Some Paradoxes, Easy and Hard

Author
Affiliation

David Sanson

Illinois State University

Published

April 15, 2014

1 The Liar

O Stranger: Philetas of Kos am I,
’Twas the Liar who made me die,
And the bad nights caused thereby.

Let (L) be the sentence ‘(L) is not true’. Then we can argue,

  1. Suppose (L) is true.
  2. If ‘(L) is not true’ is true, then (L) is not true.
  3. So (L) is not true.

And we can also argue,

  1. Suppose (L) is not true.
  2. If (L) is not true then ‘(L) is not true’ is true.
  3. So (L) is true.

(L2) and (L5) follow from a fundamental principle governing truth:

T. ‘S’ is true if and only if S.

2 The Horned Man

  1. What you have not lost you still have.
  2. You have not lost horns.
  3. So, you still have horns.

The conclusion is obviously false. Is the argument valid? Is it sound?

3 Eating Raw Meat (Buridan, Sophismata 4, Second Sophism)

Yesterday you bought a piece of raw meat, and today you ate it well-cooked.

  1. Whatever you bought yesterday is what you ate today.
  2. Yesterday you bought raw meat.
  3. So, today you ate raw meat.

Buridan considers two replies, and endorses the first:

  • (R1) is false: yesterday, you bought both the substance of the meat, and the accidents inhering in it. Hence you bought a rawness yesterday that you did not eat today.
  • The premises are true but the conclusion does not follow. In premise (R2), ‘raw’ stands for past or present raw things. In the conclusion, ‘raw’ stands only for presently raw things.

4 Are There More Men Alive or Dead?

[Alexander the Great] captured the ten gymnosophists [lit. “naked wise men”] who had been the ringleaders behind Sabbas’ rebellion and had therefore done the Macedonians a very great deal of harm. They were reputed to have the ability to answer questions cleverly and pithily, so Alexander put difficult questions to them, with the warning that he would kill the first one to give an incorrect answer, and then the rest, one after another, on the same principle. He started with the oldest one, and asked him whether, in his opinion, the living or the dead were more numerous: ‘The living,’ he said, ‘because the dead no longer exist.’ (Plutarch 2016, Alexander 64)

  1. Everything exists.
  2. The dead no longer exist.
  3. So, nothing is dead.

5 The Hooded Man1

A man walks into a bar, wearing a hood. He is your brother, but you don’t know this. So:

  1. This man is your brother.
  2. You do not know this man.
  3. So, you do not know your brother.

Response: (H2) is false. You know the man, because you know your brother, and he is your brother.

  1. This man is your brother.
  2. You do not know who this man is.
  3. So, you do not know who your brother is.

Response: (H’) is false. You know who the man is, because you know who your brother is, and he is your brother.

  1. This man is your brother.
  2. You do not know that this man was born in Megara.
  3. So, you do not know that your brother was born in Megara.

Response: (H″) is false. You do know that this man was born in Megara, because you know your brother was born in Megara, and he is your brother.

  1. This man is your brother.
  2. You do not realize that this man was born in Megara.
  3. So, you do not realize that your brother was born in Megara.

6 Hesperus and Phosphorus (Frege 1948)

  1. ‘Hesperus = Hesperus’ is known a priori and analytic.
  2. ‘Hesperus = Phosphorous’ is neither a priori nor analytic.
  3. So, identity is not a relation between objects.

7 Scott and the Author of Waverley (Russell 1905)

  1. If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other in any proposition without altering the truth or falsehood of that proposition.
  2. George IV wished to know whether Scott was the author of Waverley.
  3. Scott was the author of Waverley.
  4. So, George IV wished to know whether Scott was Scott.

“But an interest in the law of identity can hardly be attributed to the first gentleman of Europe.”

8 “I owe you a horse” (Buridan 2001, Sophismata 4.15)

“I posit the case that in return for some good service that you performed for me, I promised you one good horse, and that I obligated myself before a competent judge to give you one good horse.”

  1. I do not owe you any horse: not this horse, not that horse, nor that horse…
  2. So, I do not owe you a horse.

Buridan denies (O1). I owe you each horse, since giving it to you would discharge my debt. But I do not owe you all the horses together, since giving you just one horse is enough to discharge my debt.

9 The Sorites

  • 1 grain of wheat does not make a heap.
  • If 1 grain of wheat does not make a heap then 2 grains of wheat do not.
  • If 2 grains of wheat do not make a heap then 3 grains do not.
  • \vdots
  • If 9,999 grains of wheat do not make a heap then 10,000 do not.
  • So, 10,000 grains of wheat do not make a heap.

10 Sources

Eubulides of Megara was a 4th century BCE Greek philosopher. Seven paradoxes are attributed to him: the Liar; the Hooded Man and two variants on the Hooded Man called ‘the Electra’ and ‘the Unnoticed Man’; two versions of the Sorities: the Heap and the Bald Man; and the Horned Man. In the words of a contemporary poet,

“Eubulides the Eristic […] propounded his quibbles about horns and confounded the orators with falsely pretentious arguments…”

Jean Buridan (~1300-~1360) was a medieval philosopher. One of his most famous works is the Summulae de Dialectica. The last section discusses several “sophisms”—i.e., logical puzzles and paradoxes.

References

Buridan, Jean. 2001. Summulae de Dialectica. Translated by Gyula Klima. New Haven: Yale University Press.
Frege, Gottlob. 1948. “Sense and Reference.” Translated by Peter Geach. The Philosophical Review 57 (3): 209–30. https://doi.org/10.2307/2181485.
Plutarch. 2016. Hellenistic Lives: Including Alexander the Great. Translated by Robin Waterfield. Oxford: Oxford University Press.
Priest, Graham. 2005. Towards Non-Being: The Logic and Metaphysics of Intentionality. Oxford: Oxford University Press. http://books.google.com/books?id=lq802KBV-B4C.
Russell, Bertrand. 1905. “On Denoting.” Mind 14 (October): 479–93. http://www.jstor.org/stable/2248381.

Footnotes

  1. I borrow this presentation from Priest (2005), p. 34ff.↩︎