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18 March, 2015 - 11:20
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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.