Propositional logic

Logic helps us understand how ideas connect. One important part is called propositional logic. It is also known as statement logic or sentential calculus. This branch of math deals with propositions. A proposition is a simple statement that is either true or false.

You might say, "London is the capital of England." That is a true statement. You might say, "The sky is green." That is a false statement. Propositional logic looks at how these statements relate to one another. It does not look at the tiny details inside a single sentence. Instead, it focuses on the whole idea at once.

We can build bigger ideas by joining simple ones together. We use special words called logical connectives to do this. Some common connectives are "and," "or," and "not." There are also "if" and "if and only if."

When we join statements, we create compound sentences. For example, you could say, "It is raining, and it is cloudy." This new sentence is a compound sentence. We can also use letters to represent these statements. These letters are called propositional variables. Using letters makes it easier to study the patterns of the logic. This turns a long sentence into a short formula.

People have studied these patterns for a very long time. A man named Chrysippus is often credited with building this system. He lived during the 3rd century BC. His work was later expanded by a group called the Stoics. Most of their original writings were lost over time. The system faded away between the 3rd and 6th century CE. It was only found again in the 20th century. Later, a mathematician named Gottfried Leibniz worked on symbolic logic. He lived in the 17th and 18th centuries. His ideas helped lead to the work of George Boole and Augustus De Morgan.

Many thinkers helped create the tools we use today. Truth tables are a famous way to show how logic works. We are not quite sure who invented them first. Some people think Ludwig Wittgenstein or Emil Post created the table shape. Other names like Bertrand Russell and Gottlob Frege were very important too.

Other tools include truth trees, which were invented by Evert Willem Beth. There is also a method called natural deduction. Gerhard Gentzen and Stanisław Jaśkowski invented that system. Each of these tools helps us check if an idea is correct.

Logic is often used to build arguments. An argument has two parts: premises and a conclusion. The premises are the facts you start with. The conclusion is the idea that follows from them.

An argument is valid if the conclusion must be true when the premises are true. It is impossible for the premises to be true and the conclusion to be false. If the argument is valid and the facts are actually true, we call it sound. This is a way to make sure our thinking is steady. It helps us see if one idea truly leads to another.