Philosophical Implications of Logical Paradoxes

2007 ◽  
pp. 131-142 ◽  
Author(s):  
Roy A. Sorensen
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.


1979 ◽  
Vol 38 (2) ◽  
pp. 373-391 ◽  
Author(s):  
Laurence Goldstein

There can be no doubt that the existence of paradoxes has stimulated vigorous and highly productive activity in philosophy and in logic. Take two famous examples: The Liar Paradox, which arises from a sentence such asL This statement is false.and the Russell Paradox which arises from the sentenceR The class of all classes which are not members of themselvesis a member of itself.The first of these has been a source of anguish for over 2,000 years. The second has engaged the serious attention of logicians for over three quarters of a century. Investigation of paradoxes of this sort has spawned whole new fields of study, such as technical semantics and axiomatic set theory.It has been claimed that legal reasoning is infected with paradoxes and that these paradoxes are similar in structure to those, like the two we have cited, which are of interest to the logician. If this claim were true one of two consequences would follow. Either the jurisprudent would face what would in all likelihood be a protracted struggle with these legal paradoxes resulting, perhaps, in significant additions to legal theory, or else, if these paradoxes were sufficiently similar to those of the logician, he might try to utilise the logician's results to solve his own legal puzzles.The first alternative, though attractive to a theoretician, may appear rather dismal to those engaged in the business of law. Whereas reflection on the logical paradoxes can lead to only more refined abstractions—the philosopher's meat and drink—legal theory is rather intimately connected with practical affairs.


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