Dynamic Logic (modal Logic) - Derived Rules of Inference

Derived Rules of Inference

As for modal logic, the inference rules modus ponens and necessitation suffice also for dynamic logic as the only primitive rules it needs, as noted above. However, as usual in logic, many more rules can be derived from these with the help of the axioms. An example instance of such a derived rule in dynamic logic is that if kicking a broken TV once can't possibly fix it, then repeatedly kicking it can't possibly fix it either. Writing for the action of kicking the TV, and for the proposition that the TV is broken, dynamic logic expresses this inference as, having as premise and as conclusion . The meaning of is that it is guaranteed that after kicking the TV, it is broken. Hence the premise means that if the TV is broken, then after kicking it once it will still be broken. denotes the action of kicking the TV zero or more times. Hence the conclusion means that if the TV is broken, then after kicking it zero or more times it will still be broken. For if not, then after the second-to-last kick the TV would be in a state where kicking it once more would fix it, which the premise claims can never happen under any circumstances.

The inference is sound. However the implication is not valid because we can easily find situations in which holds but does not. In any such counterexample situation, must hold but must be false, while however must be true. But this could happen in any situation where the TV is broken but can be revived with two kicks. The implication fails (is not valid) because it only requires that hold now, whereas the inference succeeds (is sound) because it requires that hold in all situations, not just the present one.

An example of a valid implication is the proposition . This says that if is greater or equal to 3, then after incrementing, must be greater or equal to 4. In the case of deterministic actions that are guaranteed to terminate, such as, must and might have the same force, that is, and have the same meaning. Hence the above proposition is equivalent to asserting that if is greater or equal to 3 then after performing, might be greater or equal to 4.