Journal articles 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.

Abstract This study empirically demonstrates how governance-type organized crime groups (OCGs) operate as an enforcer against volume crimes in the communities they control and argues that their ability to mitigate volume crimes forms an integral component of controlling their territory in the long term. This is because the costs incurred from deterring other crimes are offset by the tangible and intangible revenues that it facilitates. Indeed, combating volume crimes fosters an environment in which OCGs can conduct their activities unfettered by other criminals and law enforcement agencies, safeguard those businesses that pay them protection and curry favour amongst the population. Consequently, the present study verifies the validity of the security governance paradigm by conducting an econometric analysis of 11 different volume crimes.

I began my last set of reviews by expressing doubts about the speculative literary prehistory in Mary Bachvarova'sFrom Hittite to Homer(G&R64 [2017], 65). Near Eastern antecedents also feature in Bruno Currie'sHomer's Allusive Art. Currie displays more methodological awareness and more intellectual suppleness: he recognizes the possibility of parallels arising independently (213–15), but denies that his examples can be coincidental, while acknowledging that this confronts us with a ‘glaring paradox’ (217). To be fair, he has a point in this instance, and in many of his other case studies; and his overarching argument is beautifully conceived. On the debit side of the account, there are methodological tautologies: that we should accept conclusions if there is ‘sufficient warrant’ (29) or the evidence is ‘sufficiently compelling’ (174), and not bring charges ‘too quickly’ (32), follows from the meaning of ‘sufficient’ and ‘too’. Adverbial IOUs of indeterminate creditworthiness like ‘arguably’ (×45) are not an adequate substitute for arguments (cf.G&R63 [2016], 235). ‘Of course’ (×50) is superfluous if it refers to what is genuinely a matter of course, and misleading if not. And, of course, Currie's use of scare quotes is arguably too extravagant. Some weaknesses are more substantive. For example, when trying to determine theIliad’s relation to a hypothetical antecedent (designated ‘*Memnonis(Aethiops)’), Currie maintains that ‘the short life of Achilleus arguably [!] has the status of “fact” [!] because the audience knows – through familiarity with an earlier version – which way Achilleus is ultimately going to make up his mind’ (62). Regardless of their familiarity with any hypothetical earlier version, the audience of theIliadknows that Achilles' life will be short because theextantversion establishes it as a fact when it makes this a presupposition of the exchange between Achilles and Thetis (Il.1.352, cf. 416–18, 505–6). From 9.410–5 we might infer that what is presupposed in Book 1 results from Achilles' prior choice: if so, the change of mind implied in his answer to Odysseus is implicitly retracted in his response to Ajax (650–5). ‘The choice that Achilleus is actually going to make only after the death of Patroklos' (62) had therefore already been made. It is disappointingly reductive to say that ‘Diomedes plays out the part of Gilgamesh in this episode ofIliadV, but for this part of theIliadDiomedes serves as a “stand-in” [!] for Achilleus, and Achilleus in theIliadmore widely plays out the part of Gilgamesh’ (197): Homer's characters are not tokens, and Diomedes is always, and distinctively, himself. The point of puttingOd. 19.96–604 alongside an alternative version manufactured to be parallel but different (47–55) eluded me entirely. ‘I do not see’, says Currie, ‘what is gained by refusing to speak of allusion to a particular poem’ (102). Nor do I; and some of his parallels seemed compelling, however hard I tried to resist. Nevertheless, we must balance the loss in refusing to speak of allusion against the risks of building on foundations that may have too high a proportion of sand. Currie has written a brilliant and subtle book. Its contents will need careful sifting.

If a sentence says of itself that it is not true, there is little choice but to take it for its word. But what if a sentence says of itself that it is true, or, in any case, provable? Logically, there is an inconsistency in proving this statement. Martin Löb and Haskell Curry were two mathematical logicians who sought to examine this question.

Abstract Based on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic that does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s.