Academic literature on the topic 'Grelling's paradox'

К.Grelling – L.Nelson’s paradox arises if the author (authors) and the addressee (addressees) use the classical linguistic model of meaning formation, which pos­tulates that self-referential signs (words), or modules “form – meaning”, are em­ployed and understood in word-containing semiotic procedures. The commu­nicative model of meaning formation, on the contrary, states the impossibility of an autonomous reference of a sign, considers an attempt at indirect influence (i.e., process) as the only possible object of reference and understanding in a given communicative act, recognizes the actional (“influential”) mode of semiotic ac­tor’s consciousness as the exclusive source of the produced “meaning”, includ­ing of sign reference. The paradox is viewed as a sequence of attempts at com­municative influences on a conceivable addressee who understands (approves or disapproves) the actions of the author of the paradox by considering his “influen­tial” cognitive states: on which objects the author focuses the addressee’s atten­tion, what are the denotations hidden by the author behind the “bodies” of signs, how justified the established classes, relationships, links, etc. The main vulnera­bility of the paradox is proposed to be seen in the fact that the author unjustifi­ably recognizes the independent out-of-communicative ability of a sign (word) to produce influences (to name, to describe, to have its own properties), does not notice the communicative constants of meaning formation in the theoretical field, although inevitably uses them practically. The paradox is eliminated to­gether with the removal of the classical concept of a sign as an autonomous module “form – meaning”.

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a well-known schema introduced by Graham Priest, namely, the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on why the Inclosure Schema is both too narrow and too broad. We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry's paradox, Validity Curry, Provability Liar, Provability Curry, Knower's paradox, Knower's Curry, Grelling-Nelson's paradox, Russell's paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes. We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution.

К. Grelling – L. Nelson’s paradox arises if the author (authors) and the addressee (addressees) use the classical linguistic model of meaning formation, which pos­tulates that self-referential signs (words), or modules “form – meaning”, are em­ployed and understood in word-containing semiotic procedures. The commu­nicative model of meaning formation, on the contrary, states the impossibility of an autonomous reference of a sign, considers an attempt at indirect influence (i.e., process) as the only possible object of reference and understanding in a given communicative act, recognizes the actional (“influential”) mode of semi­otic actor’s consciousness as the exclusive source of the produced “meaning”, including of sign reference. The paradox is viewed as a sequence of attempts at communicative influences on a conceivable addressee who understands (ap­proves or disapproves) the actions of the author of the paradox by considering his “influential” cognitive states: on which objects the author focuses the ad­dressee’s attention, what are the denotations hidden by the author behind the “bodies” of signs, how justified the established classes, relationships, links, etc. The main vulnerability of the paradox is proposed to be seen in the fact that the author unjustifiably recognizes the independent out-of-communicative ability of a sign (word) to produce influences (to name, to describe, to have its own prop­erties), does not notice the communicative constants of meaning formation in the theoretical field, although inevitably uses them practically. The paradox is elimi­nated together with the removal of the classical concept of a sign as an autono­mous module “form – meaning”.

AbstractThis paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

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 Grelling’s and Russell’s 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 Grelling’s (which I present as a paradox of satisfaction) and, by analogy with Grelling’s paradox, Russell’s paradox. I extend these into truth-functional and some non-truth-functional variations, beginning with the Epimenides, Curry’s 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 Curry’s 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 Grelling’s 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.

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 Grelling’s and Russell’s 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. ¶ ...

This chapter shows that the method that Peter Unger (1979, 1980) has developed for dealing with the sorites paradox can, and perhaps should, be extended and applied to the semantic paradoxes—specifically, to Grelling’s paradox and to the liar paradox. After carefully explicating Unger’s earlier method for treating the sorites, the chapter expands on a very brief, compact argument in which he (1979) contends that, in light of certain putatively paradoxical semantic expressions, which are not obviously soritical, there are no expressions and, hence, no languages. The concluding section of the chapter identifies some important similarities between the liar paradox and the sorites.