Logic

We explain what logic is, its history and characteristics. Also, computational and formal logic.

What is logic?

Logic is the science of reasoning. In general, logic is considered to have its origin in philosophy and its application in mathematics. However, Logic is considered an independent science while its origin occurred in parallel to that of philosophy and not as a direct consequence of it.

Those who are dedicated to logic study reasoning called “arguments” or “argument schemes.” Your task is to discover what makes a valid argument valid. Depending on which branch of logic they are dedicated to, this will be the content of the different arguments. The logic works with concepts, definitions, propositions and formal arguments. All of them are given in order to determine the validity of each of the arguments discussed.

In general, Logic can be divided into formal logic and informal logic. Formal logic, for its part, works with systems of propositional logic (which operates on propositions), first-order logic (which operates on predicates) and modal logic (which operates on truth values).

See also: Mathematical thinking

Etymology of the term “logic”

The word “logic” It has its origin in the Greek voice logike (“endowed with reason”) coming from the term logoswhich is equivalent to “word” or “thought.”

However, In everyday language we use this word as a synonym for “common sense.”. It is also used as a synonym for “way of thinking”, such as when referring to “sports logic”, “military logic”, etc.

See also: Philosophical knowledge

history of logic

The logic It has its origins in different cultures and traditions throughout the world. Already in Babylon, Greece, China or India, different philosophers and thinkers dedicated themselves to logic. However, the most worked on has been, without a doubt, Aristotelian logic.

Aristotelian logic is the tradition of thought that begins with the works of Aristotle (384-322 BC), considered the Western founder of logic and one of the most important authors of the world's philosophical tradition.

Aristotle's main works regarding logic are gathered in his Organon (from the Greek “instrument”), compiled by Andronicus of Rhodes several centuries after writing. They display a logical system that was extremely influential in Europe and the Middle East until after the Middle Ages.

In this work, Aristotle also postulated the fundamental axioms of logic:

• The principle of non-contradiction. It establishes that something cannot be and not be at the same time (A and ¬A cannot be true at the same time).
• The identity principle. It states that something is always identical to itself (A is always equal to A).
• The principle of the excluded middle. It establishes that something is or is not true and there are no possible gradations (A or then ¬A).

The Aristotelian logical system then came into contact with Megarian and Stoic logic. From the confluence of these three currents, and after the contributions of different authors, formal logic as we know it today emerged in the 20th century. Authors such as Frege, Russel, and Whitehead worked to shape mathematical logic and create the possibility of new logical developments and schools.

Arguments, argument schemes and validity

Just as logic is the science of reasoning, argumentation is the application of reasoning. Logic investigates what makes an argument valid.

An argument is a sequence of sentences in which the premises are at the beginning and the conclusion at the end. A valid argument is one in which the truth of the premises implies the truth of the conclusion. In a valid argument, if the premises are true, the conclusion must be true.

For example:

1. Juan will come home or María will come home. (premise)
2. Maria won't come home. (premise)
3. Juan will come home. (conclusion)

If we replace each of the sentences with signs, we will see that what is really important about the argument is its form. In this case we will obtain something like: “A or B (p1), B is not given (p2), A is given (conclusion)”. It does not matter that the premises are true, in fact, what matters is that if we accept that they are true, we must accept that the conclusion is true. This is what is called “argument outline.”

Types of logic

Formal and informal logic

A distinction is often made between two fields of logic: formal logic and informal logic.

• The formal logic. It pays attention to formal language, that is, to the way of expressing its contents. It uses them strictly, without ambiguity, so that the deductive path can be analyzed based on the validity of its shapes (hence its name).
• The informal logic. He studies his arguments a posteriori, distinguishing valid and invalid forms based on the information given, without paying attention to their logical form or formal language. This variant emerged in the mid-20th century as a discipline within philosophy.

Continue in: Epistemology

The formal logic

The application of logical thinking to certain areas of mathematics and science is known as formal or mathematical logic.

This involves the study of the inference process through formal representation systems such as propositional logic, modal logic or first-order logic, which allow natural language to be “translated” into logical language. Each of these systems operates on different elements.

• Propositional logic operates on propositions with propositional variables and does not use quantifiers or individual variables.
• First-order logic or predicative logic operates on predicates and uses quantifiers and individual variables.
• Modal logic operates on truth value of the different propositions and predicates.

Formal logic covers four large areas:

• Model theory. It proposes the study of axiomatic theories and mathematical logic through mathematical structures known as groups, bodies or graphs, thus attributing semantic content to the purely formal constructions of logic.
• Proof theory. It proposes demonstrations using mathematical objects and mathematical techniques as the way to verify logical problems. While model theory is concerned with giving semantics (a meaning) to the formal structures of logic, proof theory is more concerned with their syntax (their ordering).
• Theory of sets. It proposes abstract collections of objects, understood in themselves as objects, as well as their basic operations and interrelationships. This branch of mathematical logic is one of the most fundamental, since it constitutes a basic tool of any mathematical theory.
• Computability theory. It proposes a link between mathematics and computer science and studies the decision problems that an algorithm (equivalent to a Turing machine) can face. To do this, it uses set theory, and understands them as computable or non-computable sets.

Computational logic

Computational logic is the same mathematical logic but applied to the field of computing that is, at various fundamental levels of computing: computational circuits, logical programming and algorithm management. Artificial intelligence is also part of it, a relatively recent field in the area.

Broadly speaking, computational logic aspires to power a computer system through logical structures that express, in mathematical language, the different possibilities of human thought, and for that, it creates intelligent computing systems.

Continue with: Analytical method

References

• Gamut, LTF, & Durán, C. (2002). Introduction to logic. Buenos Aires, Argentina: Eudeba.