Paradox Super Shock Shadow. Our revolutionary creme-powder eyeshadow, famous for its unique bouncy texture, delivers bold colour in a soft pearl finish in one swipe without creasing, fading or fallout. This warm satin burgundy red is gonna throw you into a spiral satin warm burgundy red. Super Shock Shadow. What's my undertone? In the more technical part of the paper, Curry carefully axiomatizes the main ingredients exploited by Kleene and Rosser and carries out a lot of non-trivial work both on the logical side and the mathematical side e.
Curry notes that a twofold construction is possible. It is interesting to note that the two ways correspond to by-now standard tools, the so-called first fixed point theorem and second fixed point theorem of combinatory logic and lambda calculus Barendregt , p. Here it is enough to recall that according to him, a remedy would be to formulate within the system the very notion of proposition, and a way to avoid the contradictions would lead to a hierarchy of canonical propositions or to a theory of levels of implication, already adumbrated by Church.
Related ideas have been developed since the seventies by Scott , Aczel , Flagg and Myhill , and others. In the s, an alternative route to solve the antinomies emerged. The role of contraction was noticed by Fitch , who observed that, in order to derive the Russell paradox one considers a function of two variables, then one diagonalizes and regards such an object as a new unary propositional function.
One has to wait until the mid eighties to see contraction-free logics used systematically in proof theory and in theoretical computer science see the entry linear logic. Fitch proposed a new approach to the problem of finding consistent combinatory logic systems, which were progressively expanded and refined over many years until Truth and membership are inductively generated by iterating rules that correspond to natural logical closure conditions and can be formalized by means of positive i. This fact implies that the generation process is cumulative and becomes saturated at a certain point, thus yielding consistent non-trivial interpretations for truth and membership.
Later he was able to strengthen his approach to include forms of negation and implication, insofar as he provided a simultaneous generation of truth and falsehood , and this actually amounts to conceive truth as a partial predicate.
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To a certain extent, the ideas of Fitch can be regarded as introducing the view that the basic predicates of truth and membership have to be partial or, if you like, three-valued. His logical analysis leads to the conclusion that the paradoxes involve meaningless statements. No formula built up with the standard connectives can be valid or a tautology, i.
Bochvar describes a version of the extended type-free logical calculus of Hilbert-Ackermann , and, in order to dispose of the paradoxes, he restricts substitution and hence the comprehension schema of the form. This makes his theory quite expressive e.
Examples of Paradox
So the contradiction is ascribed to an error in the theory of definitions, namely to the use of definitions that give rise to an infinite chain of substitutions, without converging to a result. For instance, the syllogism Barbara, usually stated in the form. Nevertheless, his work has inspired work by Aczel and Feferman Lewis and Langford are led to conclusions which are not dissimilar to those of Behmann. According to them, the paradoxes show that certain expressions do not express propositions. In this case, there is no contradiction, but we become entangled in a vicious regress p.
In general, one can create arbitrary complicated cycles and check that they can lead either to contradictions or to infinite regress; but in either case, the expression fails to converge to a definite proposition. Even after the logics developed by Russell, Zermelo and Tarski had created the theoretical means to get rid of difficulties involved in the notions of class, set, truth, definability, the paradoxes have remained alive.
This probably is due to a persistent interest in alternative formal paradigms, to the controversial features and axioms of Principia Mathematica, and to the problematic place that self-reference occupies in mathematical logic. Moreover, in NF the universal set exists.
The consistency problem for NF is still open though partial results are known concerning fragments with bounds on stratification or restriction to extensionality. Remarkably, NF refutes the axiom of choice by a classical theorem of Specker. Again, a classical result of Specker establishes the existence of a model of NF in a suitable version of simple type theory with a formal counterpart of typical ambiguity.
Paradoxes are not that far from NF. ML was defined to avoid certain weaknesses of NF e. Once more, the Lyndon-Rosser result brought about the unexpected presence of a paradox in set theory and the foundations of mathematical logic. One might think that the development of logic and set theory in the 20th century has exorcized paradoxes, and that contradictions in logical systems is a phenomenon of the years of foundational crisis only.
This is not so: paradoxes have been discovered in several recent logical systems, especially systems related to computer science.
A general type-free development of the theory of constructions as a foundation for constructive provability in logic and mathematics was originally proposed by Kreisel and Goodman. It turned out to be affected by an antinomy, and has been recently reconsidered by Dean and Kurokawa Interestingly, it has been shown that closely related systems have unexpected applications to the characterization of complexity classes Girard , Terui , Eberhard and Strahm ; on the other hand, the system is computationally complete it can interpret combinatory logic, Cantini The role of uniformity is essential in previous investigations.
This has led to the study of so-called hyperuniverses. Beginning in , there was an attempt, due to K. The Berry paradox has been related to the incompleteness phenomena also because of work going back to the sixties and the seventies in the so-called Kolmogorov complexity and algorithmic information theory. In particular, Chaitin has shown in a number of papers how to exploit randomness to prove certain limitations of formal systems see Chaitin Since Mirimanoff, Finsler and others, logicians have studied universes of set theory where circular sets exist.
However, it is only since the early eighties that a genuine mathematics of non-well-founded sets has been being developed. Using the axiom AFA of anti-foundation, direct self-reference is allowed in set theory and there exist plenty of sets solving general self-referential equations AFA was introduced by Forti and Honsell in ; for systematic development and history, see Aczel or the entry on non—wellfounded set theory.
In particular, non-well founded sets are applied to the analysis of the paradoxes, to the semantics of natural languages and to theoretical computer science see Barwise and Etchemendy , Barwise and Moss The issue this construction raises, namely whether circularity and self-reference are necessary and sufficient conditions to the appearance of paradoxes, has been further considered in Yablo see Cook for a comprehensive study on this matter, and Halbach and Zhang forthcoming for a proof without diagonal lemma.
On the other hand, category theory has been used for new approaches to paradoxes since Lawvere Philosophical motivations are strongly influential in contemporary logical investigation of paradoxes. Tarski notwithstanding, since the hierarchical approach has been somewhat superseded by new ideas that have rendered the ideal of logical and semantical closure in many respect accessible especially by means of the fixed point methods used by Kripke and Martin-Woodruff — see Martin ; for a presentation, an evaluation of their impact, as well as relations to further studies, see the entries on self-reference and axiomatic theories of truth.
We also mention the approach stemming from Herzberger, Gupta and Belnap see the entry revision theory of truth , that has connections with non-elementary parts of definability theory and set theory Welch Gupta presents an application of revision theory to the concept of strategic rationality in a certain type of finite games.
Examples of Paradox
His analysis stems from considerations about circularity, being inherently connected to the common way of understanding rational choice. Gaifman about rationality being affected by paradoxes resembling the truth theoretic paradoxes like the liar paradox see Gaifman This limit rule, which is essential to address the concept of truth, turned out to be the most critical aspect of the revision-theoretic approach to circular concepts from both the conceptual, and the complexity point of view.
Field , has generated solutions of the semantical paradoxes which combine Kripkean and revision theoretic techniques. Field has consequently developed a theory of truth with a non-classical conditional operator, which allows to express a notion of determinate truth and to state that the Liar is not determinately true. In the same direction, a considerable attention has been directed in recent literature to the so called revenge problem: typical solutions, say, of the Liar paradox, rely on notions that, if expressible in the object language, give rise to new versions of the paradox.
So the solution is only an illusion. The revenge problem can be instantiated by the so-called Strengthened Liar: informally, once we have a model which makes the Liar sentence L itself neither true nor false, and we can express this very fact, L is after all not true.
But this is the claim made by L , and hence L is true. Indexical solutions of the Liar have been developed in several contributions, e. The idea is that the Liar paradox does not involve sentences, but specific occurrences of sentences , i. Besides the model-theoretic side, axiomatic investigations of truth and related paradoxes have become increasingly important since the seminal papers of Friedman and Sheard , Feferman Since the year , this research thread has been intensively studied with various aims, from proof theoretic analysis to philosophical discussion of minimalism for a survey of the varieties of truth theoretic systems and appropriate references, see the entry on axiomatic theories of truth and the recent monographs of Halbach , Horsten ; see also the papers Feferman , Fujimoto , and , Horsten and Leigh forthcoming, Leigh and Rathjen and , Leigh , a, b, Enayat and Visser The axiomatic study of epistemic notions has greatly benefited from application of techniques used for proving incompleteness and indefinability results since the early sixties: they have yielded negative results Kaplan and Montague , Montague , Thomason and established an interesting link with the Surprise test paradox.
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The situation may have also been changed by the study of possible world semantics for modal notions conceived as predicates in Halbach, Leitgeb and Welch However this is open to debate and experimentation: for instance it is argued in Halbach and Welch that the predicate approach to necessity is a viable route—insofar as the expressive power is considered—provided one resorts to languages that involve both a truth predicate and the necessity operator.
Also, a number of solutions have been proposed, which rely on the use of paraconsistent logics Priest or substructural logics see the entry paraconsistent logic and Beall for a recent work relating classical and paraconsistent theories, as well as the entry substructural logics and Mares and Paoli The investigation of semantical and set-theoretic paradoxes in infinite-valued logic—which was pioneered by Mow Shaw-Kwei and Skolem in —has received a new impulse with contributions by Hajek, Shepherdson and Paris and Hajek Typically, in these papers basic results from mathematical analysis are applied e.
It is worth mentioning that Leitgeb has given a consistency proof for a probabilistic theory of truth based upon unrestricted T-schema by making use of the Hahn-Banach Theorem. Theories of naive truth—as based on the unrestricted biconditional and on a logic without contraction—are to be found in the literature, e. Ripley , instead, presents an alternative approach based on a non-transitive logical system. Besides tools from algebra and analysis, logical investigations about paradoxes and truth in particular have recently exploited ideas from graph theory see Cook , Rabern, Rabern and Macauley , Schindler and Beringer The items occurring in this list mainly concern the primary literature on paradoxes in the period — This list contains i items cited in the final section; ii items related to developments of paradoxes after the Second World War; iii critical historical papers.
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Introduction 2. Paradoxes: early developments — 2.
- II. Examples of Paradox.
- General Paradoxes!
- Flowers: How They Changed the World.
Difficulties involving ordinal and cardinal numbers 2.