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 over 2,000 years ago. Some of the most famous philosophers of ancient Greece wrote about the nature of retention over 2,300 years ago. Ancient Chinese thinkers were writing about logical paradoxes around the same time. Though its roots go back a long way, logic is still a vibrant field of study.
Mathematical symbolic logic
You also need to be able to understand and reason, which is why special attention was paid to logical conclusions when there was no special equipment for analyzing and diagnosing various areas of life. Modern symbolic logic arose from the work of Aristotle (384-322 BC), the great Greek philosopher and one of the most influential thinkers of all time. Further successes wereby the Greek Stoic philosopher Chrysippus, who developed the foundations of what we now call propositional logic.
Mathematical or symbolic logic received active development only in the 19th century. The works of Boole, de Morgan, Schroeder appeared, in which scientists algebraized the teachings of Aristotle, thereby forming the basis for the propositional calculus. This was followed by the work of Frege and Preece, in which the concepts of variables and quantifiers were introduced, which began to be applied in logic. Thus was formed the calculation of predicates - statements about the subject.
Logic implied proof of indisputable facts when there was no direct confirmation of the truth. Logical expressions were supposed to convince the interlocutor of the veracity.
Logical formulas were built on the principle of mathematical proof. So they convinced the interlocutors of accuracy and reliability.
However, all forms of arguments were written in words. There were no formal mechanisms that would create a logical deduction calculus. 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 a different favor.
Birth of meaningfulness: solid logic in mathematics as proof of truth
Toward the end of the 18th century, mathematical or symbolic logic emerged as a science, which involved the process of studying the correctness of conclusions. They were supposed to have a logical end and a connection. But how was it to proveor justify the research data?
The great German philosopher and mathematician Gottfried Leibniz was one of the first to realize the need to formalize logical arguments. It was Leibniz's dream: to create a universal formal language of science that 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 tasks and solutions in philosophical questions. Yes, and this area of science became more significant, because then the meaningless philosophical chatter then became the bottom on which mathematics itself relies!
In our time, traditional logic is symbolic Aristotelian, which is simple and unpretentious. In the 19th century, science was faced with the paradox of sets, which gave rise to inconsistencies in 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 its transformation steps
Formal logic is now a subject that is included in the course. However, it owes its appearance to the symbolic one, the one that was originally created. Symbolic logic is a method of representing logical expressions using symbols and variables rather than ordinary language. This eliminates the ambiguity that accompanies common languages such as Russian and makes things easier.
There are many systems of symbolic logic, such as:
- Classical propositional.
- First order logic.
- Modal.
Symbolic logic as understood by Lewis Carroll would have to indicate the true and false statements in the question asked. Each can have separate characters or exclude the use of certain characters. Here are some examples of statements that close the logical chain of conclusions:
- All people who are identical to me are beings that exist.
- All heroes that are identical to Batman are creatures that exist.
- So (since Batman and I were never seen in the same place), all people identical to me are heroes identical to Batman.
This is not a valid form syllogism, but it is the same structure as the following:
- All dogs are mammals.
- All cats are mammals.
- That's why all dogs are cats.
It should be obvious that the above symbolic form is not valid in logic. However, in logic, justice is defined by this expression: if the premise were true, then the conclusion would be true. This is clearly not true. The same will be true for the hero example, which has the same shape. Validity only applies to deductive arguments that are meant to prove their conclusion with certainty, since a deductive argument cannot be valid. These "corrections" are also applied in statistics when there is a result of data error, and modern symbolic logic asthe formality of simplified data helps in many of these matters.
Induction in modern logic
An inductive argument is only meant to demonstrate its conclusion with high probability or rebuttal. Inductive arguments are either strong or weak.
As an inductive argument, the example of the superhero Batman is simply weak. It is doubtful that Batman exists, so one of the statements is already wrong with a high probability. Although you have never seen him in the same place as someone else, it is ridiculous to take this expression as evidence. To understand the essence of logic, imagine:
- You have never been seen in the same place as the native of Guinea.
- It's implausible that you and the Guinean person are the same person.
- Now imagine that you and an African have never met in the same place. It is not plausible that you and an African are the same person. But the Guinean and the African crossed paths, so you can't be both at the same time. Evidence that you are African or Guinean has dropped substantially.
From this point of view, the very idea of symbolic logic does not imply an a priori relation to mathematics. All it takes to recognize logic as a symbol is the extensive use of symbols to represent logical operations.
Carroll's Logical Theory: Entanglement or Minimalism in Mathematical Philosophy
Carroll learned some unusual wayswhich forced him to solve rather difficult problems faced by his colleagues. This prevented him from making significant progress due to the complexity of the logical notation and systems that he received as a result of his work. The raison d'être of Carroll's symbolic logic is the problem of elimination. How to find the conclusion to be drawn from a set of premises regarding the relationship between given terms? Eliminating "middle terms".
It was to solve this central problem of logic in the mid-nineteenth century that symbolic, diagrammatic, even mechanical devices were invented. However, Carroll's methods for processing such "logical sequences" (as he called them) did not always give the right solution. Later, the philosopher published two papers on hypotheses, which are reflected in the journal Mind: The Logical Paradox (1894) and What the Tortoise Said to Achilles (1895).
These papers were widely discussed by logicians of the nineteenth and twentieth centuries (Pearce, Russell, Ryle, Prior, Quine, etc.). The first article is often cited as a good illustration of material implication paradoxes, while the second leads to what is known as the inference paradox.
Simplicity of symbols in logic
The symbolic language of logic is a substitute for long ambiguous sentences. Convenient, because in Russian you 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.
- 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.
- Secondly, symbols make it easier to see and formulate logical truths. Items 1 and 2 encourage "algebraic" manipulation of logical formulas.
- Third, when logic expresses logical truths, symbolic formulation encourages study of the structure of logic. This is related to the previous point. Thus, symbolic logic lends itself to the mathematical study of logic, which is a branch of the subject of mathematical logic.
- Fourth, when repeating the answer, the use of symbols is an aid in preventing the vagueness (eg, multiple meanings) of ordinary language. It also helps ensure that the meaning is unique.
Finally, the symbolic language of logic allows for the predicate calculus introduced by Frege. Over the years, the symbolic notation for the predicate calculus itself has been refined and made more efficient, as good notation is important in mathematics and logic.
Aristotle's ontology of antiquity
Scientists became interested in the work of the thinker when they began to use Slinin's methods in their interpretations. The book presents theories of classical and modal logic. An important part of the concept was the reduction to CNF in symbolic logic of the formula of the logic of proposition. The abbreviation means conjunction or disjunction of variables.
Slinin Ya. A. suggested that complex negations, 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 abridged version. Working with negations was reduced to de Morgan's formulas. The laws that bear De Morgan's name are a pair of related theorems that make it possible to turn statements and formulas into alternative and often more convenient ones. The laws are as follows:
- The negation (or inconsistency) of a disjunction is equal to the union of the negation of alternatives – p or q is not equal to p and not q or symbolically ~ (p ⊦ q) ≡ ~p ~q.
- 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 not 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 course of lectures where the area of intersection of functions is studied. And the matrix interpretation is also based on logic formulas. What is the essence of logic in algebraic connection? This is a level linear function, when you can put the science of numbers and philosophy on the same bowl as a “soulless” and not profitable sphere of reasoning. Although E. Kant thought otherwise, being a mathematician and philosopher. He noted that philosophy is nothing until proven otherwise. And the evidence must be scientifically sound. And so it happened that philosophy began to have significance thanks tomatching with the true nature of numbers and calculations.
Application of logic in science and the material world of reality
Philosophers do not usually apply the science of logical reasoning to just some ambitious post-degree project (usually with a high degree of specialization, such as adding to social science, psychology, or ethical categorization). It is paradoxical that philosophical science "gave birth" to the method of calculating truth and falsehood, but philosophers themselves do not use it. So for whom are such clear mathematical syllogisms created and transformed?
- Programmers and engineers used symbolic logic (which is not so different from the original) to implement computer programs and even design boards.
- In the case of computers, logic has become complex enough to handle numerous function calls, as well as advance mathematics and solve mathematical problems. Much of it is based on a knowledge of mathematical problem solving and probability combined with the logical rules of elimination, extension, and reducibility.
- Computer languages cannot be easily understood to work logically within the limits of knowledge of mathematics and even perform special functions. Much of the computer language is probably proprietary or understood only by computers. Programmers now often let computers do logic tasks and solve them.
In the course of such prerequisites, many scientists assume the creation of advanced material not for the sake of science, but forease of use of media and technology. Perhaps soon logic will seep into the spheres of economics, business, and even the "two-faced" quantum, which behaves both like an atom and like a wave.
Quantum logic in modern practice of mathematical analysis
Quantum logic (QL) was developed as an attempt to build a propositional structure that would allow describing interesting events in quantum mechanics (QM). QL replaced the boolean structure, which was not enough to represent the atomic realm, although it is 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 the inclusion set, with a pair of operations representing union and disjunction.
This algebra is consistent with the discourse of both classical and relativistic phenomena, but is incompatible in a theory that forbids, 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 penetration: is truth really needed in mathematics as an exact science
During its development, quantum logic began to refer not only to traditional, but also to several areas of modern research that tried to understand mechanics from a logical point of view. Multiplequantum approaches to introduce different strategies and problems discussed in the literature of quantum mechanics. Whenever possible, unnecessary formulas are eliminated to give an intuitive understanding of concepts prior to obtaining or introducing the associated mathematics.
A perennial question in the interpretation of quantum mechanics is whether fundamentally classical explanations for quantum mechanical phenomena are available. Quantum logic has played a large role in shaping and refining this discussion, in particular allowing us to be fairly precise about what we mean by classical explanation. Now it is possible to establish with accuracy which theories can be considered reliable, and which ones are the logical conclusion of mathematical judgments.
