There are a myriad of propositions for WaterWorld, which can be grouped:
- Whether or not a location contains a pirate: A − unsafe, B − unsafe, ..., Z − unsafe.
- Whether or not a location contains no pirate: A − safe, B − safe, ..., Z − safe.
ASIDE: Yes, using the intended interpretation, these are redundant with the previous ones. Some domain axioms below will formalize this.
- Propositions indicating the number of neighboring pirates, to a location: A − has − 0, A − has − 1, A − has − 2, B − has − 0, B − has − 1, B − has − 2, ..., H − has − 0, H − has − 1, H − has − 2, H − has − 3, ..., Z − has − 0, Z − has − 1. These are all truejfalse propositions; there are no explicit numbers in the logic. A domain axiom below will assert that whenever (say) B − has − 1 is true, then B − has − 0 and B − has − 2 are both false.
ASIDE: There is no proposition A − has − 3 since location A has only two neighbors. Similarly, there is no proposition B − has − 3. We could have chosen to include those, but under the intended interpretation they'd always be false.
These propositions describe the state of the underlying board -the model -and not our particular view of it. Our particular view will be reflected in which formulas we'll accept as premises. So we'll accept A − has − 2 as a premise only when A has been exposed and shows a 2.
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