Logic

Logic is the study of correct reasoning. It helps us understand how to make good arguments and decide what is true based on facts.

There are two main types of logic: formal and informal. Formal logic looks at how conclusions follow from premises using specific rules, while informal logic examines arguments in everyday language.

Logic studies valid forms of inference like modus ponens.

Arguments are made up of premises that lead to a conclusion. For example, if it is Sunday and Sundays are days off from work, then we can conclude we do not have to work. Some arguments are correct, meaning the premises really do support the conclusion.

People have been studying logic for thousands of years. Ancient thinkers like Aristotle developed early systems of logic. Today, classical logic is the most common system, and it is used in many areas such as philosophy, mathematics, and computer science. Logic helps us think clearly and make better decisions by understanding how ideas connect together.

Definition

The word "logic" comes from the Greek word logos, meaning reason, discussion, or language. Logic is the study of correct reasoning. It helps us see if arguments are right or wrong by checking if the facts given support the conclusion.

Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid. In this example, the letter "c" represents Carmen while the letters "M" and "T" stand for "Mexican" and "teacher". The symbol "∧" has the meaning of "and".

Logic has two main parts: formal and informal. Formal logic studies reasoning using symbols and rules. It looks at whether arguments follow correct patterns. Informal logic looks at everyday arguments, helping us understand and judge them better.

Formal logic uses special languages with clear rules to check arguments. Informal logic deals with everyday speech and arguments that can be tricky. Both help us think clearly and spot mistakes in reasoning.

Basic concepts

Premises, conclusions, and truth

Premises and conclusions are important in logic. When an argument is right, the conclusion comes from the premises. For example, if we know "Mars is red" and "Mars is a planet," we can say "Mars is a red planet." In logic, premises and conclusions have a truth value — they are either true or false.

Premises and conclusions can be simple or complicated. Simple ones, like "Mars is red," have no parts. Complicated ones, like "Mars is red and Venus is white," are made by joining simple parts with words like "and" or "if...then."

Arguments and inferences

Logic looks at whether arguments are right. An argument has premises and a conclusion. An inference is when we move from premises to a conclusion. Arguments are right when the premises support the conclusion.

Young America's dilemma: Shall I be wise and great, or rich and powerful? (poster from 1901). This is an example of a false dilemma: an informal fallacy using a disjunctive premise that excludes viable alternatives.

There are two main types of right arguments: deductive and ampliative. Deductive arguments make the conclusion true if the premises are true. For example, "All frogs are amphibians. No cats are amphibians. Therefore, no cats are frogs" is deductively right. Ampliative arguments give more information and make the conclusion more likely, but do not promise it.

Fallacies

Sometimes arguments have mistakes in thinking, called fallacies. These are not about whether the conclusion is true, but about problems in the thinking. For example, saying "It is sunny today, therefore spiders have eight legs" is wrong because the thinking is flawed, even though the conclusion happens to be true. Fallacies can be about the shape of the argument or its meaning and situation.

Truth table of various expressions
p	q	p ∧ q	p ∨ q	p → q	¬p → ¬q	p ↑ {\displaystyle \uparrow } q
T	T	T	T	T	T	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	F	T	T	T

Systems of logic

Systems of logic are ways to check if reasoning and arguments make sense. For over two thousand years, one type of logic created by Aristotle was the main way people thought about logic. But now, there are many different systems of logic.

Aristotelian logic looks at simple statements with a subject and a predicate, like "Socrates is wise." It checks if these statements follow correct patterns.

Classical logic is another type with rules like every statement must be true or false. It was mainly used for math. Extended logics add more ideas to classical logic to use in areas like ethics.

There are also other types of logic that change some rules of classical logic. Informal logic looks at arguments in everyday language.

Areas of research

Logic is studied in many fields. Sometimes, its rules are used to study other topics, like computer science. Other times, logic itself becomes the focus of study in different subjects. This can happen in many ways, such as looking at the basic ideas behind logic or using math to understand how logical systems work.

Philosophy of logic and philosophical logic

Main articles: Philosophy of logic and Philosophical logic

Philosophy of logic looks at what logic is and how it works. It asks questions about the basic ideas used in logic and how logical systems should be organized. Philosophical logic uses logic to solve problems in areas like ethics and understanding how we know things.

Metalogic

Main article: Metalogic

Bertrand Russell made various contributions to mathematical logic.

Metalogic studies the properties of formal logical systems. It looks at what can be proven in these systems, whether every true statement can be proven, and how different logical systems compare. Metalogicians also study whether logical systems are complete, sound, and consistent.

Mathematical logic

Main article: Mathematical logic

Mathematical logic uses logic within mathematics. It includes areas like model theory, proof theory, set theory, and computability theory. These areas study the mathematical properties of logical systems and how logic can help understand mathematical reasoning.

Computational logic

Conjunction (AND) is one of the basic operations of Boolean logic. It can be electronically implemented in several ways, for example, by using two transistors.

Computational logic looks at how to use computers to perform mathematical reasoning and logical tasks. This includes creating programs that can prove mathematical statements automatically and designing computer languages that use logic to express facts and draw conclusions.

Formal semantics of natural language

Main article: Formal semantics (natural language)

Formal semantics studies the meaning of language using logical and mathematical tools. It looks at when a sentence is true or false and how the meaning of complex expressions depends on the meanings of their parts.

Epistemology of logic

The epistemology of logic studies how we know that an argument is valid or that a statement is logically true. Some believe this knowledge comes from the mind's ability to understand pure ideas, while others think it comes from observing the world.

History

Main article: History of logic

Logic started a long time ago in many places. One of the first big thinkers was Aristotle. He made a way to study how we think and reason, using ideas like terms and syllogisms. His work was important for many years in Europe and the Middle East.

Later, a thinker named Ibn Sina made new ways to study logic that became popular in the Islamic world. He also helped develop methods that are important for science today. During the Middle Ages, many people translated and explained Aristotle's ideas, helping others learn from them.

In other parts of the world, like China and India, people also studied how we think and reason. They looked at how language works and how we can know things for sure.

Eventually, new ways of studying logic were developed in the 19th century. These new methods used symbols and math to make logic clearer and more exact. This helped advance both logic and mathematics.