The logical paradoxes

2020 ◽  
Vol 15 (2) ◽  
pp. 61-92
Author(s):  
Hristo Manev
Keyword(s):  
1916 ◽  
Vol 25 (1) ◽  
pp. 16 ◽  
Author(s):  
Theodore de Laguna
Keyword(s):  

1953 ◽  
Vol 18 (3) ◽  
pp. 233-233
Author(s):  
Robert L. Stanley
Keyword(s):  

The usual logical paradoxes, which stem from Russell's class R0 (i.e., and its relatives R1, R2, and so on (i.e., , , and so on, correspondingly), are contradictions (Ri ϵ Ri) ≡ ~(Ri ϵ Ri), derivable under unrestricted rules of inference. No contradiction appears to obtain.


2003 ◽  
Vol 9 (1) ◽  
pp. 3-25 ◽  
Author(s):  
György Serény

The fact that Gödel's famous incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of the community of logicians. Indeed, almost every more or less formal treatment of the theorem makes a reference to this connection. Gödel himself remarked in the paper announcing his celebrated result (cf. [7]):The analogy between this result and Richard's antinomy leaps to the eye;there is also a close relationship with the ‘liar’ antinomy, since … we are… confronted with a proposition which asserts its own unprovability.In the light of the fact that the existence of this connection is commonplace it is all the more surprising that very little can be learnt about its exact nature except perhaps that it is some kind of similarity or analogy. There is, however, a lot more to it than that. Indeed, as we shall try to show below, the general ideas underlying the three central theorems concerning internal limitations of formal deductive systems can be taken as different ways to resolve the Liar paradox. More precisely, it will turn out that an abstract formal variant of the Liar paradox, which can almost straightforwardly inferred from its original ordinary language version, is a possible common generalization of (both the syntactic and semantic versions of) Gödel's incompleteness theorem, the theorem of Tarski on the undefinability of truth, and that of Church concerning the undecidability of provability.


1981 ◽  
Vol 31 (3) ◽  
pp. 382
Author(s):  
Douglas Dunsmore Daye ◽  
Douglas M. Burns
Keyword(s):  

Philosophy ◽  
2015 ◽  
Author(s):  
Matthew A. Benton

Herbert Paul Grice (b. 1913–d. 1988) was a British philosopher and linguist, and one of the pivotal figures in philosophy during the 20th century. He wrote in many areas of philosophy, including the metaphysics of personal identity, logical paradoxes, the analytic/synthetic distinction, the philosophy of perception, philosophical psychology, and ethics. He also wrote on historical figures such as Aristotle, Descartes, Hume, and Kant. But his most significant contributions came in philosophy of language and mind, on meaning, intention, presupposition, conversation, and the theory of communication. Grice argued for an intention-based theory of meaning, and he was the first to illustrate the distinction between what came to be called semantic and pragmatic meaning, that is, between what a speaker’s utterance (or its utterance “type”) means in the abstract, and what else a speaker can mean by uttering it in a particular context. Grice highlighted this by an appeal to his framework of the Cooperative Principle and its Conversational Maxims, which are plausibly assumed by conversational participants and provide mechanisms for the ways in which speakers can “conversationally implicate” something beyond the literal meaning of what they say, and for how hearers can recover those “implicatures.’” Grice’s enduring influence on these topics helped found the burgeoning discipline in philosophy of language and linguistics now known as “pragmatics” (compare the Oxford Bibliographies in Philosophy article on “Pragmatics”).


2002 ◽  
Vol 8 (2) ◽  
pp. 185-245 ◽  
Author(s):  
Fairouz Kamareddine ◽  
Twan Laan ◽  
Rob Nederpelt

AbstractIn this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910–1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Church's own simply typed λ-calculus (λ→C) and we finish by comparing RTT, STT and λ→C.


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