Three barbers and the impossible if-then
Three barbers, Allen, Brown, and Carr, share a shop that always has at least one of them minding it. Allen is so nervous he never leaves without Brown coming too. A logician tells you: "If Carr is out, then if Allen is out, Brown is in." From these ordinary-sounding rules, an airtight-looking argument concludes Carr can never step outside. Something in the chain of if-then reasoning has gone wrong, but what?
Reveal the answer
The conclusion is a logical illusion, not a fact about the barbers. Carroll's argument invalidly treats a nested conditional the same way as a flat one; Bertrand Russell's 1903 analysis in The Principles of Mathematics gave the standard resolution. Carroll devised the puzzle to expose a genuine gap in Victorian symbolic logic's theory of hypothetical propositions.