogisms created and transformed?

1. Programmers and engineers used symbolic logic (which is not so different from the primary) to implement computer programs and even design boards.
2. In the case of computers, logic has become complex enough to handle numerous function calls, as well as advance math and solve math problems. Most of it is based on knowledge of the mathematical solution of problems and probability, combined with the logical rules of exclusion, expansion, and reducibility.
3. Computer languages cannot be easily understood in order to work logically within the knowledge of mathematics and even perform special functions. Most of the computer language is probably patented or understood only by computers. Now, programmers often allow computers to do logical tasks and solve them.

During such assumptions, many scientists suggest the creation of advanced material not for the sake of science, but for the convenience of using media and technology. Perhaps soon logic will leak into the spheres of economics, business, and even a “two-faced” quantum, which behaves both as an atom and as a wave.

Quantum logic in the modern practice of mathematical analysis

Quantum logic (QL) was developed as an attempt to construct a propositional structure that would describe interesting events in quantum mechanics (QM). QL replaced the Boolean structure, which was insufficient to represent the atomic kingdom, although it was suitable for the discourse of classical physics.

The mathematical structure of a propositional language about classical systems is a set of powers partially ordered by a set of inclusions, with a pair of operations representing unification and disjunction.

This algebra is consistent with the discourse of both classical and relativistic phenomena, but is incompatible in a theory that prohibits, for example, giving simultaneous truth values. The proposal of the founding fathers of QL was created to replace the Boolean structure of classical logic with a weaker structure that would weaken the distributive properties of conjunction and disjunction.

Weakening of the established symbolic introduction: is truth needed in mathematics as an exact science

During its development, quantum logic began to refer not only to the traditional, but also to several areas of modern research that tried to understand mechanics from a logical point of view. Multiple quantum approaches have been presented to introduce the different strategies and problems discussed in the literature of quantum mechanics. Whenever possible, unnecessary formulas are eliminated to give an intuitive understanding of concepts before obtaining or introducing related mathematics.

The long-standing question in the interpretation of quantum mechanics is whether fundamentally classical explanations are available for quantum-mechanical phenomena. Quantum logic has played a large role in shaping and refining this discussion, in particular by allowing us to be reasonably accurate with respect to what we mean by classical explanation. Now it is possible to establish with accuracy which theories can be considered reliable, and which the logical conclusion of mathematical judgments.