The correct conclusion to draw from the incompatibility of the two "hypotheticals" is that what is hypothesized in them (that Allen is out) must be false under our assumption that Carr is out.
The paradox arose out of a disagreement between Carroll and his Oxford colleague, Wykeham Professor of Logic John Cook Wilson, the two of whom had a long-running antagonism.
The problem was also discussed by others with whom Carroll corresponded, and was addressed in later articles published by John Venn, Alfred Sidgwick and Bertrand Russell among others.
Cook Wilson's view is represented in the story by the character of Uncle Joe, who attempts to prove that Carr must always remain in the shop.
As Carroll noted, "I am in correspondence with about a dozen logicians on this curious point; & so far, opinions appear equally divided as to C's freedom".
Carroll presents this intuition-defying result as a paradox, hoping that the contemporary ambiguity would be resolved.
First let's break down one of the two offending conditionals: Substituting this into Which yields, with continued application of the law of implication, And finally, (on the right we are distributing over the parentheses) So the two statements which become true at once are: "One or more of Allen, Brown or Carr is in", which is simply Axiom 1, and "Carr is in or Allen is in or Brown is out".
He attempts to clarify the issue by arguing that the protasis and apodosis of the implication "If Carr is in ..." are "incorrectly divided".
However, application of the Law of Implication removes the "If ..." entirely (reducing to disjunctions), so no protasis and apodosis exist and no counter-argument is needed.