Zebra Puzzle

1. There are five houses.
2. The Englishman lives in the red house.
3. The Spaniard owns the dog.
4. Coffee is drunk in the green house.
5. The Ukrainian drinks tea.
6. The green house is immediately to the right of the ivory house.
7. The Old Gold smoker owns snails.
8. Kools are smoked in the yellow house.
9. Milk is drunk in the middle house.
10. The Norwegian lives in the first house.
11. The man who smokes Chesterfields lives in the house next to the man with the fox.
12. Kools are smoked in the house next to the house where the horse is kept.
13. The Lucky Strike smoker drinks orange juice.
14. The Japanese smokes Parliaments.
15. The Norwegian lives next to the blue house.

Now, who drinks water? Who owns the zebra? In the interest of clarity, it must be added that each of the five houses is painted a different color, and their inhabitants are of different national extractions, own different pets, drink different beverages and smoke different brands of American cigarets [sic]. One other thing: in statement 6, right means your right.

— Life International, December 17, 1962

Wine barrels solution

Two half-full barrels are dumped into one of the empty barrels. Two more half-full barrels are dumped into another one of the empty barrels. This results in nine full barrels, three half-full barrels, and nine empty barrels. Each son gets three full barrels, one half-full barrel, and three empty barrels.

Zeno’s Achilles and the Tortoise Paradox

Zeno of Elea (ca. 490–430 BCE) was the first person in history to show that the concept of infinity is problematic. In Zeno’s Achilles and the Tortoise Paradox, Achilles races to catch a slower runner–for example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones.[1]

Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished “things infinite in respect of divisibility” (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension (“with respect to their extremities”).^[2]

Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Modern calculus achieves the same result, using more rigorous methods. These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.[3]

[1] Internet Encyclopedia of Philosophy.

[2] Aristotle. Physics 6.9; 6.2, 233a21-31.

[3] Boyer, Carl (1959). The History of the Calculus and Its Conceptual Development. Dover Publications. p. 295.

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Theseus’s paradox

Theseus is remembered in Greek mythology as the slayer of the Minotaur. For years, the Athenians had been sending sacrifices to be given to the Minotaur, a half-man, half-bull beast who inhabited the labyrinth of Knossos. One year, Theseus braved the labyrinth, and killed the Minotaur.  The ship in which he returned was long preserved. As parts of the ship needed repair, it was rebuilt plank by plank.

The Ship of Theseus, also known as Theseus’s paradox, is a paradox that raises a question of identity – whether an object which has had all its components replaced remains fundamentally the same object. The paradox is most notably recorded by Plutarch in Life of Theseus from the late 1st century. Plutarch asked whether a ship which was restored by replacing all and every of its wooden parts, remained the same ship.

The paradox had been discussed by more ancient philosophers such as Heraclitus, Socrates, and Plato prior to Plutarch’s writings; and more recently by Thomas Hobbes and John Locke. There are several variants, notably “my grandfather’s axe”, and in the UK “Trigger’s Broom”. This thought experiment is “a model for the philosophers”; some say, “it remained the same,” some saying, “it did not remain the same”.^[1]

George Washington’s axe (sometimes “my grandfather’s axe”) is the subject of an apocryphal story of unknown origin in which the famous artifact is “still George Washington’s axe” despite having had both its head and handle replaced.

“…as in the case of the owner of George Washington’s axe which has three times had its handle replaced and twice had its head replaced!” [2]

 
References:

[1] Rea, M., (1995) The Problem of Material Constitution, The Philosophical Review, 104: 525-552.

[2] Ray Broadus Browne, Objects of Special Devotion: Fetishism in Popular Culture, p. 134

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Wine barrels puzzle

A man is the owner of a winery who recently passed away. In his will, he left 21 barrels (seven of which are filled with wine, seven of which are half full, and seven of which are empty) to his three sons. However, the wine and barrels must be split so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels. Note that there are no measuring devices handy. How can the barrels and wine be evenly divided? The solution is here.

Introduction

Welcome to Tim Harding’s blog of writings and talks about logic, rationality, philosophy and skepticism. There are also some reblogs of some of Tim’s favourite posts by other writers, plus some of his favourite quotations and videos.  This blog has a Facebook connection at The Logical Place.

There are over 2,600 posts here about all sorts of topics – please have a good look around before leaving.

If you are looking for an article about Skepticism, Science and Scientism published in The Skeptic magazine titled ”A Step Too Far?’, it is available here.

If you are looking for an article about the Birth of Experimental Science published in The Skeptic magazine titled ‘Out of the Dark’, it is available here.

If you are looking for an article about the Dark Ages published in The Skeptic magazine titled ‘In the Dark’, it is available here.

If you are looking for an article about the Traditional Chinese Medicine vs. Endangered Species published in The Skeptic magazine titled ‘Bad Medicine’, it is available here.

If you are looking for an article about the rejection of expertise published in The Skeptic magazine titled ‘Who needs to Know?’, it is available here.

If you are looking for an article about Charles Darwin published in The Skeptic magazine titled ‘Darwin’s Missing Link“, it is available here.

If you are looking for an article about the Astronomical Renaissance published in The Skeptic magazine titled ‘Rebirth of the Universe‘, it is available here.

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The Grandfather Paradox

The Grandfather Paradox is one of several metaphysical arguments that attempt to prove that time travel is logically impossible (whether it is physically possible is a question left by philosophers to the physicists).  These arguments all have the same basic form:

Premise 1: If time travel is possible, then X must be possible.

Premise 2: X is not possible.

Conclusion: Therefore, time travel is impossible.

The Grandfather Paradox is described as follows: a time machine is invented enabling a time traveller to go back in time to before his grandfather had fathered offspring.  At that time, the time traveller kills his grandfather, and therefore, one of the time traveller’s parents would never exist and thus the time traveller himself would never exist either.  If he is never born, then he is unable to travel back through time and kill his grandfather, which means he would be born, and so on.

The paradox is also described in this video cartoon.

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Straw Man Fallacy

by Tim Harding B.Sc., B.A.

A speaker commits the Straw Man Fallacy (known in the UK as an ‘Aunt Sally’) whenever she falsely attributes a weak position to her opponent that he wouldn’t have proposed himself and then proceeds to attack the weak position. The opponent is a real man with a real argument; the weak position is an artificial one held by an artificial person —the “straw man” or a scarecrow that the speaker has created as a debating tactic. It’s easier to attack a straw man; nevertheless, the attack is irrelevant. It is a diversion from the main issue.[1]

You are not committing the straw man fallacy simply by drawing a consequence from what the man says that is not what he himself would draw. It must be clear that you are also misrepresenting what he did say.

The straw man fallacy occurs in the following pattern of argument:

1. Debater 1 has position X.
2. Debater 2 disregards certain key points of X and instead misrepresents X as the superficially similar position Y.
3. Debater 2 attacks position Y, concluding that X is false/incorrect/flawed.

This reasoning is fallacious because attacking a distorted version of a position does not address the actual position.  This argument doesn’t make sense; it is a non sequitur. Debater 2 relies on the audience not noticing this.

Christopher Tindale presents, as an example, the following passage from a draft of a bill (HCR 74) considered by the Louisiana State Legislature in 2001:^[2]

Whereas, the writings of Charles Darwin, the father of evolution, promoted the justification of racism, and his books On the Origin of Species and The Descent of Man postulate a hierarchy of superior and inferior races. . . .Therefore, be it resolved that the legislature of Louisiana does hereby deplore all instances and all ideologies of racism, does hereby reject the core concepts of Darwinist ideology that certain races and classes of humans are inherently superior to others, and does hereby condemn the extent to which these philosophies have been used to justify and approve racist practices.

Tindale comments that “the portrait painted of Darwinian ideology is a caricature, one not borne out by any objective survey of the works cited”. That similar misrepresentations of Darwinian thinking have been used to justify and approve racist practices is beside the point: the position that the legislation is attacking and dismissing is a Straw Man. In subsequent debate this error was recognized, and the eventual bill omitted all mention of Darwin and Darwinist ideology.

[1] Bradley H. Dowden (2012) Logical Reasoning. public domain/fair use. pp. 247-249.
[2] Christopher W. Tindale (2007). Fallacies and Argument Appraisal. Cambridge University Press. pp. 19–28.