Rule of replacement

In logic, a rule of replacement[1][2][3] is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions.

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

Common rules of replacement include de Morgan's laws, commutation, association, distribution, double negation,[lower-alpha 1] transposition, material implication, material equivalence, exportation, and tautology.

Notes

not admitted in intuitionistic logic

References

Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.
Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.
Moore and Parker

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