Don't Ask Alice - Part the Rhyme

This is the ninth installment on Dodgson's unsolved sorites - for the previous ones go here:
Don't ask Alice, I don't think she'll know [from Dodgson's Symbolic Logic, p. 186]
Don't ask Alice - Part the Second [from Dodgson's Symbolic Logic, p. 187]
Don't ask Alice - Part the Third [from Dodgson's Symbolic Logic, p. 188]
Don't ask Alice - Part the Fourth [from Dodgson's Symbolic Logic, p. 190]
Don't ask Alice - Part the Fifth [from Dodgson's Symbolic Logic, p. 191]
Don't ask Alice - Part the Sixth [from Dodgson's Symbolic Logic, p. 192]
Don't ask Alice - Part the Seventh [from Dodgson's Symbolic Logic, p. 193]
Don't ask Alice - Part the Eighth [from Dodgson's Symbolic Logic, p. 194
The ninth problem is not one of his compositions, but rather the nursery rhyme:

"Jack Sprat could eat no fat:
His wife could eat no lean:
And so, between them both,
They licked the platter clean."

Solve this as a Sorites-Problem, taking lines 3 and 4 as the Conclusion to be proved. It is permitted to use, as Premisses, not only all that is here asserted, but also all that we may reasonably understand to be implied.

It is difficult to understand what Dodgson had in mind as a sorites proof of this verse. Begin by assuming that the universe is types of food, and define the following classes : J – the food that Jack Sprat eats, W – the food his wife eats, F – foods which are fatty, and L – foods that are lean. It may reasonably be assumed that the classes F and L are complementary, i.e., F = L^~ and vice versa. The premises are interpreted as: firstly, that "None of the food Jack Sprat eats is fatty' or ρ(J^~F^~), and secondly, that 'None of the food his wife eats is lean' or ρ(W^~L^~) = ρ(W^~F). Writing the conjunction of these premises and using the theorem to find the conclusion :
ρ(J^~F^~) · ρ(W^~F) ⇒ ρ(J^~W^~)
But this conclusion is only that Jack ate nothing his wife ate and vice-versa - the two classes are disjoint. The conclusion 'They licked the platter clean' is not valid, since it is possible they did not eat all the food. The purported conclusion follows only on the additional assumptions that Jack ate all the lean and his wife ate all the fat, or that together they ate all the food, in either case begging the question.

Perhaps Dodgson intended this exercise to show that the assumptions did not support the purported conclusion, or perhaps he was simply mistaken - we'll never know.