This is a homework for first order logic. The task itself is easy. Only the third point on the list “prove by contradiction” is a bit more difficult to interpret, as the term prove is not defined exactly in a mathematical context.
In the domain of game theory, to prove something means to find a walk-through for the game. If something was proven, a plan was found to bring the system from the initial state into the goal state. In the case of propositional logic the possible numbers of plans is low, so in most cases it is easy to find a plan. In a real life problem, finding the plan is equal to search in the state space.
At first, the sentences have to converted into boolean expressions:
- A: Owner walks with dog
- B: Dog is happy
- C: Dog chews up the furniture
- D: Dog mess up the apartment
- E: Owner is annoyed
Then the rules of the game have to be formalized:
- if A then B
- if not A then C
- if (B or C) then D
- if D then E
The rules are equal to the game-mechanics, that means the game engine calculates the future state for a given input. As mentioned above, to prove is equal to finding a plan. That means we have a goal state ("Dog made a mess"), and now it is possible to search with backtracking for a plan which will produce “D=true”.