Introduction to Logic

Logic is the science of correct thinking and reasoning. It helps determine whether an argument is valid, a proof is correct, or a statement is true or false. It is the foundation of almost every area of mathematics and computer science.

What is a Statement?

In logic, a statement (proposition) is any declaration about which it can be unambiguously determined whether it is true or false.

• “2+2=4” → true statement
• “The Sun is blue” → false statement
• “Open the window!” → not a statement, because it has no truth value

Truth Value

We assign a truth value to every statement: TRUE (1) or FALSE (0). This is the basis of two-valued logic, developed by George Boole in the 19th century. With truth values, we can examine more complex statements.

The symbol p denotes a statement, ∈ expresses 'is an element of', and {0,1} is the set of truth values (0 = false, 1 = true). Thus, p ∈ {0,1} means that every statement's truth value can only be one of two kinds: true or false.

Why is Logic Important?

• In mathematics: every proof rests on logical foundations.
• In computer science: conditional instructions in programs are built on logical expressions.
• In everyday life: our decisions often rely on inferences.

Simple Examples

Consider the statement: 'If it rains, the road will be wet.' This is a logical connection whose validity we can examine based on truth values. With the tools of logic, we can clearly describe the consequences.

Summary

The basics of logic are statements and their truth values. These form the foundation for all further topics: logical operations, truth tables, proof methods, predicate logic, and quantifiers. This introductory chapter shows why logical thinking is indispensable in mathematics and computer science.

Practice Exercise