Deductive system

A deductive system is a formal set of rules that specifies how conclusions may be derived from premises. Its derivations make explicit which conclusions follow under its chosen rules.

Formal language: A formally specified set of symbols and rules for forming expressions. Its syntax supplies the formulas that a deductive system manipulates.
Hilbert system: A proof system built from axiom schemas and a small number of inference rules. It exemplifies derivations made from axioms and repeated rule applications.
Soundness: A property ensuring that every conclusion derivable in a system is semantically valid under its intended interpretation. Soundness guarantees that the system's rules do not derive false consequences from valid premises.
Semantic consequence: A relation in which premises entail a conclusion because every interpretation satisfying them also satisfies it. It defines consequence by truth across interpretations rather than by formal derivation.
Axiom: A sentence accepted as a starting point within a formal theory or deductive system. Axioms can begin derivations without being derived within the system.
Natural deduction: A proof system whose rules model the introduction and elimination of logical connectives. Its rules organize proofs around assumptions and the logical operators in conclusions.
Completeness: A property ensuring that every semantically valid conclusion is derivable in a formal system. Completeness measures whether the system can derive all consequences its semantics recognizes.
Inductive reasoning: Reasoning that supports generalizations or predictions from observed cases without guaranteeing their truth. Unlike deduction, its conclusions can be informative without following necessarily from premises.
Inference rule: A rule specifying which conclusions may be drawn from given premises. Inference rules are the system's basic steps for extending a derivation.
Sequent calculus: A proof system that derives sequents, expressions of relationships between sets of assumptions and conclusions. It makes structural properties of derivations explicit through sequent rules.

Linked from 7 pages