Formal system

A formal system consists of symbols, axioms, and rules for deriving theorems. Its derivations depend on stated rules rather than informal interpretation.

Formal language: A set of strings formed from an alphabet according to specified syntactic rules. Its alphabet and formation rules define the expressions a system can manipulate.
Derivation: A sequence of formulas in which each formula follows from axioms or earlier formulas by inference rules. It is the explicit process by which a formal system yields a theorem.
Consistency: The property of a theory that it does not derive a contradiction. A system deriving contradictions can undermine the reliability of its theorems.
Informal logic: The study of reasoning in ordinary language, including argument structure and informal fallacies. It evaluates arguments without requiring every step to be encoded in a formal calculus.
Axiom: A statement accepted as a starting point in a deductive system without proof within that system. Axioms supply the initial premises from which theorems are derived.
Proof calculus: A formal collection of inference rules for constructing proofs in a logical language. A calculus specifies the permitted moves within many formal systems.
Completeness (logic): The property that every semantically valid sentence is derivable in a given proof system. It measures whether the system's rules capture all truths in the intended semantics.
Intuitionistic logic: A logic that treats proof as central and rejects unrestricted use of the law of excluded middle. It changes which inferences and theorems are accepted compared with classical systems.
Inference rule: A rule specifying which conclusions may be drawn from given premises. Inference rules license each step in a formal derivation.
Sequent calculus: A proof system that represents entailment with sequents and derives them by inference rules. It offers a structured alternative for presenting derivations and proof structure.

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