Is x ∨ y ∨ z a tautology? Clearly not. Setting the three propositions each to false, the formula is false. But now consider: Is A − has − 0 ∨ A − has − 1 ∨ A − has − 2 a tautology? The answer here is " yes of course, ... well, as long we're interpreting those propositions to refer to a WaterWorld board. " We'll capture this notion by listing a bunch of domain axioms for WaterWorld: formulas which are true for all WaterWorld boards.
There are a myriad of domain axioms which express the rules of WaterWorld. Here are a few of them:
- A − has − 0 ⇒ B − safe ∧ G − safe
- A − has − 2 ⇒ B − unsafe ∧ G − unsafe
A more complete list is here (Section 6.5.2: The domain axioms). Whenever we deal with WaterWorld, we implicitly take all these domain axioms as given.