Formal logic

any well-defined system of abstract thought based on the model of mathematics

study of propositions, statements, and deductive arguments; practice of deriving conclusions from premises

Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

formal systems of logic that significantly differ from standard logical systems