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Paradoxes: Their Roots, Range, and Resolution
Paradoxes—sets of propositions that are individually plausible but collectively inconsistent—range from transparent tricks of word-play to more puzzling issues. For example, if the barber, an adult male in the village, shaves all adult males who do not shave themselves, then he both shaves himself and does not shave himself. In this fresh and uncluttered approach, Nicholas Rescher introduces the subject, surveys the entire range of types of paradoxes, and introduces an integrated theory of paradoxes. He explains and analyzes over 130 paradoxes, showing how they can all be handled by one approach.
- GenresPhilosophyLogic
320 pages, Paperback
First published April 19, 2001
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August 20, 2018
Inconceivable! How to Defeat the Sicilian
'Alice laughed. "There's no use trying," she said. "One can't believe impossible things."
"I dare say you haven't had much practice," said the queen. "When I was your age, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." ' - Lewis Carrol, Alice in Wonderland.
This is an absolutely brilliant book - the best and most complete coverage of self referential conundrums that I have ever come across. Rescher reaches back to Aristotle for the origin of the category (aporia) and gives a more than able description of these puzzles, both classic and modern, and provides a set of clever tools with which to resolve them. Though the occassional use of symbolic logic and set notation might put off some readers no proofs are involved and the symbolism is also also explained by the text.
Whereas ignorance stems from not having enough information, "paradox", from the Greek para (beyond) doxa (belief) is often the problem of having too much information, some of it contradictory. Rescher makes it clear that self reference alone is not necessarily bad, rather the problem may be in how it is introduced.
One of the ways out of a paradox is to reduce the number of premises to mutually consistant subsets, and when having a choice of subsets pick either the largest or most powerful group, or one which conforms most to evidence. Another strategy is to relax absolute true/false splits to degrees of plausibility or multivalued logic. The Liar's Paradox group falls away to this - most Cretans are likely to lie all the time, but may introduce a truth to confuse you. A third resolution may be in realizing that the initial frame of the premise does not hold in the conclusion - an example being the "Suprise Test" which will be held next week. In the time frame of the announcement, it's always a surprise. The only time it is not a surprise is on the last day - but that is a different frame than the announcement.
Some paradoxes as simply language problems, such as Berry's Paradox of the smallest number that can be named in no less than 14 (or n) words - and here is the problem - we are actually discussing two different classes of notation - a direct representation and an indirect one Directly one can name the digits; indirectly one can call it X, therefore there is no such number using an indirect reference - the question is only revelant within a framework of directly naming. Other classical langauge oriented paradoxes turn out to be nothing more than rhetorical sophistry rather than logical tricks, such at the Horns of Eubulides "Have you lost two horns. No! Then you must still have them", which is comical sophistry as well as a double entendre of cuckoldry. In the Mikado Paradox, Rescher muses, can one execute the Official Lord High Executioner? (My answer - of course - appoint another one.)
There are only a few examples where Rescher dips into theology, so he doesn't get into a discussion of whether God could be powerful enough create an immovable object - though in such cases I always imagined that it is the metric of the universe that would change, not the object itself. Here the discussion plumbs Kierkegard's exploration as to whether rational Man can believe in what he can't properly understand. Kierkegard did not take the atheist's route, instead he took the approach that faith is transcendant. Omnicience may be an impossible concept, so arguing that God is or is not omniscient may either be moot, or require a different definition. Rescher classifies this approach, which he deems legitimate, as "Abandonment", arguing that there are cases where self refence cannot apply. Indeed, logic was only one of the classic schools of reasoning, something we modernists tend to forget. IMHO most decisions, even by rational individuals, are based on rhetoric and desire, not logic, so the classical models or reasoning still hold.
Paralleling this is the consideration of recursive sets, such as Russell's barber who shaves everyone in the village who does not shave himself. Reschers applies his Successful Identification Principle (SIP) - we can't beg the question by introducing a premise that encompasses itself - he sees this a referential singularity, the logical equivalent of division by zero.
As you probably guessed, I really liked the book. There are occassional bits of humour but it's generally serious and not written for a popular audience. It's not whimsical like Martin Gardner or Raymond Smullyan; a university level background would be recommended, IMHO it has all the ingredients of a classic and I'd highly recommend it people who've seriously studied any philosophy, math, theology, engineering or other professions involving rhetoric, logic and critical thinking. Heck, if philosophy or political punditry were a licenced profession I'd make mastering this book a requirement before graduation!
10 out of 5 stars and highly recommended!!!
'Alice laughed. "There's no use trying," she said. "One can't believe impossible things."
"I dare say you haven't had much practice," said the queen. "When I was your age, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." ' - Lewis Carrol, Alice in Wonderland.
This is an absolutely brilliant book - the best and most complete coverage of self referential conundrums that I have ever come across. Rescher reaches back to Aristotle for the origin of the category (aporia) and gives a more than able description of these puzzles, both classic and modern, and provides a set of clever tools with which to resolve them. Though the occassional use of symbolic logic and set notation might put off some readers no proofs are involved and the symbolism is also also explained by the text.
Whereas ignorance stems from not having enough information, "paradox", from the Greek para (beyond) doxa (belief) is often the problem of having too much information, some of it contradictory. Rescher makes it clear that self reference alone is not necessarily bad, rather the problem may be in how it is introduced.
One of the ways out of a paradox is to reduce the number of premises to mutually consistant subsets, and when having a choice of subsets pick either the largest or most powerful group, or one which conforms most to evidence. Another strategy is to relax absolute true/false splits to degrees of plausibility or multivalued logic. The Liar's Paradox group falls away to this - most Cretans are likely to lie all the time, but may introduce a truth to confuse you. A third resolution may be in realizing that the initial frame of the premise does not hold in the conclusion - an example being the "Suprise Test" which will be held next week. In the time frame of the announcement, it's always a surprise. The only time it is not a surprise is on the last day - but that is a different frame than the announcement.
Some paradoxes as simply language problems, such as Berry's Paradox of the smallest number that can be named in no less than 14 (or n) words - and here is the problem - we are actually discussing two different classes of notation - a direct representation and an indirect one Directly one can name the digits; indirectly one can call it X, therefore there is no such number using an indirect reference - the question is only revelant within a framework of directly naming. Other classical langauge oriented paradoxes turn out to be nothing more than rhetorical sophistry rather than logical tricks, such at the Horns of Eubulides "Have you lost two horns. No! Then you must still have them", which is comical sophistry as well as a double entendre of cuckoldry. In the Mikado Paradox, Rescher muses, can one execute the Official Lord High Executioner? (My answer - of course - appoint another one.)
There are only a few examples where Rescher dips into theology, so he doesn't get into a discussion of whether God could be powerful enough create an immovable object - though in such cases I always imagined that it is the metric of the universe that would change, not the object itself. Here the discussion plumbs Kierkegard's exploration as to whether rational Man can believe in what he can't properly understand. Kierkegard did not take the atheist's route, instead he took the approach that faith is transcendant. Omnicience may be an impossible concept, so arguing that God is or is not omniscient may either be moot, or require a different definition. Rescher classifies this approach, which he deems legitimate, as "Abandonment", arguing that there are cases where self refence cannot apply. Indeed, logic was only one of the classic schools of reasoning, something we modernists tend to forget. IMHO most decisions, even by rational individuals, are based on rhetoric and desire, not logic, so the classical models or reasoning still hold.
Paralleling this is the consideration of recursive sets, such as Russell's barber who shaves everyone in the village who does not shave himself. Reschers applies his Successful Identification Principle (SIP) - we can't beg the question by introducing a premise that encompasses itself - he sees this a referential singularity, the logical equivalent of division by zero.
As you probably guessed, I really liked the book. There are occassional bits of humour but it's generally serious and not written for a popular audience. It's not whimsical like Martin Gardner or Raymond Smullyan; a university level background would be recommended, IMHO it has all the ingredients of a classic and I'd highly recommend it people who've seriously studied any philosophy, math, theology, engineering or other professions involving rhetoric, logic and critical thinking. Heck, if philosophy or political punditry were a licenced profession I'd make mastering this book a requirement before graduation!
10 out of 5 stars and highly recommended!!!
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