There Are Non-circular Paradoxes (But Yablo’s Isn't One of Them!)

Cook, Roy T.

doi:10.5840/monist200689137

Cited by 31 publications

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“…, kn / ∈ R, we say that ϕ (strongly) represents R in Th. 3 The original, in German, reads: "Wir haben also einen Satz vor uns, der seine eigene Unbeweisbarkeit behauptet." (Gödel [5,p.…”

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“…One might feel inclined to believe that the naïve view on self-reference is the only kind of view on self-reference we can have in arithmetic, as Cook [3] seems to suggest. Since "the notion of stating one's own provability in the original question cannot be eliminated by the notion of being a fixed point" (Halbach and Visser [10, p. 672]), the question about the status of Henkin-Rosser sentences would be ill-posed.…”

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“…Thus, unlike Yablo's version, Visser's is not only intended to show that paradox is possible in the absence of self-reference, but also in the presence of typed truth. 10 See, for instance, Priest [24], Sorensen [29], and Cook [3].…”

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Reference in Arithmetic

Picollo

2018

The Review of Symbolic Logic

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Abstract. Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference, is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn't give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. §1. To prove his famous first incompleteness result for arithmetic, Gödel [5] developed a technique called "arithmetization" or "gödelization". It consists in codifying the expressions of the language of arithmetic with numbers, so that the language can 'talk' about its own formulae. Then, he constructed a sentence in the language that he described as stating its own unprovability in a system satisfying certain conditions, 1 and showed this sentence to be undecidable in the system. His method led to enormous progress in metamathematics and computer science, but also in philosophical logic and other areas of philosophy where formal methods became popular. Let's take a closer look.Let L be the language of first-order Peano arithmetic (PA). L contains =, ¬, ∧, ∨, →, ∀, and ∃ as logical constants, 0 as the only individual constant, S as a monadic function symbol, + and × as dyadic function symbols, and a stock of extra function symbols for recursive functions to be specified. All other logical and non-logical symbols are taken to be the usual abbreviations. We assume PA I am deeply indebted to Volker Halbach, with whom I had countless fruitful discussions on reference and self-reference over the last five years. I would also like to particularly thank Thomas Schindler, for great suggestions and encouragement, especially when it came to proofs.

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Reference in Arithmetic

Picollo

2018

The Review of Symbolic Logic

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“…The amount of implicit agreement, underlying most of its discussions, fails in the face of more complex examples or, perhaps, of more involved notions of circularity. For instance, although Yablo's paradox appears at first sight uncontroversially non-circular, this has been challenged and disputed by a series of authors, e.g., [30,32,4,10], some claiming it to possess a sort of circularity. One can construe circularity so that it applies to Yablo's paradox, but this is then a different notion from the simple one, which does not apply to it.…”

Section: Circularitymentioning

confidence: 99%

Propositional discourse logic

Dyrkolbotn

Walicki

2013

Synthese

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A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdl allows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses.

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“…We note that Y(i) ≡ ¬Tr( Y(i + 1) )∧Y(i + 2) holds for any i ∈ ω. Technically speaking, we can construct a Yablo sequence in truth theories with (nearly) full T-schema in a quite complicated way [Co06]:…”

Section: Definition 6 (Yablo's Sequence) Y(n)mentioning

confidence: 99%