Mathematical logic . It is the discipline that deals with methods of reasoning. At an elementary level, logic provides rules and techniques for determining whether or not a given argument is valid. Logical reasoning is used in mathematics to prove theorems; in computer science to verify whether or not the programs are correct; in the physical and natural sciences, to draw conclusions from experiments; and in the social sciences and in everyday life, to solve a multitude of problems. Certainly logical reasoning is constantly used to carry out any activity.
Summary
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- 1 Home
- 1 Plato
- 2 Aristotle
- 2 Objectives
- 3 Algebra of logic
- 4 References
- 5 Sources
Start
Logic studies the form of reasoning, it is a discipline that through rules and techniques determines if an argument is valid. Logic is widely applied in Philosophy , Mathematics , Computing , Physics . In philosophy to determine if a reasoning is valid or not, since a phrase can have different interpretations, however logic allows us to know the correct meaning. In mathematics to prove theorems and infer mathematical results that can be applied in research.
The evolution of logic is linked to the intellectual evolution of the human being, since as a science of reasoning it can be affirmed that its history represents the very history of man. Logic arises from the moment when man, when facing nature, begins to observe, experience, deduce and reason.
During the period 600 BC to 300 BC the formal principles of mathematics were developed in Greece , this period was called the classical period where its main representatives are: Plato who introduced his ideas and abstractions; Aristotle who presented the ductive and systematic reasoning and Euclid who was the most influential since this established the axiomatic method.
Plato
Plato proposes to establish in Syracuse a utopian republic led by philosophers. It creates the Academy of Athens that was not only a philosophical institution, but a center for political training for young aristocrats. According to some specialists, Plato builds his theory of knowledge in order to justify the emerging power of the figure of the philosopher. It supports the existence of two worlds – the world of ideas and the physical world of objects. According to Plato, the concrete is perceived in terms of the abstract and therefore the sensible world exists thanks to the world of ideas. Plato chooses the dialogue format as a form of transmission of thought.
Aristotle
Aristotle’s treatises on logic , known as Organon, contain the first systematic treatise on the laws of thought for the acquisition of knowledge. They represent the first serious attempt to found logic as science.
objectives
The objective of mathematical logic is to question the concepts and deduction rules that are used in mathematics and this constitutes logic as true mathematics.
Algebra of logic
Part of mathematical logic based on the application of algebraic methods to the study of logical objects: classes and propositions. On the one hand, the proposition expresses a meaning (judgment); on the other, it designates a truth (V) or a lie (M). Thus, the propositions “The Volga empties into the Caspian Sea” and “2 x 2 = 4” express a different meaning, but both designate a truth (they have the meaning of V). [one]
The algebra of logic examines propositions only from the point of view of their meaning, with the particularity that those that have the same meaning of truthfulness are considered equivalent. [one]
The algebra of logic uses symbolic notation (Logical Symbolism). In addition to the symbols of the propositions, symbols are used for the operations: conjunction, disjunction, implication, negation, with which the algebra of logic forms some expressions starting from others. [one]
An expression will be composed if it has been formed by others by means of logical algebraic operations; otherwise it will be simple. Two expressions are called equivalents if they have equal meanings in every possible combination of meanings of the simple expressions contained in them. Thus A ® B is equivalent to Aœ Ë B, given that in the four possible combinations of meanings of V and M for A and B: VV, VM, MV, MM, A ® B has the same meaning as Aœ Ë B. [ one]
In relation to the concepts introduced, a series of problems are raised in the algebra of logic to which this discipline is applied. Historically, the algebra of logic emerged as algebra of classes (Boole) and only later was it interpreted as algebra of propositions. With the works of VI Shestakov and Claude Shannon , the algebra of logic finds wide application in the theory of electrical diagrams and contact relay diagrams.
