Notion of Safeness of Inference Rules

Safeness is a name for property of logical inference rules. It is in use now for more than a dozen of years at least. It is not yet an established standard such as soundness and completeness of a set of inference rules. Roughly speaking, a rule is safe if its inverse rule is sound. More precisely, a rule is sound iff the validity of its preconditions implies the validity of its conclusion; a rule is safe iff the invalidity of one of its preconditions implies the invalidity of its conclusion. For reductive inference systems, soundness means that a goal is solved if we can solve the sub-goals it is reduced to. Safeness means that if one of the sub-goals is unsolvable, the goal is unsolvable, too. Safeness is useful for disproving theorems. If we disprove a sub-goal generated from a goal or from one of its transitive sub-goals, then, if all reduction steps are safe, we know that the original goal was invalid already. Lecture Notes on Logic, incl. Safeness
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Last modified 2005/05/02/11.15