Law of Excluded Middle

Main thing

The Law of Excluded Middle is a principle in logic.

This principle states that for any given statement, it is either true or its opposite is true. There is no third option where the statement could be neither true nor false. This law is one of the foundational concepts in classical logic, which helps to establish clear rules for reasoning and argumentation.

Example: If the statement is "It is raining," according to the Law of Excluded Middle, the statement must be either true (it is indeed raining) or false (it is not raining). There is no in-between state where the statement could be considered neither true nor false.

Terms

• Proposition: A statement that can be either true or false. Example: "The sky is blue."

• Negation: The opposite of a proposition. Example: If the proposition is "The cat is on the mat," its negation is "The cat is not on the mat."

• Classical Logic: A system of logic that includes the Law of Excluded Middle. Example: In classical logic, the statement "This apple is either red or not red" must be true.

An analogy

The Law of Excluded Middle is like a coin toss. Just as a coin will land either heads or tails (ignoring the extremely rare edge case), a proposition in logic is either true or false, with no middle ground.

A main misconception

Many people confuse the Law of Excluded Middle with the idea that every question has a clear answer. In reality, while the law states that every proposition is either true or false, it does not guarantee that we can always determine which one it is.

Example: The question "Is there life on other planets?" has a true or false answer, but we may not currently have the means to determine it.

The history

1. Ancient times: Aristotle discusses early forms of the law.

2. 19th century: The law becomes formalized in classical logic.

3. Early 20th century: Mathematicians like L.E.J. Brouwer challenge the law in the context of mathematical intuitionism.

"The principle of excluded middle is essential for the science of non-contradictory reasoning," stated by Aristotle, who is famous for laying the groundwork for formal logic.

Three cases how to use it right now

1. Legal Decision Making: In a court of law, when a judge faces a case, they apply the Law of Excluded Middle by ruling that the defendant is either guilty or not guilty based on the evidence. There's no verdict that says "somewhat guilty." For instance, in a theft case, evidence such as video footage or fingerprints is analyzed to conclusively determine the defendant's guilt or innocence.

2. Programming Logic: In computer science, when a programmer writes a conditional statement, they use the Law of Excluded Middle to dictate the flow of the program. For example, if a user inputs their age, the program might check if the user is above 18. The condition is either true (user is adult) or false (user is not adult), and based on this, different actions are taken, such as allowing or denying access to certain content.

3. Policy Implementation: In government or corporate decision-making, when considering the adoption of a new policy, the decision is ultimately binary: the policy is either implemented or not. For example, a company deciding on a remote work policy must choose to either allow remote work or not. The decision process involves evaluating benefits such as increased employee satisfaction against potential drawbacks like reduced team cohesion, leading to a clear yes or no decision.

Interesting facts

• The Law of Excluded Middle is one of the three classic laws of thought.

• It is crucial for the binary system that underpins computer operations.

• Some alternative logics, like intuitionistic logic, do not accept the law.

• The law is often used in proofs by contradiction in mathematics.

• It has been a subject of philosophical debate regarding its applicability to real-world scenarios.