A Cognitive Illusion

Image: Flickr

Given these premises, what can you infer?

  1. If there is a king in the hand then there is an ace, or if there isn’t a king in the hand then there is an ace, but not both.
  2. There is a king in the hand.

Practically everyone draws the conclusion “There is an ace in the hand.” But this is wrong: We’ve been told that one of the conditional assertions in the first premise is false, so it may be false that “If there is a king in the hand, then there is an ace.”

But almost no one sees this. Princeton psychologist Philip Johnson-Laird writes, “[Fabien] Savary and I, together with various colleagues, have observed it experimentally; we have observed it anecdotally — only one person among the many distinguished cognitive scientists to whom we have given the problem got the right answer; and we have observed it in public lectures — several hundred individuals from Stockholm to Seattle have drawn it, and no one has ever offered any other conclusion.” Johnson-Laird himself thought he’d made a programming error when he first discovered the illusion in 1995.

Why it happens is unclear; in puzzling out problems like this, we seem to focus on what’s true and neglect what might be false. Computers are much better at this than we are, which ironically might lead a competent computer to fail the Turing test. In order to pass as human, writes researcher Selmer Bringsjord, “the machine must be smart enough to appear dull.”

(Philip N. Johnson-Laird, “An End to the Controversy? A Reply to Rips,” Minds and Machines 7 [1997], 425-432.)

10/18/2016 UPDATE: Readers Andrew Patrick Turner and Jacob Bandes-Storch point out that if we take the first premise to mean material implication (and also allow double negation elimination), then not only can we not infer that there must be an ace, but we can in fact infer that there cannot be an ace in the hand — exactly the opposite of the conclusion that most people draw! Jacob offers this explanation (XOR means “or, but not both”, and ¬ means “not”):

I’ll use the shorthand “HasKing” to be a logical variable indicating whether there is a king in the hand.
Similarly, “HasAce” is a variable which indicates whether there is an ace in the hand.

We’re given two statements:

#1: (HasKing → HasAce) XOR ((¬HasKing) → HasAce).

#2: HasKing.

#2 has just told us that our “HasKing” variable has the value “true”.

So, we can fill this in to #1, which becomes “(true → HasAce) XOR (false → HasAce)”.

I’ll call the sub-clauses of #1 “1a” & “1b”, so #1 is “1a XOR 1b”.

1a: “(true → HasAce)” is a logical expression that’s equivalent to just “HasAce”.

1b: “(false → HasAce)” is always true — because the antecedent, “false”, can never be satisfied, the consequent is effectively disregarded.

Recall what statement #1 told us: (1a XOR 1b). We now know 1b is true, so 1a must be false. Thus “HasAce” is false: there is not an ace in the hand.

Jacob also offered this demonstration in Prolog. Many thanks for Andrew and Jacob for the patience in explaining this to me.

Black and White

dawson half-move chess puzzle

One more chess curiosity by T.R. Dawson: How can White mate in two half moves?

The answer is to play the first half of Bg1-f2, and the second half of Bf1-g2, thus getting the white bishop from g1 to g2 and giving mate.

A fair-minded reader might ask why Black can’t pull the same trick, transferring his bishop from b8 to b7 to block the check. The answer, Dawson argues, is that some of the constituent moves are illegal: Black can’t combine Bb8-c7 and Bc8-b7 because a bishop on c8 would put the white king in an unreal check on h3; and he can’t combine Bb8-a7 and Ba8-b7 because a8 is occupied.

From Caissa’s Fairy Tales (1947).



A memory of Lewis Carroll by Lionel A. Tollemache:

He was, indeed, addicted to mathematical and sometimes to ethical paradoxes. The following specimen was propounded by him in my presence. Suppose that I toss up a coin on the condition that, if I throw heads once, I am to receive 1d.; if twice in succession, 2d.; if thrice, 4d.; and so on, doubling for each successful toss: what is the value of my prospects? The amazing reply is that it amounts to infinity; for, as the profit attached to each successful toss increases in exact proportion as the chance of success diminishes, the value (so to say) of each toss will be identical, being in fact, 1/2d.; so that the value of an infinite number of tosses is an infinite number of half-pence. Yet, in fact, would any one give me sixpence for my prospect? This, concluded Dodgson, shows how far our conduct is from being determined by logic.

Actually this curiosity was first noted by Nicholas Bernoulli; Carroll would have met it in his studies of probability. Tollemache wrote, “The only comment that I will offer on his astounding paradox is that, in order to bring out his result, we must suppose a somewhat monotonous eternity to be consumed in the tossing process.”

(Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” Literature, Feb. 5, 1898.)

Podcast Episode 120: The Barnes Mystery


In 1879 a ghastly crime gripped England: A London maid had dismembered her employer and then assumed her identity for two weeks, wearing her clothes and jewelry and selling her belongings. In this week’s episode of the Futility Closet podcast we’ll describe the murder of Julia Thomas and its surprising modern postscript.

We’ll also discover the unlikely origins of a Mary Poppins character and puzzle over a penguin in a canoe.


Early airplanes were sometimes attacked by confused eagles.

Alberta, Canada, has been rat-free for 50 years.

Sources for our feature on the murder of Julia Thomas:

Elliott O’Donnell, ed., Trial of Kate Webster, 1925.

Transcript of Kate Webster’s trial at the Old Bailey.

“The Richmond Murder,” Glasgow Herald, May 29, 1879.

“Kate Webster Hanged,” Reading [Pa.] Eagle, July 31, 1879.

Matt Blake, “Attenborough Skull Mystery Finally Solved,” Independent, July 5, 2011.

Cigdem Iltan, “The Skull in the Backyard,” Maclean’s 124:28 (July 25, 2011), 37.

Image: Wikimedia Commons

Park Road, Richmond, today. At left is the site of the former Mayfield Cottages, where the murder took place. At center is the home of naturalist Sir David Attenborough. At right is the site of the former Hole in the Wall pub. Thomas’ skull was discovered in 2010 at the site of the pub’s stables.

Listener mail:

GitHub, “System Bus Radio” (retrieved Sept. 2, 2016).

Catalin Cimpanu, “Emitting Radio Waves from a Computer with No Radio-Transmitting Hardware,” Softpedia, March 2, 2016.

A 40-second rendition of the discarded Mary Poppins song “Admiral Boom.”

Wikipedia, Mary Poppins (film)” (retrieved Sept. 2, 2016).

This week’s lateral thinking puzzles were contributed by listeners Matt Sargent and Jacob Bandes-Storch.

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at http://feedpress.me/futilitycloset.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation on the Support Us page of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Here and There


In 1976, Australian monkey trainer Alex Brackstone declared his four-hectare property northeast of Adelaide to be the independent Province of Bumbunga and named himself its governor-general. He was concerned that Australia was drifting toward republicanism and wanted to be sure that at least a part of the continent would always be loyal to the British Crown.

To underscore his devotion to the queen he drew a huge scale model of Great Britain using strawberry plants. He planned to sprinkle each county with authentic soil imported from Britain, but customs authorities put a stop to that, and the strawberries eventually died in a drought. Full points for effort, though.

Related: In her 1981 book The Emperor of the United States of America, Catherine Caufield says that British eccentric John Alington laid out a giant street map of London on the grounds of his estate at Letchworth, to rehearse his laborers who were traveling there to see the Great Exhibition. This article repeats the story, noting that Alington was greatly taken with giant maps: “In 1855, he had a reproduction of the fortifications of Sebastopol built so his workers could better understand the progress of the Crimean War. He also had a pond remodelled into a map of the world, which the men toured in rowboats as he lectured them in geography.” I haven’t been able to confirm this elsewhere, though.

Crossing Guard

Suppose some 2n points in the plane are chosen so that no three are collinear, and then half are colored red and half blue. Will it always be possible to connect each red point with a blue one, in pairs, so that none of the connecting lines intersect?

Click for Answer

Best Efforts


A portrait of a Civil War field hospital in 1863, written by a Union colonel wounded at Port Hudson:

I never wish to see another such time as the 27th of May. The surgeons used a large Cotton Press for the butchering room & when I was carried into the building and looked about I could not help comparing the surgeons to fiends. It was dark & the building lighted partially with candles: all around on the ground lay the wounded men; some of them were shrieking, some cursing & swearing & some praying; in the middle of the room was some 10 or 12 tables just large enough to lay a man on; these were used as dissecting tables & they were covered with blood; near & around the tables stood the surgeons with blood all over them & by the side of the tables was a heap of feet, legs & arms. On one of these tables I was laid & being known as a Col. the Chief Surgeon of the Department was called (Sanger) and he felt of my mouth and then wanted to give me cloriform: this I refused to take & he took a pair of scissors & cut out the pieces of bone in my mouth: then gave me a drink of whiskey & had me laid away.

In 1918, after a half-century of medical advances, one federal surgeon looked back on the war:

We operated in old blood-stained and often pus-stained coats, the veterans of a hundred fights. … We used undisinfected instruments from undisinfected plush-lined cases, and still worse, used marine sponges which had been used in prior pus cases and had been only washed in tap water. If a sponge or an instrument fell on the floor it was washed and squeezed in a basin of tap water and used as if it were clean. Our silk to tie blood vessels was undisinfected. … The silk with which we sewed up all wounds was undisinfected. If there was any difficulty in threading the needle we moistened it with … bacteria-laden saliva, and rolled it between bacteria-infected fingers. We dressed the wounds with clean but undisinfected sheets, shirts, tablecloths, or other old soft linen rescued from the family ragbag. We had no sterilized gauze dressing, no gauze sponges. … We knew nothing about antiseptics and therefore used none.

In The Life of Billy Yank, historian Bell I. Wiley writes, “Little wonder that gangrene, tetanus and other complication were so frequent and that slight wounds often proved mortal.”

Big News


The Soviet Union took propaganda to a ludicrous extreme in the 1930s with the Maksim Gorki, a multimedia communications empire in the sky. With a wingspan of 206 feet and a takeoff weight of 42 tons, it was the largest land aircraft ever built at the time, requiring eight huge 900-horsepower engines to get aloft.

Aboard were a complete printing plant, capable of printing 10,000 copies per hour of an illustrated 12″ x 16″ newspaper, a photographic darkroom, and a high-speed radio apparatus and telegraph. On the ground, a projection room could cast movies onto a folding screen for up to 10,000 spectators through a window in the fuselage.


“The aircraft also contained a cafe, its own internal telephone exchange, and sleeping quarters and toilets,” notes James Gilbert in The World’s Worst Aircraft. “Four auxiliary engines were required to generate the power to run the huge loudspeakers that broadcast the Soviet message down upon the astonished peasants over which the aircraft flew, and at night to power a system of lights along the underside flashing slogans.” Whether anyone wanted to hear all this is another question.