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Journal articles on the topic 'Russell's paradox'

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Published: 4 June 2021

Last updated: 29 January 2023

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1

Coury, Aline Germano Fonseca, and Denise Silva Vilela. "Russell's Paradox: A Historical Study about the Paradox in Frege's Theories." Revista Brasileira de História da Matemática 19, no. 37 (October 16, 2020): 95–116. http://dx.doi.org/10.47976/rbhm2019v19n3795-116.

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For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work was done, Bertrand Russell sent him a letter pointing out a paradox, known as Russell’s paradox. It is often considered that Russell identified the paradox in Frege’s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributed significantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor’s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege’s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell’s paradox.

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2

Levine, James. "On Russell's vulnerability to Russell's paradox." History and Philosophy of Logic 22, no. 4 (December 2001): 207–31. http://dx.doi.org/10.1080/01445340210154312.

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3

Orilia, Francesco. "A Contingent Russell's Paradox." Notre Dame Journal of Formal Logic 37, no. 1 (January 1996): 105–11. http://dx.doi.org/10.1305/ndjfl/1040067319.

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4

Pal, Jagat. "Balzer's solution to Russell's Paradox." Journal of Value Inquiry 27, no. 3-4 (December 1993): 539–40. http://dx.doi.org/10.1007/bf01087703.

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5

Moorcroft, Francis. "Why Russell's Paradox Won't Go Away." Philosophy 68, no. 263 (January 1993): 99–103. http://dx.doi.org/10.1017/s0031819100040080.

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In ‘The Mind's I is Illiterate’, G. S. Miller discusses several paradoxes and paradoxical sentences which Miller claims are related by a common abuse of language. The Whiteley sentence ‘Lucas cannot consistently believe this sentence’ fails to be meaningful for want of a referent outside of the sentence for the phrase ‘this sentence’; the Liar Paradox when formulated as ‘I am lying’ is similarly disposed of when it is seen that the verb is defective and the sentence fails to refer to anything outside of itself. The same point is made concerning the Russell Paradox of the set of all sets that do not belong to themselves. The moral made is that philosophers are simply to be more careful about the laneuaee that thev are usine and then the paradoxes will go away.

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6

Urbaniak, Rafal. "Leśniewski and Russell's Paradox: Some Problems." History and Philosophy of Logic 29, no. 2 (May 2008): 115–46. http://dx.doi.org/10.1080/01445340701550817.

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7

Pinheiro, I. M. R. "Concerning the Solution to the Russell's paradox." E-LOGOS 19, no. 1 (June 1, 2012): 1–15. http://dx.doi.org/10.18267/j.e-logos.335.

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8

Cocchiarella, Nino B. "Russell's paradox of the totality of propositions." Nordic Journal of Philosophical Logic 5, no. 1 (January 2000): 25–37. http://dx.doi.org/10.1080/08066200050217977.

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9

Orilia, Francesco. "Type-free property theory, exemplification and Russell's paradox." Notre Dame Journal of Formal Logic 32, no. 3 (June 1991): 432–47. http://dx.doi.org/10.1305/ndjfl/1093635839.

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10

Fuhrmann, André. "Russell's way out of the paradox of propositions." History and Philosophy of Logic 23, no. 3 (September 2002): 197–213. http://dx.doi.org/10.1080/01445340210161017.

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11

Pollanen, Michael S. "On Balzer's small set solution to Russell's Paradox." Journal of Value Inquiry 27, no. 3-4 (December 1993): 541. http://dx.doi.org/10.1007/bf01087704.

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12

Esser, Olivier. "Inconsistency of the Axiom of Choice with the positive theory." Journal of Symbolic Logic 65, no. 4 (December 2000): 1911–16. http://dx.doi.org/10.2307/2695086.

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AbstractThe idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .

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13

Landini, G. "Zermelo and Russell's Paradox: Is There a Universal set?" Philosophia Mathematica 21, no. 2 (November 11, 2012): 180–99. http://dx.doi.org/10.1093/philmat/nks027.

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14

JOLLEY, KELLY DEAN. "LOGIC's CARETAKER-WITTGENSTEIN, LOGIC, AND THE VANISHMENT OF RUSSELL's PARADOX." Philosophical Forum 35, no. 3 (September 2004): 281–309. http://dx.doi.org/10.1111/j.1467-9191.2004.00175.x.

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15

Pitts, Andrew M., and Paul Taylor. "A note on russell's paradox in locally cartesian closed categories." Studia Logica 48, no. 3 (September 1989): 377–87. http://dx.doi.org/10.1007/bf00370830.

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16

Ahmad, Rashed. "A Recipe for Paradox." Australasian Journal of Logic 19, no. 5 (December 20, 2022): 254–81. http://dx.doi.org/10.26686/ajl.v19i5.7887.

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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.

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17

RUJEVIĆ, GORAN. "GARDNER'S PARADOX AND THEORY OF DESCRIPTIONS." Arhe 27, no. 34 (March 17, 2021): 47–60. http://dx.doi.org/10.19090/arhe.2020.34.47-60.

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Martin Gardner's two-children paradox posits two scenarios, in one we know that of two children one is a girl, and in the other we know that of two children the older one is a girl. The chances of the other child being a girl is not the same in these two scenarios, in the first being 1 in 3 while in the second they are 1 in 2. Gardner himself believed that the problem of this paradox lies in the ambiguous way the scenarios are articulated. However, it is possible to show that the original version of the paradox provides sufficient content for a meaningful explanation of these unexpected results. Inspired by comments by Leonard Mlodinow, we attempt to provide a comprehensible explanation for this counterintuitive change with help of Bertrand Russell's theory of descriptions. The difference between the two scenarios then boils down to the difference between indefinite and definite descriptions.

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18

Landini, G. "Gregory Landini. Zermelo and Russell's Paradox: Is There a Universal Set?" Philosophia Mathematica 22, no. 1 (February 1, 2014): 142. http://dx.doi.org/10.1093/philmat/nku002.

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19

Pleitz, Martin. "Paradox as a Guide to Ground." Philosophy 95, no. 2 (April 2020): 185–209. http://dx.doi.org/10.1017/s0031819120000078.

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AbstractI will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.1

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20

STEVENS, GRAHAM. "From Russell's Paradox to the Theory of Judgement: Wittgenstein and Russell on the Unity of the Proposition." Theoria 70, no. 1 (February 11, 2008): 28–61. http://dx.doi.org/10.1111/j.1755-2567.2004.tb00979.x.

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21

NOMURA, Yasushi. "Russell's Paradox and the Theory of Classes in The Principles of Mathematics." Journal of the Japan Association for Philosophy of Science 41, no. 1 (2013): 23–36. http://dx.doi.org/10.4288/kisoron.41.1_23.

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22

Klement, Kevin C. "Russell's Paradox in Appendix B of the Principles of Mathematics : Was Frege's response adequate?" History and Philosophy of Logic 22, no. 1 (March 2001): 13–28. http://dx.doi.org/10.1080/01445340110112887.

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23

Parra Moyano, JosÉ. "On the Continuity and Origin of Identity in Distributed Ledgers: Learning from Russell's Paradox." Metaphilosophy 48, no. 5 (October 2017): 687–97. http://dx.doi.org/10.1111/meta.12272.

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24

CASTAÑEDA, HECTOR-NERI. "Ontology and grammar: I. Russell's paradox and the general theory of properties in natural language." Theoria 42, no. 1-3 (February 11, 2008): 44–92. http://dx.doi.org/10.1111/j.1755-2567.1976.tb00678.x.

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25

Kanamori, Akihiro. "The Mathematical Import of Zermelo's Well-Ordering Theorem." Bulletin of Symbolic Logic 3, no. 3 (September 1997): 281–311. http://dx.doi.org/10.2307/421146.

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Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science.

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26

Yourgrau, Palle. "Sets, Aggregates, and Numbers." Canadian Journal of Philosophy 15, no. 4 (December 1985): 581–92. http://dx.doi.org/10.1080/00455091.1985.10715877.

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Frege's definition of the natural number n in terms of the set of n-membered sets has been treated rudely by history. It has suffered not one but two crippling blows. The discovery of Russell's Paradox revealed a fatal flaw in the ‘naive’ conception of set. In spite of its intuitive appeal, Frege's Basic Law V (in the context of the rest of his theory) turned out to be impermissible, leaving us only with the etiolated concept of set that survives in the axiomatic treatments initiated by Zermelo. The independence results, however, of Godel and Cohen, concerning Cantor's Continuum Hypothesis, have left us adrift in choosing between Cantorian and non-Cantorian set theories, which has induced in some logicians a skepticism in regard to the very idea of set-theoretic platonism.

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27

Orilia, Francesco. "Property theory and the revision theory of definitions." Journal of Symbolic Logic 65, no. 1 (March 2000): 212–46. http://dx.doi.org/10.2307/2586533.

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§1. Introduction. Russell's type-theory can be seen as a theory of properties, relations, and propositions (PRPs) (in short, a property theory). It relies on rigid type distinctions at the grammatical level to circumvent the property theorist's major problem, namely Russell's paradox, or, more generally, the paradoxes of predication. Type theory has arguably been the standard property theory for years, often taken for granted, and used in many applications. In particular, Montague [27] has shown how to use a type-theoretical property-theory as a foundation for natural language semantics.In recent years, it has been persuasively argued that many linguistic and ontological data are best accounted for by using a type-free property theory. Several type-free property theories, typically with fine-grained identity conditions for PRPs, have therefore been proposed as potential candidates to play a foundational role in natural language semantics, or for related applications in formal ontology and the foundations of mathematics (Bealer [6], Cocchiarella [18], Turner [35], etc.).Attempts have then been made to combine some such property theory with a Montague-style approach in natural language semantics. Most notably, Chierchia and Turner [15] propose a Montague-style semantic analysis of a fragment of English, by basically relying on the type-free system of Turner [35]. For a similar purpose Chierchia [14] relies on one of the systems based on homogeneous stratification due to Cocchiarella. Cocchiarella's systems have also been used for applications in formal ontology, inspired by Montague's account of quantifier phrases as, roughly, properties of properties (see, e.g., Cocchiarella [17], [19], Landini [25], Orilia [29]).

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28

Boolos, George. "Frege's Theorem and the Peano Postulates." Bulletin of Symbolic Logic 1, no. 3 (September 1995): 317–26. http://dx.doi.org/10.2307/421158.

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Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous.

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29

UZQUIANO, GABRIEL. "A NEGLECTED RESOLUTION OF RUSSELL’S PARADOX OF PROPOSITIONS." Review of Symbolic Logic 8, no. 2 (March 31, 2015): 328–44. http://dx.doi.org/10.1017/s1755020315000106.

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AbstractBertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell’s paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the uses to which modern descendants of Russell’s paradox of propositions have been put in recent literature.

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30

Koshlakov, Dmitriy, Marina Khokhlova, Galina Tsareva, and Galina Garbuzova. "Eponyms in science terms (Epistemological aspect)." SHS Web of Conferences 72 (2019): 01016. http://dx.doi.org/10.1051/shsconf/20197201016.

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The paper is devoted to eponyms used in scientific discourse. The concept of the eponym is borrowed from linguistic research. The term is understood from epistemological standpoint. It is stated that eponyms realize two functions in the language of science – cognitive and communicative. It is also stressed that to some extend eponyms connect two worlds – the world of ideas and the world of people, or, more specifically, the world of abstract concepts and the world of scientists, who study these abstract concepts. Historical examples (cases) demonstrating some features of functioning eponyms are given and discussed. The main historical example for the study is the history of discovering Lorentz’s transformations, which had a significant impact on forming the theory of special relativity. In addition, the paper gives the analysis of some other examples, in particular, related to such terms as Halley's comet, L'Hospital rule, Russell's paradox. It is noted that the fact of discovering some scientific object by one or another scientist in general is not the only reason for forming an eponym containing the name of this scientist. The formation of eponyms is influenced by many other factors, including social and political ones.

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31

Abdy, Muhammad, Awi Dassa, and Sri Julia Nensi. "Konsep Himpunan Fuzzy pada Paradoks Russel." Journal of Mathematics, Computations, and Statistics 2, no. 2 (May 12, 2020): 189. http://dx.doi.org/10.35580/jmathcos.v2i2.12582.

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Himpunan fuzzy menggunakan dasar logika fuzzy untuk menyatakan suatu objek menjadi anggota dengan derajat keanggotaan ( ), tetapi Logika fuzzy melanggar hukum logika biner sehingga muncul anggapan bahwa logika fuzzy memiliki masalah yang sama dengan paradoks. Tetapi nilai kebenarana logika fuzzy tergantung dari derajat keanggotaan yang dimilikinya sehingga dapat ditarik sebuah kesimpulan dari besar darajat keanggotaan tersebut, sedangkan paradoks nilai kebenarannya tidak dapat ditarik kesimpulan apapun. Paradoks merupakan bentuk kritik landasan yang bertujuan untuk mengungkapkan dan menentukan inkonsistensi atau kontradiksi yang dihasilkan dari beberapa eksperimen mental dalam matematika, salah satu paradoks yang terkenal dalam kritik landasan teori himpunan adalah paradok Russel Pemecahan paradoks Russel dengan menggunakan konsep teori himpunan fuzzy diperoleh derajat keanggotaan adalah 0.5 merupakan pernyataan setengah benar (half true) dan adalah 0.5 jugan merupakan pernyataan setengah benar (half true). Kata kunci: Logika fuzzy, himpunan fuzzy, paradoks, paradoks Russel.Fuzzy sets use the basis of fuzzy logic to declare an object to be a member with the degree of membership ( ), but fuzzy logic violates the law of binary logic so that the assumption arises that fuzzy logic has the same problem with paradox. But the true value of fuzzy logic depends on the degree of membership it has so that a conclusion can be drawn from the large membership ranks, while the paradox of its value cannot be drawn any conclusions. The paradox is a form of ground criticism that aims to express and determine the inconsistencies or contradictions that result from several mental experiments in mathematics, one of the paradoxes that is well-known in critics of set theory is Russel's paradox . The paradoxical solution of Russell by using fuzzy set theory concepts is that the degree of membership is 0.5 and is 0.5.Keywords: Fuzzy Logic, fuzzy set, paradox, Russel paradox.

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32

Thrush, Michael. "Do Meinong’s Impossible Objects Entail Contradictions?" Grazer Philosophische Studien 62, no. 1 (January 24, 2001): 157–73. http://dx.doi.org/10.1163/18756735-06201009.

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Meinong’s theory of objects commits him to impossiblia: objects which have contradictory properties. Russell famously objected that these impossiblia were apt to infringe the law of non-contradiction. Meinong’s defenders have often relied upon the distinction between internal and external negation, a defense that only works against less exotic impossiblia. The more exotic impossiblia fall victim to an argument that uses an intuitively attractive logical principle similar to the abstraction principle, but which is not subject to Russell’s paradox. The upshot is that things are not as bad as Russell claims. Some impossiblia don’t entail contradictions. Nevertheless, things are still disastrous for Meinong. Some of his impossiblia do entail contradictions.

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33

Coury, Aline Germano Fonseca, and Denise Silva Vilela. "Russell’s Paradox: a historical study about the paradox in Frege’s theories." Revista Brasileira de História da Matemática 18, no. 35 (October 22, 2020): 1–22. http://dx.doi.org/10.47976/rbhm2018v18n351-22.

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For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work wasdone, Bertrand Russell sent him a letter pointing out a paradox, known as Russell‟s paradox. It is often considered that Russell identified the paradox in Frege‟s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributedsignificantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor‟s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege‟s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell‟s paradox.

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34

Ladov, Vsevolod. "Is the Liar Paradox a semantic paradox?" ΣΧΟΛΗ. Ancient Philosophy and the Classical Tradition 13, no. 1 (2019): 285–93. http://dx.doi.org/10.25205/1995-4328-2019-13-1-285-293.

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The Liar Paradox has been widely discussed from the ancient times and preserved its importance in contemporary philosophy of logic and mathematics. At the beginning of the 20th century, F.P. Ramsey asserted that the Liar Paradox is different from pure logical paradoxes such as Russell’s paradox. The Liar Paradox is connected with language and can be considered a semantic paradox. Ramsey's point of view has become widespread in the logic of the 20th century. The author of the article questions this view. It is argued that the Liar Paradox cannot be unequivocally attributed to the semantic paradoxes and therefore Ramsey's point of view should be revised.

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35

Sutrop, U. "WITTGENSTEIN’S TRACTATUS 3.333 AND RUSSELL’S PARADOX." Trames. Journal of the Humanities and Social Sciences 13, no. 2 (2009): 179. http://dx.doi.org/10.3176/tr.2009.2.06.

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36

Sbardolini, Giorgio. "Aboutness Paradox." Journal of Philosophy 118, no. 10 (2021): 549–71. http://dx.doi.org/10.5840/jphil20211181038.

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The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then extend to propositions a modal predicative response to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles.

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37

Doets, Kees. "Relatives of the Russell Paradox." Mathematical Logic Quarterly 45, no. 1 (1999): 73–83. http://dx.doi.org/10.1002/malq.19990450107.

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38

Klement, Kevin C. "Introduction to G.E. Moore's Unpublished Review of The Principles of Mathematics." Russell: the Journal of Bertrand Russell Studies 38 (January 27, 2019): 131–7. http://dx.doi.org/10.15173/russell.v38i2.3845.

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Several interesting themes emerge from G. E. Moore’s previously unpublished review of The Principles of Mathematics. These include a worry concerning whether mathematical notions are identical to purely logical ones, even if coextensive logical ones exist. Another involves a conception of infinity based on endless series neglected in the Principles but arguably involved in Zeno’s paradox of Achilles and the Tortoise. Moore also questions the scope of Russell’s notion of material implication, and other aspects of Russell’s claim that mathematics reduces to logic.

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39

FLORIO, SALVATORE, and GRAHAM LEACH-KROUSE. "WHAT RUSSELL SHOULD HAVE SAID TO BURALI–FORTI." Review of Symbolic Logic 10, no. 4 (February 27, 2017): 682–718. http://dx.doi.org/10.1017/s1755020316000484.

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AbstractThe paradox that appears under Burali–Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be–absurdly–an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali–Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali–Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.

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PISTONE, PAOLO. "POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE." Bulletin of Symbolic Logic 24, no. 1 (March 2018): 1–52. http://dx.doi.org/10.1017/bsl.2017.43.

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AbstractThe investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

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OMORI, HITOSHI. "REMARKS ON NAIVE SET THEORY BASED ONLP." Review of Symbolic Logic 8, no. 2 (February 12, 2015): 279–95. http://dx.doi.org/10.1017/s1755020314000525.

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AbstractDialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view by showing that there is a way to handle classical negation.

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Linsky, Bernard. "The Resolution of Russell’s Paradox in Principia Mathematica." Noûs 36, s16 (October 2002): 395–417. http://dx.doi.org/10.1111/1468-0068.36.s16.15.

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Slater, Hartley. "Frege’s Hidden Assumption." Crítica (México D. F. En línea) 38, no. 113 (December 6, 2006): 27–37. http://dx.doi.org/10.22201/iifs.18704905e.2006.479.

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This paper is concerned with locating the specific assumption that led Frege into Russell’s Paradox. His understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking one could invariably replace a two-place relation with a one-place property.

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44

Bobzien, Susanne. "How to give someone Horns." History of Philosophy and Logical Analysis 15, no. 1 (April 5, 2012): 159–84. http://dx.doi.org/10.30965/26664275-01501007.

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This paper discusses ancient versions of paradoxes today classified as paradoxes of presupposition and how their ancient solutions compare with contemporary ones. Sections 1–4 air ancient evidence for the Fallacy of Complex Question and suggested solutions, introduce the Horn Paradox, consider its authorship and contemporary solutions. Section 5 reconstructs the Stoic solution, suggesting the Stoics produced a Russellian-type solution based on a hidden scope ambiguity of negation. The difference to Russell’s explanation of definite descriptions is that in the Horn Paradox the Stoics uncovered a hidden conjunction rather than existential sentence. Sections 6 and 7 investigate hidden ambiguities in “to have” and “to lose” and ambiguities of quantification based on substitution of indefinite plural expressions for indefinite or anaphoric pronouns, and Stoic awareness of these. Section 8 considers metaphorical readings and allusions that add further spice to the paradox.

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DEAN, WALTER. "INCOMPLETENESS VIA PARADOX AND COMPLETENESS." Review of Symbolic Logic 13, no. 3 (May 23, 2019): 541–92. http://dx.doi.org/10.1017/s1755020319000212.

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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.

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Monteiro, Gisele de Lourdes, and Fabiane Mondini. "Sobre a Inconsistência Lógica das Antinomias." Educação Matemática em Revista 25, no. 67 (June 30, 2020): 11–20. http://dx.doi.org/10.37001/emr.v25i67.1747.

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O texto apresenta um estudo histórico sobre os paradoxos do tipo antinomia. Como exemplo, apresentamos o paradoxo de Russell que ficou conhecido como o “paradoxo do conjunto de todos os conjuntos” ou simplesmente “paradoxo de Russell” . Trata-se de uma pesquisa qualitativa, de cunho exploratório, desenvolvida no âmbito da Educação Matemática, com o intuito de produzir conhecimento sobre esse tipo de asserção lógica, dada a complexidade que apresentam à mente humana, mediante a impossibilidade de compreensão das mesmas como verdadeiras ou falsas. A intenção é contribuir com a comunidade acadêmica mediante a exposição de um texto que se propõe a expor uma síntese compreensiva sobre o assunto, vislumbrando contribuir com discussões sobre o tema.

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Hansen, Casper Storm. "The Temperature Paradox and Russell’s Analysis of the Definite Determiner." Linguistic Inquiry 47, no. 4 (October 2016): 695–705. http://dx.doi.org/10.1162/ling_a_00227.

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Lasersohn (2005) has argued that the use of Russell’s analysis of the definite determiner in Montague Grammar, which is responsible for giving the correct prediction in the case of the temperature paradox, is also responsible for giving the wrong prediction in the case of the Gupta syllogism. In this article, I argue against this claim and show that the problem with the Gupta syllogism can be solved by making a minor addition to Montague Grammar. This solution is one that Lasersohn discusses but rejects. I show that his critique of it is ill- founded.

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TUCKER, DUSTIN, and RICHMOND H. THOMASON. "PARADOXES OF INTENSIONALITY." Review of Symbolic Logic 4, no. 3 (September 2011): 394–411. http://dx.doi.org/10.1017/s1755020311000128.

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We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink the commonly accepted methods for dealing with the logical paradoxes.

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Hansen, Kaj Børge. "CONCEPTUAL FOUNDATIONS OF OPERATIONAL SET THEORY." DANISH YEARBOOK OF PHILOSOPHY 45, no. 1 (August 2, 2010): 29–50. http://dx.doi.org/10.1163/24689300_0450103.

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I formulate the Zermelo-Russell paradox for naive set theory. A sketch is given of Zermelo’s solution to the paradox: the cumulative type structure. A careful analysis of the set formation process shows a missing component in this solution: the necessity of an assumed imaginary jump out of an infinite universe. Thus a set is formed by a suitable combination of concrete and imaginary operations all of which can be made or assumed by a Turing machine. Some consequences are drawn from this improved analysis of the concept of set, for the theory of sets and for the philosophy and foundations of mathematics.

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Weber, K. Matthias. "The Success and Failure of Combined Heat and Power (CHP) in the UK, Germany and the Netherlands: Revisiting Stewart Russell’s Perspective on Technology Choices in Society." Science & Technology Studies 27, no. 3 (January 1, 2014): 15–46. http://dx.doi.org/10.23987/sts.55313.

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Stewart Russell’s research work on combined heat and power / district heating (CHP/DH) in the UK was among the first empirical contributions to demonstrate that technological change is not just determined by seemingly objective technical and economic performance characteristics, but rather the result of social choices. His rich conceptual thinking is reconstructed in a coherent framework, and its explanatory power explored by analysing the innovation diff usion paradox of CHP/DH: in spite of very similar technical and economic characteristics, the patterns of innovation and diff usion diff er signifi cantly across countries. To this end, the evolution of CHP/DH in the UK, Germany and the Netherlands is compared. Russell’s ideas can be regarded as a predecessor of recent multi-level approaches to the analysis of socio-technical change. He put much emphasis on studying power relations for explaining the (non-) occurrence of socio-technical change; an issue that is still debated today.

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