The Free On-line Dictionary of Computing (30 December 2018):

axiomatic semantics

    A set of assertions about properties of a system and
   how they are effected by program execution.  The axiomatic
   semantics of a program could include pre- and post-conditions
   for operations.  In particular if you view the program as a
   state transformer (or collection of state transformers), the
   axiomatic semantics is a set of invariants on the state which
   the state transformer satisfies.

   E.g. for a function with the type:

   	sort_list :: [T] -> [T]

   we might give the precondition that the argument of the
   function is a list, and a postcondition that the return value
   is a list that is sorted.

   One interesting use of axiomatic semantics is to have a
   language that has a finitely computable sublanguage that is
   used for specifying pre and post conditions, and then have the
   compiler prove that the program will satisfy those conditions.

   See also operational semantics, denotational semantics.

   (1995-11-09)