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


in logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). for example, the rule of inference called modus ponens takes two premises, one in the form "if p then q" and another in the form "p", and returns the conclusion "q". the rule is valid with respect to the semantics of classical logic (as well as the semantics of many other nonclassical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion.
typically, a rule of inference preserves truth, a semantic property. in manyvalued logic, it preserves a general designation. but a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. usually only rules that are recursive are important; i.e. rules such that there is an effective procedure for determining whether any given formula is the conclusion of a given set of formulae according to the rule. an example of a rule that is not effective in this sense is the infinitary ωrule.^{[1]}
popular rules of inference in propositional logic include modus ponens, modus tollens, and contraposition. firstorder predicate logic uses rules of inference to deal with logical quantifiers.