Logic is the study of the principles of valid demonstration and inference . Logic is a branch of philosophy, a part of the classical trivium, as well as a branch of mathematics. The word derives from Greek λογική (logike), fem. of λογικός (logikos), "possessed of reason, intellectual, dialectical, argumentative", from λόγος logos, "word, thought, idea, argument, account, reason, or principle".

Logic concerns the structure of statements and arguments, in formal systems of inference and natural language. Topics include validity, fallacies and paradoxes, reasoning using probability and arguments involving causality. Logic is also commonly used today in argumentation theory.

Nature of logic

The concept of logical form is central to logic; it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics.

• Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato are a good example of informal logic.
• Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The works of Aristotle contain the earliest known formal study of logic, which were incorporated in the late nineteenth century into modern formal logic. In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuance of natural language.)
• Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. For a more modern treatment, see A. G. Hamilton, Logic for Mathematicians, Cambridge, 1980, ISBN 0-521-29291-3 Symbolic logic is often divided into two branches, propositional logic and predicate logic.
• Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

Consistency, soundness, and completeness

Among the important properties that logical systems can have are:

• Consistency, which means that no theorem of the system contradicts another.
• Soundness, which means that the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.
• Completeness, which means that there are no true sentences in the system that cannot, at least in principle, be proved in the system.