Paradox and Logic

Check Out These Optical Illusions

Illusions And Paradoxes: Seeing Is Believing?

This page illustrates that our visual perception cannot always be trusted.

Optical Illusions

A Selection Of The Most Popular Optical illusions

Kids Page: Optical Illusions

What are "illusions"? Illusions trick us into perceiving something differently than it actually exists, so what we see does not correspond to physical reality.

Logic Paradoxes

Liar Paradox

The Liar Paradox

The Liar Paradox is among the simplest of paradoxes. It can be traced back at least as far as Eubulides of Miletus, a fourth-century B.C. Greek philosopher.

Eubulides' version of the paradox is this: A man says that he is lying; is what he says true or false?

Zeno's Paradox of the Tortoise and Achilles

Achilles and the Tortoise

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

The Barber Paradox

The Barber Paradox

The Barber paradox is attributed to the British philosopher Bertrand Russell. It highlights a fundamental problem in mathematics, exposing an inconsistency in the basic principles on which mathematics is founded.

The barber paradox asks us to consider the following situation:

In a village, the barber shaves everyone who does not shave himself, but no one else.

The question that prompts the paradox is this:

Who shaves the barber?

Heavy Rock Paradox Solved!

The Paradox of the Stone

God is all-powerful, or as theologians put it, "omnipotent"; there is nothing that he cannot do. This is part of the definition of "God".

So can God create a stone that is so heavy that he cannot lift it?

Ship of Theseus

Ship of Theseus

It seems like you can replace any component of a ship, and it will still be the same ship. So you can replace them all, or one at a time, and it will still be the same ship. But then you can take all the original pieces, and assemble them into a ship. That, too, is the same ship with which you started.

Newcomb's Paradox

Newcomb's Paradox

You and a friend are presented with two boxes. The first box is transparent and contains $1,000. The other box is opaque and either contains nothing or $1 million. A mysterious benefactor offers you this choice and tells you that you may choose to take both boxes or just the opaque box.
"However," your generous benefactor cautions, "If I expected you to take both boxes, I have left the opaque box empty -- you get only the $1,000." The mysterious person continues. "If I predicted that you would take only the opaque box, then I have placed $1 million in that box. You will get it all."

You and your friend begin to discuss what to do. Your friend wants to take just the opaque box. You argue that the benefactor has already made his prediction -- the million dollars is either in the opaque box or it is not. It is not going to change. Whose argument is more correct?

Double Liar Paradox (Jourdain's paradox)

Double Liar Paradox (Jourdain's paradox)

This version of the famous paradox was presented by an English mathematician P. E. B. Jourdain in 1913.
The following inscriptions are on a paper:

Back side
Inscription on the other side is true

Face side
Inscription on the other side is not true

Mathematical and Physics Paradoxes

Galileo's Paradox of the Infinite

Galileo's Paradox

This is a demonstration of one of the surprising properties of infinite sets.
In his final scientific work, the Two New Sciences, Galileo Galilei made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect squares (i.e., the square of some integer, in the following just called a square), while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.

The Two Envelopes Paradox

The Two Envelopes

The setup: The player is given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. The player may select one envelope and keep whatever amount it contains, but upon selection, is offered the possibility to take the other envelope instead.
The switching argument:
Denote by A the amount in the selected envelope.
The probability that A is the smaller amount is 1/2, and that it's the larger also 1/2
The other envelope may contain either 2A or A/2
If A is the smaller amount, the other envelope contains 2A
If A is the larger amount, the other envelope contains A/2
Thus, the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2
So the expected value of the money in the other envelope is:
This is greater than A, so swapping is favored
After the switch, reason in exactly the same manner as above, but denote the second envelope's contents as B
It follows that the most rational thing to do is to swap back again
This line of reasoning dictates that envelopes be swapped indefinitely
As it seems more rational to open just any envelope than to swap indefinitely, the player is left with a paradox.

Irresistible Force Paradox

Irresistible Force Paradox

The Irresistible force paradox, also the unstoppable force paradox, is a classic paradox formulated as follows:
What happens when an irresistible force meets an immovable object?
Common responses to this paradox resort to logic and semantics.
Logic: if such a thing as an irresistible force exists, then no object is immovable, and vice versa. It is logically impossible to have these two entities (a force that cannot be resisted and an object that cannot be moved by any force) in the same universe.
Semantics: if there is such a thing as an irresistible force, then the phrase immovable object is meaningless in that context, and vice versa, and the issue amounts to the same thing as, for example, asking for a triangle that has four sides.

Monty Hall Paradox

Let's Make a Deal!

Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.

The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.

After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch?

Schroedinger's Cat Paradox

A cat is penned up in a steel chamber, along with a Geiger counter in which there is a tiny bit of radioactive substance, so small that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The Psi function for the entire system would express this by having in it the living and the dead cat mixed or smeared out in equal parts.

Can't Get Enough?

Check Out These Paradox and Logic Theories

The Prisoner's Dilemma

Is it better to rat on your fellow prisoner or not?

Russell's Paradox

A paradox concerning set theory.

Banach-Tarski paradox

It is possible to take a solid sphere, cut it up into a finite number of pieces, rearrange them and re-assemble them into two identical copies of the original sphere.

Classic Math Problems

Classic Problems from the Dr. Math Archives

Cantor's Infinities

Are there more odd or even integers?

Time Travel Paradoxes

Einstein's Twin Paradox

The Twin Paradox of Einstein is an interesting thought experiment involving two twins (who are nearly exactly the same age), one of whom sets out on a journey into space and back. Because of the time dilation effect of relativity, the twin who left experiences a slowing down of time and will actually be much younger than the twin that stayed behind. The reason that this is considered a paradox is that Special Relativity seems to imply that either one can be considered at rest, with the other moving. It does, and it doesn't.

Grandfather Paradox

Imagine you build a time machine. It is possible for you to travel back in time, meet your grandfather before he produces any children (i.e. your father/mother) and kill him. Thus, you would not have been born and the time machine would not have been built, a paradox.