Paradoxes appear in all shapes and forms. Certain are uncomplicated paradoxes of reasoning with minimal potential for investigation, while others sit atop icebergs of full scale scientific disciplines. Many may be solved by mindful consideration of their hidden assumptions, one or more of them could be faulty. These, strictly stating, really should not be referred to as paradoxes at all, because as soon as a puzzle is solved it stops to be a paradox.
“One should not think slightingly of the paradoxical; for the paradox is the source of the thinker’s passion, and the thinker without a paradox is like a lover without feeling: a paltry mediocrity.” – Søren Kierkegaard.
A real paradox is a statement that results in a circular and self-contradictory argument, or refers to something which is logically not possible. However the term “paradox” does tend to be utilized more broadly to incorporate what I choose to call “perceived paradoxes.” For this kind of puzzles there is a way out. It might be that the paradox has hidden within it a key or sleight of hand that intentionally misleads the listener or reader. As soon as the trick is exposed, the contradiction or logical absurdity vanishes. Another kind of perceived paradox is certainly one wherein the statement and the conclusions, while at first appearing absurd or at the minimum counterintuitive, turn out on more careful consideration not to be so, even though the outcome stays somewhat shocking.
Like in Gary Allen’s book “None Dare Call it Conspiracy” – http://amzn.to/2wGRP11 where he describes the experience as trying to discover the “hidden picture” of a donkey, a cart or a boy within another picture in a children’s magazine. The picture painters of the mass media are artfully creating landscapes for us which deliberately hide the real picture. Once you know how to discover the “hidden picture” in the landscapes presented to us daily through newspapers, radio, television and once you can see through the camouflage, you will see the donkey, the cart and the boy who have been there all along …
Take the simple assertion: “This statement is a lie.” On first reading, I believe the words by themselves will probably appear straightforward enough. Consider their meaning, however, and the logical paradox may become obvious as you work cautiously through the statement’s implications. You see, “This statement is a lie” is telling you that in announcing itself to be a lie it must itself be a lie, and so it is not a lie—in which case it is true, which is to say that it really is a lie, which means it is not a lie, and so on in an infinite loop.
In contrast, a falsidical paradox, begins perfectly sensibly, yet somehow ends in a ridiculous outcome. Nevertheless, in this case the apparently absurd end result is definitely bogus, because of some subtly misleading or erroneous step in the proof. Examples of falsidical paradoxes are the mathematical tricks that, by following a few steps of algebra, “prove” something like 2 = 1. Suffice to stipulate that the computation causing the “solution” often involves a step in which an amount is divided by zero—something any self-respecting mathematician understands to avoid at all costs.
Can you puzzle the next falsidical paradox out?
Three tourists check into a hotel for the night.
The youngster at the reception desk charges them 30€ for a room with 3 beds in it. They accept to cut the cost of the room equally, every one of them giving 10€. They take the key and head up to the room to settle in.
After a couple of minutes the receptionist realizes he has made a fault. The hotel has a special offer on all week and he should only have charged them 25€ for the room. In order not to get into trouble with his supervisor, he rapidly takes a bill of 5€ from the till and rushes up to correct his mistake. Toward the room he realizes that he simply cannot split the 5€ equally between the three men, therefore he chooses to give each of them one Euro and keep 2 for himself. Doing this, he argues, everybody is happy.
Here, then, is the problem we are left with: every one of the three buddies may have contributed 9€ toward the room. That makes 27€ that the hotel has made, and the receptionist has a further 2€, which makes 29€. What has happened to the last Euro out of the original 30€?You may well be able to see the solution to this immediately; I definitely didn’t when I first heard it. Think about it a little before you read on to the explanation below.
Have you worked it out? The thing is, this puzzle only may seem paradoxical due to the deceiving method it is stated. The error in the reasoning is the fact the story added the 27€ to the 2€ taken by the receptionist, and there is absolutely no reason to do that, because there is simply no longer a full 30€ that must be accounted for. The receptionist’s 2€ should be subtracted from the 27€ paid by the friends, leaving 25€, which is the amount in the till.