Systems of logic

Main thing

Systems of logic are frameworks for reasoning and argumentation that define rules for drawing valid conclusions from premises.

The main types of logic systems include:

1. Informal logic: Analyzes arguments presented in everyday language, considering their content, context, and potential fallacies.

   • Traditional or Aristotelian logic: A type of informal logic that focuses on syllogisms, which are arguments containing three statements: a major premise, a minor premise, and a conclusion.
     Example: "All mammals are animals. All dogs are mammals. Therefore, all dogs are animals."

2. Formal logic: Deals with the form of arguments using precise rules and symbols, abstracting from content and focusing on validity.

   • Classical logic: A type of formal logic based on principles such as the law of excluded middle and the law of non-contradiction.
     Example: "If A is true, and A implies B, then B is true."

   • Mathematical logic: A branch of formal logic that uses mathematical symbols and techniques to analyze the structure of arguments and the foundations of mathematics.
     Example: "∀x (P(x) → Q(x))" (For all x, if P(x) is true, then Q(x) is true.)

   • Non-classical logics: Formal logical systems that reject or extend some principles of classical logic.

     • Intuitionistic logic: A type of non-classical logic that rejects the law of excluded middle.
       Example: In intuitionistic logic, the statement "Either A is true, or not-A is true" is not always valid.

     • Many-valued logics: Non-classical logics that allow for truth values beyond "true" and "false."
       Example: In a three-valued logic, a proposition can be "true," "false," or "unknown."

These systems provide different approaches to analyzing arguments and reasoning, ranging from informal analysis of everyday language to highly formal, symbolic representations of arguments and mathematical structures.

Terms

• Logic – the study of correct reasoning, focusing on the principles and methods for distinguishing valid from invalid arguments. Example: Using logical principles to determine whether a conclusion follows from a set of premises.

• Argument – a series of statements, called premises, that are used to support or justify a conclusion. Example: "All dogs are mammals. All mammals are animals. Therefore, all dogs are animals."

• Premise – a statement in an argument that provides evidence or reasons for accepting the conclusion. Example: In the argument "All dogs are mammals. Spot is a dog. Therefore, Spot is a mammal," the premises are "All dogs are mammals" and "Spot is a dog."

• Conclusion – the main claim or assertion in an argument, which is supposed to follow from the premises. Example: In the argument "All dogs are mammals. Spot is a dog. Therefore, Spot is a mammal," the conclusion is "Spot is a mammal."

• Syllogism – a form of logical argument that consists of a major premise, a minor premise, and a conclusion. Example: "All A are B. All C are A. Therefore, all C are B."

• Validity – a property of an argument where the conclusion necessarily follows from the premises, regardless of the truth of the premises. Example: "All dogs are purple. Spot is a dog. Therefore, Spot is purple." (Valid but unsound)

• Law of excluded middle – the principle in classical logic stating that for any proposition, either that proposition is true, or its negation is true. Example: For the proposition "Socrates is mortal," either "Socrates is mortal" is true, or "Socrates is not mortal" is true.

• Law of non-contradiction – the principle in classical logic stating that contradictory propositions cannot both be true at the same time and in the same sense. Example: The propositions "Socrates is mortal" and "Socrates is not mortal" cannot both be true simultaneously.

• Quantifiers – symbols used in mathematical logic to express the extent to which a predicate is true for elements in a domain. The two main quantifiers are the universal quantifier (∀) meaning "for all," and the existential quantifier (∃) meaning "there exists." Example: "∀x (Dog(x) → Mammal(x))" means "For all x, if x is a dog, then x is a mammal."

An analogy

Systems of logic are like different tools in a toolbox. Just as a carpenter might use a hammer for some tasks and a screwdriver for others, logicians use different systems of logic depending on the type of argument they want to analyze or the problem they want to solve. Each tool has its own strengths and limitations, and the key is to know when and how to use them effectively.

Example: When analyzing a complex argument with many steps, a logician might use formal logic to break it down into its constituent parts and ensure that each step follows validly from the previous ones.

A main misconception

Many people believe that logic is just about following rules and that all logical arguments are necessarily true. However, the validity of an argument doesn't guarantee the truth of its conclusion. An argument can be logically valid but still have false premises, leading to a false conclusion.

Example: "All cats are reptiles. All reptiles are mammals. Therefore, all cats are mammals." This argument is logically valid but unsound because its premises are false.

The history

1. Traditional or Aristotelian logic: Developed by Aristotle in the 4th century BCE, focusing on syllogisms and deductive reasoning.

2. Mathematical or classical logic: Pioneered by Gottlob Frege, Bertrand Russell, and Alfred North Whitehead in the late 19th and early 20th centuries.

3. Informal logic: Emerged in the 1950s and 1960s as a response to the limitations of formal logic, focusing on analyzing everyday arguments.

4. Formal logic: Developed alongside mathematical logic in the late 19th and early 20th centuries by figures like Gottlob Frege, Bertrand Russell, and Alfred North Whitehead.

5. Non-classical logics: Developed in the 20th century, challenging assumptions of classical logic.

   • Logics rejecting the law of excluded middle: Intuitionistic logic, which rejects the law of excluded middle, was introduced by L.E.J. Brouwer in the early 20th century.

   • Many-valued logics: Allowing for truth values beyond "true" and "false," many-valued logics were introduced by Jan Łukasiewicz and Emil Post in the 1920s.

"Logic takes care of itself; all we have to do is to look and see how it does it." - Ludwig Wittgenstein, one of the most influential philosophers of the 20th century, known for his work on logic, language, and the foundations of mathematics.

Three cases how to use it right now

1. When evaluating a political argument, use informal logic to identify any fallacies or unsupported claims, ensuring that the conclusion follows from the premises.

2. When designing a computer program, use mathematical logic to define the rules and constraints that the program must follow, ensuring that it behaves correctly and efficiently.

3. When analyzing a philosophical argument, use formal logic to break it down into its constituent parts and evaluate its validity, identifying any hidden assumptions or inconsistencies.

Interesting facts

• The study of logic dates back to ancient civilizations like Greece and India, with philosophers like Aristotle and Gautama laying the foundations for the field.

• In the 20th century, the development of non-classical logics challenged assumptions that had been taken for granted for over 2,000 years.

• Logic has played a crucial role in the development of computer science, with concepts like Boolean algebra and logical gates forming the basis for digital circuits and programming languages.

• The famous "barber paradox" is an example of a logical paradox that arises from self-referential statements: "The barber is the one who shaves all those, and those only, who do not shave themselves. Does the barber shave himself?"

• Kurt Gödel's incompleteness theorems, published in 1931, showed that there are limits to what can be proven within any consistent formal system, revolutionizing the foundations of mathematics and logic.