Academic literature on the topic 'Curry's paradox'

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.

It is well known that combinatory logic with unrestricted introduction and elimination rules for implication is inconsistent in the strong sense that an arbitrary term Y is provable. The simplest proof of this, now usually called Curry's paradox, involves for an arbitrary term Y, a term X defined by X = Y(CPy).The fact that X = PXY = X ⊃ Y is an essential part of the proof.The paradox can be avoided by placing restrictions on the implication introduction rule or on the axioms from which it can be proved.In this paper we determine the forms that must be taken by inconsistency proofs of systems of propositional calculus based on combinatory logic, with arbitrary restrictions on both the introduction and elimination rules for the connectives. Generally such a proof will involve terms without normal form and cut formulas, i.e. formulas formed by an introduction rule that are immediately removed by an elimination with at most some equality steps intervening. (Such a sequence of steps we call a “cut”.)The above applies not only to the strong form of inconsistency defined above, but also to the weak form of inconsistency defined by: all propositions are provable, if this can be represented in the system.Any inconsistency proof of this kind of system can be reduced to one where the major premise of the elimination rule involved in the cut and its deduction must also appear in the deduction of the minor premise involved in the cut.We can, using this characterization of inconsistency proofs, put appropriate restrictions on certain introduction rules so that the systems, including a full classical propositional one, become provably consistent.

In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it

Consider the following sentence schema:This sentence entails that ϕ.Call a sentence which is obtained from this schema by the substitution of an arbitrary, contingent sentence, s, for ϕ, the sentence CS (for ‘Curry’s Sentence’). Thus,(CS) This sentence entails that s.Now ask the following question: Is CS true?One sentence classically entails a second if and only if it is impossible for both the first to be true and the second to be false. Thus ‘Xanthippe is a mother’ entails ‘Xanthippe is female’ if and only if it is impossible for both ‘Xanthippe is a mother’ to be true and ‘Xanthippe is female’ to be false. CS makes a claim about a purported entailment. Thus, CS is true if and only if it is impossible for both the sentence it mentions as entailing a second to be true and the sentence it mentions as being entailed by the first to be false. In other words, CS is true if and only if it is impossible for both CS to be true and s to be false. In yet other words, CS is false if and only if it is possible for both CS to be true and s to be false.

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.

Se busca relacionar a la paradoja de Curry con la de El Mentiroso y la de Bertrand Russell. Para ello, se presenta cada paradoja acompañada de una de sus múltiples soluciones. Después de esta presentación, se realizan las comparaciones entre las paradojas expuestas para constar que si bien todas estas paradojas hacen uso de la autorreferencia o del predicado de ser miembro de sí mismo (autopertenecencia) se distinguen en que la de Curry no usa negaciones ni deriva en contradicciones. Además, solo la de Curry hace uso del principio de contracción. Finalmente, previa distinción entre solución y disolución (la primera consiste en elaborar una herramienta y explicar las causas de la aparición del problema; mientras que la segunda busca limitar o prohibir que ciertas expresiones puedan ser siquiera elaboradas), en el caso de la paradoja de Curry se expone la disolución (o bloqueo) presentada por Łukowski (2011) que consiste básicamente en conjuntar la expresión de Curry con un enunciado falso. Se sitúa esta propuesta algo artificiosa pero ingeniosa dentro del campo de la lógica clásica pues solo apela a tablas de verdad y ciertas leyes básicas de conectores lógicos. En la segunda parte, se parte tratando la paradoja de Curry desde un enfoque clásico. Se llegan a interesantes resultados en este punto como, por ejemplo que A (A B) puede reducirse a A∧B o que básicamente la paradoja de Curry pide algo en contra de las leyes de la tabla de verdad del condicional pues en ningún caso un condicional se reduce a su antecedente. Asimismo, se encuentra cierta repetición cíclica cuando se opera el condicional de Curry siendo A 0 B equivalente a A y siendo A n+1 B equivalente a la expresión (A n B) B. De tal modo que, cuando n=0, A 1 B sería equivalente a (A 0 B) B, es decir, a A B, y cuando n=1, A 2 B sería equivalente a (A 1 B) B, es decir, a (A B) B. Y así sucesivamente se obtiene: A B, (A B) B, [(A B) B] B, etc. Después, se presentan las soluciones que se han planteado desde la lógica no-clásica, en especial, las lógicas paracompleta y paraconsistente. Para el primer caso se sigue la propuesta de Field (2008), el cual a su vez toma resultados previos de Kripke (1975) y Gupta y Belnap (1993), llegando a constatar la no-verdad y no-falsedad del enunciado de Curry y para el segundo caso en base a Priest (2006a) se modifica la relación condicional haciendo uso del concepto de mundos no normales que son aquellos en los cuales, de acuerdo a Priest (1992) los teoremas lógicos no son verdaderos. Ambos de estos intentos son elaboraciones técnicas y con alto grado de complejidad. Sin embargo, se advierte con Beall y Shapiro (2018) que ambos intentos se han visto frustrados pues no logran eludir del todo a la escurridiza paradoja de Curry. Finalmente, en la parte tercera, se plantea las bases para una comprensión clarificadora sobre la paradoja de Curry. Como se sabe, la paradoja de Curry se plantea al suponer que si un condicional es cierto entonces B. Con esto se consigue probar cualquier proposición. Pues bien, primero, se realizan las observaciones sobre las diversas partes de la paradoja de Curry. De este modo, cuestionamos premisa, desarrollo y conclusión de esta paradoja. En cuanto a la premisa, Curry es acusado de ser un enunciado un tanto ambiguo (o incluso irrelevante); en cuanto al desarrollo, se observa la obsesión por desarrollar a Curry solo usando la prueba condicional y no las otras dos; finalmente, en cuanto a la conclusión, se indica la posibilidad de que tal vez no se llegue a probar cualquier cosa habida cuenta que, después de todo, la conclusión B también está incluida en el enunciado original de Curry. Enseguida, se utiliza el recurso de la lógica relevante que parecía ser un buen aliciente para acabar con esta paradoja. Así, se sostiene que esta lógica relevante buscaba que los razonamientos válidos compartan variables entre premisas y conclusiones y, además, que la premisa sea usada para derivar tal conclusión. Se observa cómo esta lógica le hace frente a las paradojas de la implicación y este tema particularmente nos interesa pues hay cierta semejanza entre el enunciado de Curry y estas paradojas. Sin embargo, después de todo, la esperanza era en vano: la lógica relevante tampoco logra frenar a la paradoja de Curry. Al final, se aborda el tema de la pragmática para tratar de interpretar esta paradoja. Se utiliza el marco teórico de Paul Grice (1975) para intentar plantear un enfoque propio de la paradoja de Curry. Se llega a la conclusión de que se trata de una implicatura conversacional que burla la máxima de cantidad y que, al parecer, solo busca indicar la seguridad que tenemos en un cierto enunciado B. También se señala que este aparente condicional se trataría de una conjunción encubierta.
Tesis

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. ¶ ...

Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrinsic interest of their own. These include a transfinite paradox of denotation, and variations on the Liar paradox, including the Truth-Teller, Curry’s paradox, and paradoxical Liar loops. The transfinite paradox of denotation shows the need to accommodate limit ordinals. The Truth-Teller, like the Liar, exhibits semantic pathology-but, unlike the Liar, it does not produce a contradiction. The distinctive challenge of the Curry paradox is that it seems to allow us to prove any claim we like (for example, the claim that 2+2=5). Paradoxical Liar loops, such as the Open Pair paradox, extend the Liar paradox beyond single self-referential sentences. The chapter closes with the resolution of paradoxes that do not exhibit circularity yet still generate contradictions. These include novel versions of the definability paradoxes and Russell’s paradox, and Yablo’s paradox about truth.