My thesis aims at contributing to classifying the Liar-like paradoxes (and related Truth-teller-like expressions) by clarifying distinctions and relationships between these expressions and arguments. Such a classification is worthwhile, firstly, because it makes some progress towards reducing a potential infinity of versions into a finite classification; secondly, because it identifies a number of new paradoxes, and thirdly and most significantly, because it corrects the historically misplaced distinction between semantic and set-theoretic paradoxes. I emphasize the third result because the distinction made by Peano [1906] and supported by Ramsey [1925] has been used to warrant different responses to the semantic and set-theoretic paradoxes. I find two types among the paradoxes of truth, satisfaction and membership, but the division is shifted from where it has historically been drawn. This new distinction is, I believe, more fundamental than the Peano-Ramsey distinction between semantic and set-theoretic paradoxes. The distinction I investigate is ultimately exemplified in a difference between the logical principles necessary to prove the Liar and those necessary to prove Grellings and Russells paradoxes. The difference relates to proofs of the inconsistency of naive truth and satisfaction; in the end, we will have two associated ways of proving each result. ¶
Another principled division is intuitively anticipated. I coin the term 'hypodox' (adj.: 'hypodoxical') for a generalization of Truth-tellers across paradoxes of truth, satisfaction, membership, reference, and where else it may find applicability. I make and investigate a conjecture about paradox and hypodox duality: that each paradox (at least those in the scope of the classification) has a dual hypodox.¶
In my investigation, I focus on paradoxes that might intuitively be thought to be relatives of the Liar paradox, including Grellings (which I present as a paradox of satisfaction) and, by analogy with Grellings paradox, Russells paradox. I extend these into truth-functional and some non-truth-functional variations, beginning with the Epimenides, Currys paradox, and similar variations. There are circular and infinite variations, which I relate via lists. In short, I focus on paradoxes of truth, satisfaction and some paradoxes of membership. ¶
Among the new paradoxes, three are notable in advance. The first is a non-truth functional variation on the Epimenides. This helps put the Epimenides on a par with Currys as a paradox in its own right and not just a lesser version of the Liar. I find the second paradox by working through truth-functional variants of the paradoxes. This new paradox, call it the ESP, can be either true or false, but can still be used to prove some other arbitrary statement. The third new paradox is another paradox of satisfaction, distinctly different from Grellings paradox. On this basis, I make and investigate the new distinction between two different types of paradox of satisfaction, and map one type back by direct analogy to the Liar, and the other by direct analogy to Russell's paradox.
| Identifer | oai:union.ndltd.org:ADTP/202478 |
| Date | January 2008 |
| Creators | Eldridge-Smith, Peter, peter.eldridge-smith@anu.edu.au |
| Publisher | The Australian National University. Faculty of Arts |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | http://www.anu.edu.au/legal/copyrit.html), Copyright Peter Eldridge-Smith |
Page generated in 0.0026 seconds