Logic
Logic studies valid reasoning, using precise rules to analyze arguments, statements, proofs, truth, and consequence.
What logic studies
Logic is the study of good reasoning. It separates the structure of an argument from the topic being argued about, then asks whether the conclusion must follow if the premises are true. This makes logic useful in both everyday critical thinking and highly formal mathematical proof.
Arguments and validity
An argument has premises and a conclusion. A valid argument is one whose conclusion cannot be false when all its premises are true. Validity does not say that the premises are actually true; it says the reasoning pattern preserves truth. Soundness adds the extra requirement that the premises are true.
Propositional logic
Propositional logic treats whole statements as units that can be true or false. Connectives such as and, or, not, if-then, and if-and-only-if build compound statements. Truth tables show how the truth of a compound statement depends on the truth values of its parts.
Predicate logic
Predicate logic looks inside statements by using predicates, variables, and quantifiers. It can express claims such as every number has a property or some object satisfies a condition. This extra structure makes it much more powerful than propositional logic for mathematics, language, and formal reasoning.
Proof systems
A proof system gives formal rules for deriving conclusions from assumptions. Natural deduction, sequent calculi, axiomatic systems, and resolution methods organize proof in different ways. The point is not to make reasoning mechanical for its own sake, but to make each step checkable.
Logic and computation
Computers use logic at many levels. Boolean logic shapes digital circuits, formal languages describe programs, type systems restrict what programs can do, and automated theorem provers search for proofs. In artificial intelligence, logic has long been used to represent knowledge and reason from explicit rules.
Limits of formal reasoning
Formal logic is powerful, but it does not make every question simple. Some systems are incomplete, undecidable, or too complex for practical proof search. Informal reasoning also involves context, relevance, ambiguity, and judgment. Logic clarifies these issues rather than eliminating them.
Why it matters
Logic matters because mistakes in reasoning can survive even when facts are correct. It gives people tools to test arguments, build proofs, design reliable systems, and understand the boundary between evidence and conclusion. In a world full of claims, logic is a way to ask what actually follows.