We introduce  a class of restrictions for the
ordered paramodulation and superposition calculi 
(inspired by the basic strategy for narrowing), in 
which paramodulation inferences are forbidden at 
terms introduced by substitutions from previous 
inference steps.  In addition we introduce 
restrictions based on term selection rules and redex 
orderings, which are general criteria for delimiting
the terms which are available for inferences.  These 
refinements are compatible with standard ordering 
restrictions and are complete without paramodulation
into variables or using functional reflexivity axioms. 
We prove refutational completeness in the context of
deletion rules, such as simplification by rewriting 
(demodulation) and subsumption, and of techniques for 
eliminating redundant inferences.