HOMEWORK #8: F.O.L. vs. MENTAL MODELS

1. Formalize the following sentence in FOL:
   A triangle is on the right of a circle that is above a square that is to the left of a dot.

   and prove, using resolution, that:

   The triangle is above the dot.

   Use the following predicates, with the following semantics:

   [[Circle(x)]] = [[x]] is a circle.
   [[Dot(x)]] = [[x]] is a dot.
   [[Square(x)]] = [[x]] is a square.
   [[Triangle(x)]] = [[x]] is a triangle.
   [[Above(x,y)]] = [[x]] is above [[y]]
   
   i.e.:

   [[x]]

   [[y]]

   
   
   [[Below(x,y)]] = [[x]] is below [[y]]
   
   i.e.:

   [[y]]

   [[x]]

   
   
   [[Left(x,y)]] = [[x]] is to the left of [[y]] (i.e.: [[x]] [[y]])
   [[Right(x,y)]] = [[x]] is to the right of [[y]] (i.e.: [[y]] [[x]])
   

   If you need to use any other predicates, be sure to give their syntax and semantics. You will need to devise an appropriate set of axioms that relate these spatial relations to each other (e.g., if x is above y, then y is below x; etc.). However, you only need to state those axioms that you use in your proof, not all possible ones.

2. Show, by translating into propositional logic (using the syntax and semantics given below), and by means of a resolution proof, that the following inference is valid:
   If there is a Jack in the hand, then there is a King in the hand, or else if there's no Jack in the hand, then there's a King in the hand.

   There is a Jack in the hand.

   Therefore, there is no King in the hand.

   [[J]] = There is a Jack in the hand.
   [[K]] = There is a King in the hand.
   Note: "or else" is to be interpreted as exclusive "or" ("xor"),
   where (P xor Q) <-> ((P v Q) ^ -(P ^ Q)).

3. Using truth tables, show why it would be incorrect to infer from the previous 2 premises that there is a King in the hand.