Question

You are given three boxes filled with chocolates. One box contains only dark chocolates, one box contains only milk chocolates, and one contains both dark and milk chocolates. However, the boxes are labeled incorrectly. You know that you have exactly one box of each, and you also know that each box is definitely labeled wrong. Given the incorrect labeling of the chocolate boxes and the three observations, use Propositional Logic to derive the correct labeling of the second box. You can begin by writing down a knowledge base including all facts you know. Use propositional symbols in the following form: O1M means a milk chocolate was drawn (observed) from box 1, L1M means box 1 was initially labeled Milk, and C1B means box 1 actually contains both types of chocolates.

a)Prove that box 2 contains milk chocolates (i.e. generate the sentence C2M), using natural deduction (i.e. using rules of inference)

b)Prove that box 2 contains milk chocolates via a Resolution Refutation proof. milk dark both

Answer 1

PART-1
i	O3D	Given, dark chocolate is observed from box 3
ii	C3D Ѵ C3B	The box 3 contains either dark chocolate or both chocolates
iii	C3B	The box 3 does not contain both chocolates, labels are wrong
iv	C3D	From sentence (ii) & (iii), box 3 need to include only dark chocolates
v	C1D ^ C2D	From sentence (v), conjunction property
vi	C1D	Given, dark chocolate is observed from box 1
vii	O1D	From sentence (vi), box 1 does not include only dark chocolate
viii	C1D Ѵ C1B	From sentence (vi) & (viii), box 1 include both chocolate
ix	C1B	From sentence (vi) & (viii), box 1 include both chocolate
x	O2M	Given, milk chocolate is observed from box 2
xi	C2M Ѵ C2B	From sentence (x), box 2 contain either milk chocolate or both
xii	C2B ^ C3B	From sentence (ix), neither box 2 nor box 3 includes both chocolate
xiii	C2B	From sentence (xii), conjunction property
xiv	C2M	From sentence (xi) & (xiii), box 2 include only milk chocolates
PART-2
i	C2M	The negotiation of the statement
ii	C1M ^ C2M	From sentence (i), if milk chocolate is not in box 2, then it will
	C1M Ѵ C3M	either be in box 1 or box 3
iii	C1M	The box 1 does not includes only milk chocolates, labels are wrong
iv	C3M	From sentence (ii) and (iii), box 3 includes only milk chocolates
v	O3D	Given, dark chocolate is observed from box 3
vi	C3D Ѵ C3B	Box 3 contains either dark chocolate or both chocolates
vii	C3B	Box 3 does not includes both chocolates, labels are wrong
viii	C3D	From sentence (vi) and (vii), box 3 need to include only dark chocolates