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Axiomatic semantics are semantic expressions of the relationships inherent in a piece of code. These expressions can be helpful in describing how some piece of software works.
An interesting thing about axiomatic semantics as contrasted to other types of expressions is that they are fairly agnostic of specific results and conditions. Rather, axiomatic semantics describe the way that a system works. One way to think of this is using the root word, axiom, which implies some broader truism about a system. For example, an axiomatic semantical statement about a certain function would describe what it is meant to do, what sort of argument it takes, and what sort of result it returns. This would not require knowledge of the actual variables involved.