A template or schema that permits deriving a conclusion formula from one or more premise formulas within a deductive system; rules of inference specify allowable steps in formal proofs.

Principle

A rule of inference transforms premises into a conclusion while preserving a designated relation (typically truth or derivability); sound rules preserve truth from premises to conclusion under the intended semantics.

Demonstration

Modus ponens is a rule of inference: from 'P' and 'If P then Q' infer 'Q'. In propositional calculus, applying modus ponens to the formulas P and P→Q yields Q as a valid deductive step.

Misapplication

Applying a rule of inference outside its formal context (e.g., using modal rules in a purely classical derivation) or using an unsound rule (one that does not preserve truth) leads to invalid proofs or fallacious inferences.

Consequence

Rules of inference generate the structure of proofs and determine what counts as a derivation; together with axioms they define provability, enable theorem proving, and underlie metatheoretic properties like soundness and completeness.

Reversal

The reversal contrasts syntactic rules with semantic consequence: instead of deriving conclusions by rules, one can check whether conclusions are entailed by premises in all models (semantic entailment rather than syntactic derivation).

Boundary

A rule of inference is specified with respect to a chosen formal language and deductive calculus; it is not itself a semantic claim, and different systems may adopt different rules (natural deduction, sequent calculi, Hilbert systems).

Semantic Tension

There is tension between algorithmic proof search (rules as operational steps in automated deduction) and normative justification (rules as preserving truth); practical search strategies may favor different rule formulations than philosophical accounts of inference.

Synthesis

A rule of inference is the formal mechanism that, given premises, produces permissible conclusions according to the deductive apparatus; it operationalizes deduction, links axioms to theorems, and ensures that proofs respect the intended semantic relation.