Propositional logic

Some sentences are true or false.

These are called statements. You can join them with special words. You can use the word "and." You can use the word "or." You can even use "not." These words help make longer sentences. These long sentences are called compound sentences. They help us build arguments. An argument uses facts to reach a conclusion. This helps us find the truth.

Logic is a way to solve puzzles.

Some sentences are either true or false. We call these statements.

You can join these statements together. We use special words called connectives to do this. For example, you can use "and" or "or." You can also use "not" to change a statement. These new, longer sentences are called compound sentences.

You can use these sentences to build an argument. An argument has two parts. The first part is a set of facts called premises. The second part is a conclusion. The conclusion is supposed to follow from the premises.

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 premises are actually true, we call it sound. People use letters to stand for statements in math. These letters are called propositional variables. This helps us study the logic without using long sentences. This way of thinking is very old. A man named Chrysippus helped build this system long ago. He lived in the 3rd century BC.

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.

Propositional logic is a fundamental branch of classical logic. It is also known by several other names, such as statement logic, sentential calculus, or propositional calculus. Some scholars refer to it as zeroth-order logic. This field studies propositions and the relationships between them. A proposition is a declarative sentence that carries a truth value. This means the statement must be either true or false.

For example, saying "London is the capital of England" is a proposition because it is true. However, asking "What is Wikipedia?" is not a proposition. Questions and commands, known as imperative statements, do not have truth values. Therefore, they fall outside the scope of standard propositional logic.

At its core, the mechanism of this logic involves combining simple statements into complex ones. Simple statements are called atomic formulas or atomic sentences. We build compound sentences, also called molecular sentences, by using logical connectives. These connectives act as truth-functors that determine the truth of the new sentence. The primary types of connectives include negation, conjunction, disjunction, implication, and biconditional. In English, these correspond to "not," "and," "or," "if," and "if and only if."

When we use these tools, we create a formal language. In this language, we represent propositions with letters called propositional variables. This allows mathematicians to study the structure of an argument without getting distracted by the specific meaning of the words.

There are several distinct ways to categorize the components of this system. First, we distinguish between atomic and compound propositions. Atomic propositions are the indivisible building blocks of the logic. Compound propositions are formed when connectives join these blocks together. Second, we distinguish between different types of logical consequences. We use the term semantic consequence to describe when a conclusion follows from premises based on truth values. We use syntactic consequence to describe when a conclusion is derived through formal rules of a system.

This distinction is vital for understanding how formal proofs actually function.

The history of propositional logic spans thousands of years. While earlier philosophers hinted at these ideas, Chrysippus is often credited with developing a formal deductive system in the 3rd century BC. His work was expanded by the Stoics, but most of their writings were lost. Between the 3rd and 6th century CE, this logic faded into oblivion. It was eventually resurrected in the 20th century. During the 17th and 18th centuries, Gottfried Leibniz developed symbolic logic through his calculus ratiocinator. Although his work was not widely known then, later logicians like George Boole and Augustus De Morgan independently recreated many of his advances.

Significant mathematical tools have emerged to help us analyze these logical structures. Truth tables are a common method used to evaluate the truth of compound sentences. While the exact inventor is uncertain, the tabular structure is often credited to Ludwig Wittgenstein or Emil Post. Other thinkers like Bertrand Russell and Gottlob Frege provided ideas that led to their use. Other specialized tools include truth trees, invented by Evert Willem Beth, and natural deduction, developed by Gerhard Gentzen and Stanisław Jaśkowski.

Each of these methods provides a different way to verify if a logical statement holds up under scrutiny.

We can use propositional logic to evaluate the strength of arguments. An argument consists of a set of premises and a conclusion. The premises are intended to support the conclusion. We call an argument valid if it is impossible for the premises to be true while the conclusion is false. In a valid argument, the conclusion is a necessary logical consequence of the premises. However, validity is not the same as soundness. An argument is only sound if it is valid and all of its premises are actually true in reality. This distinction helps us separate the structure of an idea from the facts of the world.

Propositional logic serves as a vital foundation for more complex mathematical systems. It is included in the machinery of first-order logic and higher-order logics. While propositional logic does not deal with non-logical objects or quantifiers, it provides the basic rules that those higher systems rely on. Gottlob Frege's predicate logic, for instance, builds directly upon these principles. By combining the features of syllogistic logic and propositional logic, Frege ushered in a new era of logical study. Today, the study of these connections remains essential for mathematics, computer science, and philosophy.