A paradox is a statement or problem that either appears to produce
two entirely contradictory (yet possible) outcomes, or provides proof
for something that goes against what we intuitively expect. Paradoxes
have been a central part of philosophical thinking for centuries, and
are always ready to challenge our interpretation of otherwise simple
situations, turning what we might think to be true on its head and
presenting us with provably plausible situations that are in fact just
as provably impossible. Confused? You should be.
1. ACHILLES AND THE TORTOISE
The Paradox of Achilles and the Tortoise is one of a number of
theoretical discussions of movement put forward by the Greek philosopher
Zeno of Elea in the 5th century BC. It begins with the great hero
Achilles challenging a tortoise to a footrace. To keep things fair, he
agrees to give the tortoise a head start of, say, 500m. When the race
begins, Achilles unsurprisingly starts running at a speed much faster
than the tortoise, so that by the time he has reached the 500m mark, the
tortoise has only walked 50m further than him. But by the time Achilles
has reached the 550m mark, the tortoise has walked another 5m. And by
the time he has reached the 555m mark, the tortoise has walked another
0.5m, then 0.25m, then 0.125m, and so on. This process continues again
and again over an infinite series of smaller and smaller distances, with
the tortoise
always moving forwards while Achilles
always plays catch up.
Logically, this seems to prove that Achilles can never overtake the
tortoise—whenever he reaches somewhere the tortoise has been, he will
always have some distance still left to go no matter how small it might
be. Except, of course, we know intuitively that he
can overtake
the tortoise. The trick here is not to think of Zeno’s Achilles Paradox
in terms of distances and races, but rather as an example of how any
finite value can always be divided an infinite number of times, no
matter how small its divisions might become.
2. THE BOOTSTRAP PARADOX
The Bootstrap Paradox is a paradox of time travel that questions how
something that is taken from the future and placed in the past could
ever come into being in the first place. It’s a common trope used by
science fiction writers and has inspired plotlines in everything from
Doctor Who to the
Bill and Ted
movies, but one of the most memorable and straightforward examples—by
Professor David Toomey of the University of Massachusetts and used in
his book
The New Time Travellers—involves an author and his manuscript.
Imagine that a time traveller buys a copy of
Hamlet from a
bookstore, travels back in time to Elizabethan London, and hands the
book to Shakespeare, who then copies it out and claims it as his own
work. Over the centuries that follow,
Hamlet is reprinted and
reproduced countless times until finally a copy of it ends up back in
the same original bookstore, where the time traveller finds it, buys it,
and takes it back to Shakespeare. Who, then, wrote
Hamlet?
3. THE BOY OR GIRL PARADOX
Imagine that a family has two children, one of whom we know to be a
boy. What then is the probability that the other child is a boy? The
obvious answer is to say that the probability is 1/2—after all, the
other child can only be
either a boy
or a girl, and the chances of a baby being born a boy or a girl are (
essentially)
equal. In a two-child family, however, there are actually four possible
combinations of children: two boys (MM), two girls (FF), an older boy
and a younger girl (MF), and an older girl and a younger boy (FM). We
already know that one of the children is a boy, meaning we can eliminate
the combination FF, but that leaves us with three equally possible
combinations of children in which
at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child
is a boy—MM—must be 1/3, not 1/2.
4. THE CARD PARADOX
Imagine you’re holding a postcard in your hand, on one side of which
is written, “The statement on the other side of this card is true.”
We’ll call that Statement A. Turn the card over, and the opposite side
reads, “The statement on the other side of this card is false”
(Statement B). Trying to assign any truth to either Statement A or B,
however, leads to a paradox: if A is true then B must be as well, but
for B to be true, A has to be false. Oppositely, if A is false then B
must be false too, which must ultimately make A true.
Invented by the British logician Philip Jourdain in the early 1900s,
the Card Paradox is a simple variation of what is known as a “liar
paradox,” in which assigning truth values to statements that purport to
be either true or false produces a contradiction. An
even more complicated variation of a liar paradox is the next entry on our list.
5. THE CROCODILE PARADOX
A crocodile snatches a young boy from a riverbank. His mother pleads
with the crocodile to return him, to which the crocodile replies that he
will only return the boy safely if the mother can guess correctly
whether or not he will indeed return the boy. There is no problem if the
mother guesses that the crocodile
will return him—if she is right, he is returned; if she is wrong, the crocodile keeps him. If she answers that the crocodile will
not
return him, however, we end up with a paradox: if she is right and the
crocodile never intended to return her child, then the crocodile has to
return him, but in doing so breaks his word and contradicts the mother’s
answer. On the other hand, if she is wrong and the crocodile actually
did intend to return the boy, the crocodile must then keep him even
though he intended not to, thereby also breaking his word.
The Crocodile Paradox is such an ancient and enduring logic problem
that in the Middle Ages the word "crocodilite" came to be used to refer
to any similarly brain-twisting dilemma where you admit something that
is later used against you, while "crocodility" is an equally ancient
word for captious or fallacious reasoning
6. THE DICHOTOMY PARADOX
Imagine that you’re about to set off walking down a street. To reach
the other end, you’d first have to walk half way there. And to walk half
way there, you’d first have to walk a quarter of the way there. And to
walk a quarter of the way there, you’d first have to walk an eighth of
the way there. And before that a sixteenth of the way there, and then a
thirty-second of the way there, a sixty-fourth of the way there, and so
on.
Ultimately, in order to perform even the simplest of tasks like
walking down a street, you’d have to perform an infinite number of
smaller tasks—something that, by definition, is utterly impossible. Not
only that, but no matter how small the first part of the journey is said
to be, it can always be halved to create another task; the only way in
which it
cannot be halved would be to consider the first part
of the journey to be of absolutely no distance whatsoever, and in order
to complete the task of moving no distance whatsoever, you can’t even
start your journey in the first place.
7. THE FLETCHER’S PARADOX
Imagine a fletcher (i.e. an arrow-maker) has fired one of his arrows
into the air. For the arrow to be considered to be moving, it has to be
continually repositioning itself from the place where it is now to any
place where it currently isn’t. The Fletcher’s Paradox, however, states
that throughout its trajectory the arrow is actually not moving at all.
At any given instant of no real duration (in other words, a snapshot in
time) during its flight, the arrow cannot move to somewhere it isn’t
because there isn’t time for it to do so. And it can’t move to where it
is now, because it’s already there. So, for that instant in time, the
arrow must be stationary. But because all time is comprised entirely of
instants—in every one of which the arrow must also be stationary—then
the arrow must in fact be stationary the entire time. Except, of course,
it isn’t.
8. GALILEO’S PARADOX OF THE INFINITE
In his final written work,
Discourses and Mathematical Demonstrations Relating to Two New Sciences
(1638), the legendary Italian polymath Galileo Galilei proposed a
mathematical paradox based on the relationships between different sets
of numbers. On the one hand, he proposed, there are square numbers—like
1, 4, 9, 16, 25, 36, and so on. On the other, there are those numbers
that are
not squares—like 2, 3, 5, 6, 7, 8, 10, and so on. Put
these two groups together, and surely there have to be more numbers in
general than there are
just square numbers—or, to put it another way, the total number of square numbers must be less than the total number of square
and
non-square numbers together. However, because every positive number has
to have a corresponding square and every square number has to have a
positive number as its square root, there cannot possibly be more of one
than the other.
Confused? You’re not the only one. In his discussion of his paradox,
Galileo was left with no alternative than to conclude that numerical
concepts like
more,
less, or
fewer can only
be applied to finite sets of numbers, and as there are an infinite
number of square and non-square numbers, these concepts simply cannot be
used in this context.
9. THE POTATO PARADOX
Imagine that a farmer has a sack containing 100 lbs of potatoes. The
potatoes, he discovers, are comprised of 99% water and 1% solids, so he
leaves them in the heat of the sun for a day to let the amount of water
in them reduce to 98%. But when he returns to them the day after, he
finds his 100 lb sack now weighs just 50 lbs. How can this be true?
Well, if 99% of 100 lbs of potatoes is water then the water must weigh
99 lbs. The 1% of solids must ultimately weigh just 1 lb, giving a ratio
of solids to liquids of 1:99. But if the potatoes are allowed to
dehydrate to 98% water, the solids must now account for 2% of the
weight—a ratio of 2:98, or 1:49—even though the solids must still only
weigh 1lb. The water, ultimately, must now weigh 49lb, giving a total
weight of 50lbs despite just a 1% reduction in water content. Or must
it?
Although not a true paradox in the strictest sense, the
counterintuitive Potato Paradox is a famous example of what is known as a
veridical paradox, in which a basic theory is taken to a logical but
apparently absurd conclusion.
10. THE RAVEN PARADOX
Also known as Hempel’s Paradox, for the German logician who proposed
it in the mid-1940s, the Raven Paradox begins with the apparently
straightforward and entirely true statement that “all ravens are black.”
This is matched by a “logically contrapositive” (i.e. negative and
contradictory) statement that “everything that is
not black is
not
a raven”—which, despite seeming like a fairly unnecessary point to
make, is also true given that we know “all ravens are black.” Hempel
argues that whenever we see a black raven, this provides evidence to
support the first statement. But by extension, whenever we see anything
that is
not black, like an apple, this too must be taken as
evidence supporting the second statement—after all, an apple is not
black, and nor is it a raven.
The paradox here is that Hempel has apparently proved that seeing an
apple provides us with evidence, no matter how unrelated it may seem,
that ravens are black. It’s the equivalent of saying that you live in
New York is evidence that you don’t live in L.A., or that saying you are
30 years old is evidence that you are not 29. Just how much information
can one statement actually imply anyway?
Don't be fooled by stories about 19th-Century movie audiences 
