It is familiarly said that beer … is an acquired taste; one gradually trains oneself — or just comes — to enjoy that flavor. What flavor? The flavor of the first sip? No one could like that flavor, an experienced beer drinker might retort: ‘Beer tastes different to the experienced beer drinker. If beer went on tasting to me the way the first sip tasted, I would never have gone on drinking beer! Or to put the same point the other way around, if my first sip of beer had tasted to me the way my most recent sip just tasted, I would never have had to acquire the taste in the first place! I would have loved the first sip as much as the one I just enjoyed.’ If we let this speech pass, we must admit that beer is not an acquired taste. No one comes to enjoy the way the first sip tasted. Instead, prolonged beer drinking leads people to experience a taste they enjoy, but precisely their enjoying the taste guarantees that it is not the taste they first experienced.

— Daniel Dennett, “Quining Qualia,” from Consciousness in Contemporary Science, 1988

If ever a person A deceives a person B into believing that something, p, is true, A knows or truly believes that p is false while causing B to believe that p is true. So when A deceives A (i.e., himself) into believing that p is true, he knows or truly believes that p is false while causing himself to believe that p is true. Thus, A must simultaneously believe that p is false and believe that p is true. But how is this possible?

— Alfred R. Mele, “Two Paradoxes of Self-Deception,” in Self-Deception and Paradoxes of Rationality, 1998

Henry is driving in the countryside with his son. For the boy’s edification, Henry identifies various objects on the landscape as they come into view. ‘That’s a cow,’ says Henry. ‘That’s a tractor,’ ‘That’s a silo,’ ‘That’s a barn,’ etc. Henry has no doubt about the identity of these objects; in particular, he has no doubt that the last-mentioned object is a barn, which indeed it is. Each of the identified objects has features characteristic of its type. Moreover, each object is fully in view, Henry has excellent eyesight, and he has enough time to look at them reasonably carefully, since there is little traffic to distract him.

Given this information, would we say that Henry knows that the object is a barn? Most of us would have little hesitation in saying this, so long as we were not in a certain philosophical frame of mind. Contrast our inclination here with the inclination we would have if we were given some additional information. Suppose we are told that, unknown to Henry, the district he has just entered is full of papier-mâché facsimiles of barns. These facsimiles look from the road exactly like barns, but are really just façades, without back walls or interiors, quite incapable of being used as barns. They are so cleverly constructed that travelers invariably mistake them for barns. Having just entered the district, Henry has not encountered any facsimiles; the object he sees is a genuine barn. But if the object on that site were a facsimile, Henry would mistake it for a barn. Given this new information, we would be strongly inclined to withdraw the claim that Henry knows the object is a barn. How is this change in our assessment to be explained?

— Alvin I. Goldman, “Discrimination and Perceptual Knowledge,” Journal of Philosophy, November 1976

“I am fond of the businessman’s paradox due to Lisa Collier: The president of a certain company offered a reward of $100 to any employee who could offer a suggestion which would save the company money. One employee suggested: ‘Eliminate the reward.'” — Raymond Smullyan

Let’s play a game. You’ll flip a coin, and if it comes up heads I’ll give you $1. If you flip heads again I’ll give you $2, then $4, then $8, and so on. When the coin comes up tails, the game is over and you can keep your winnings.

Because I’m taking a risk, I ought to charge you an entrance fee. What’s a fair fee? Surprisingly, it seems I should charge you an infinite amount of money. With each new flip your chance of success is 1/2 but your prospective earnings double, so your total expected earnings — the earnings times their chance of being realized — is infinite:

E = (1/2 × 1) + (1/4 × 2) + (1/8 × 4) + … = ∞

Nicholas Bernoulli first described this problem in 1713. One proposed resolution is that it ignores psychology — we’re considering the monetary value of the prize rather than its value to us. Gold shines more brightly for a beggar than for a billionaire; once we’ve amassed a certain sum, the appeal of greater riches begins to diminish. “The mathematicians estimate money in proportion to its quantity,” wrote Gabriel Cramer, “and men of good sense in proportion to the usage that they may make of it.”

(Thanks, Ross.)

Take a whole stick and cut it in half. Half a minute later, cut each half in half. A quarter of a minute after that, cut each quarter in half, and so on ad infinitum.

What will remain at the end of a minute? An infinite number of infinitely thin pieces? Writes Oxford philosopher A.W. Moore, “Do we so much as understand this?”

Does each piece have any width? If so, couldn’t we reassemble them to form an infinitely long stick? If not, how can we assemble them to form anything at all?

Caltech number theorist Tom Apostol devised this elegant proof of the irrationality of .

Suppose the number is rational. Then there must be an isosceles right triangle with minimum integer sides (here, triangle ABC with sides n and hypotenuse m).

By drawing two arcs as shown, we can immediately establish triangle FDC — a smaller isosceles right triangle with integer sides.

This leads to an infinite descent. Hence n and m can’t both be integers, and is irrational.