| Type | Book |
|---|---|
| Date | 2011 |
| Pages | 194 |
| Series | Trends in Logic (31) |
| Tags | nonfiction, logic, paradoxes |
Lukowski divides paradoxes into four classes:
These are paradoxes that arise when logical reasoning yields surprising, counter-intuitive results.
Robert Louis Stevenson's 1891 story The Bottle Imp posits a bottle containing an imp that grants wishes. The bottle can only be sold to another at a loss, and if you die while owning it, you go to hell. The paradox is that no one should be willing to buy it for a penny, and knowing this, no one should be willing to buy it for two cents, either. By induction, no one should be willing to buy it for any price, although intuitively we feel like if the price is high enough, it should be safer to buy.
Lukowski 'solves' Newcomb's problem by declaring that the solution is to take only one box, because the flow of causation simply doesn't match the flow of time. Taking two boxes and finding cash in both is not an available option, so there is no conflict.
Many words have been spent on this problem that amount to attempts to fool the oracle. This, I think, is the sort of thing Lukowski means when he says that "Newcomb's problem really touches on human frailties: greed, underestimating other people, tendency to cheat, etc."
On the birthday problem.
Two unintuitive results from geometry.
An improper application of inductive reasoning yields an absurd result.
If seeing a black raven supports the claim that all ravens are black, then so does seeing any non-black non-raven, such as a white shoe.
On the paradox of Aristotle's Wheel.
Lukowski constructs some sequences that have relations similar to those ascribed to the holy trinity.
What is more, we have demonstrated that the concept of Trinity can be conceived of not only in theology but also in the most precise of sciences that is available for man, i.e., mathematics.
Zach Weber damns the book with faint praise, writing that "If Łukowski’s monograph were the only book to collect together most of the well-known paradoxes, then there would be much to recommend it" (Weber, 2012). This sums up my thoughts, so far.
| Name | Role |
|---|---|
| Marek Gensler | Translator |
| Piotr Łukowski | Author |
| 1: Introduction | 1 |
| References | 3 |
| 2: Paradoxes of Wrong Intuition | 5 |
| 2.1: Bottle Imp Paradox (Stevenson's Bottle), or the Unintuitive Character of a Conclusion Following a Sufficient Multiplication of a Simple Reasoning | 5 |
| 2.2: Newcomb's Paradox, or the Unintuitive Character of a Conclusion Drawn from Premises with Non-intuitive Assumption | 7 |
| 2.3: Paradox of Common Birthday, or the Unintuitive Character of Some Results in Probability Calculus | 11 |
| 2.4: Paradox of Approximation and Paradox of the Equator, or the Unintuitive Character of Some Results in Euclidean Geometry | 13 |
| 2.5: Horses' Paradox, or the Intuitive Character of an Erroneous Application of Mathematical Induction | 15 |
| 2.6: Hempel's Paradox (Raven, Confirmation), or the Unintuitive Character of Some Inductive Reasoning Results | 16 |
| 2.7: Paradoxes of Infinity, or Unintuitive Character of Some Results Obtained in Set Theory | 19 |
| 2.7.1: Aristotelian Circles Paradox, or the Definition of an Infinite Set | 19 |
| 2.7.2: Holy Trinity Paradox, or Application of the Definition of Infinite Set in Theology | 27 |
| 2.8: Fitch's Paradox, or the Conflict of Two Intuitions | 32 |
| References | 35 |
| 3: Paradoxes of Ambiguity | 37 |
| 3.1: Protagoras' (Law Teacher's) Paradox | 37 |
| 3.2: Electra's (the Veiled One's) Paradox and Other Equivocations | 48 |
| 3.3: The Horny One's Paradox | 51 |
| 3.4: Nameless Club Paradox | 52 |
| 3.5: Paradox of a Stone, or an Attempt at Disproving God's Omnipotence | 54 |
| References | 73 |
| 4: Paradoxes of Self-Reference | 75 |
| 4.1: Möbius Ribbon and Klein Bottle, or Self-Reference in Mathematics | 75 |
| 4.2: Great Semantical Paradoxes | 80 |
| 4.2.1: Liar Antinomy | 80 |
| 4.2.2: Buridan's Paradox | 106 |
| 4.2.3: Generalized Form of Liar Antinomy | 108 |
| 4.2.4: Curry's Paradox | 109 |
| 4.3: Other Semantic Paradoxes | 110 |
| 4.3.1: Barber's Antinomy | 111 |
| 4.3.2: Richard's and Berry's Antinomies | 113 |
| 4.3.3: Grelling's Antinomy (Vox Non Appellans Se) | 118 |
| 4.4: Unexpected Examination (Hangman's) Paradox, or Self-Reflexive Reasoning | 119 |
| 4.5: Crocodile's Paradox, or Baron Münchhausen Fallacy | 124 |
| References | 128 |
| 5: Ontological Paradoxes | 131 |
| 5.1: Paradoxes of Difference - Paradox of a Heap | 131 |
| 5.1.1: What is Vagueness? | 141 |
| 5.1.2: Proposals Substituting Preciseness for Vagueness | 151 |
| 5.1.3: Vagueness Respecting Proposals | 166 |
| 5.2: Paradoxes of Change | 171 |
| 5.2.1: Paradox of the Moment of Death (Change of State Paradox) | 171 |
| 5.2.2: Paradoxes of Identity | 172 |
| 5.2.3: The Paradoxes of Motion | 175 |
| 5.3: Diagnosis of Paradoxes of Vagueness and Change | 181 |
| References | 185 |
| Names Index | 189 |
| Subject Index | 193 |