Lycan, What, exactly, is a paradox? Quine offered his classic characterization of the notion of paradox, a taxonomy for paradoxical arguments and some vocabulary for discussing them. The simpler characterization will have the virtue or the flaw as might be of making paradox a matter of degree.
For Quine, a paradox is an apparently successful argument having as its conclusion a statement or proposition that seems obviously false or absurd.
What is paradoxical is of course that, if the argument is indeed successful as it seems to be, its conclusion must be true. On this view, to resolve the paradox is i to show either that and why despite appearances the conclusion is true after all, or that the argument is fallacious, and ii if the former, to explain away the deceptive appearances. Quine divides paradoxes into three groups. We decide that a paradox is veridical when we look carefully at the argument and it convinces us, i.
We decide that a paradox is falsidical when we look carefully at the argument and spot the fallacy. Oddly, Quine does not mention a third related category, the obverse of a veridical paradox: the argument in question could have an obviously false or self-contradictory conclusion, yet rest on no error of reasoning however subtle — so long as it has a premiss that looks for all the world true until we let the argument itself show us that, and how, the premiss is false after all.
There is still no fallacy, but only the innocent-seeming premiss that there is a barber who shaves all and only those who do not shave themselves. Let me explain. All a valid deductive argument shows, just in virtue of its validity, is that a certain set of propositions is internally inconsistent.
If all we know about an argument. For unless we are independently moved to accept those premisses, their jointly implying C is of little interest; and even if we do already accept them, seeing that they imply C may make us reconsider one or more of them, rather than inclining us any the more strongly to endorse C as well.
More generally, then, a Paradox I mark my own proposed usage with the capital letter is an inconsistent set of propositions, each of which is very plausible. There is a further complication. De Morgan divides through by a number that is covertly equal to 0; elsewhere Lycan , I have argued that when one is reasoning in English rather than in a truth-functional calculus, reliance on the rule Modus Ponens can lead from perfectly acceptable premisses to contradictory conclusions and so must be rejected.
Now, to circumvent this complication and keep the terminology neat, let us dispense with every even faintly dubious principle of reasoning. And, owing to the semantical completeness of the propositional calculus, any Paradox is provably as well as model-theoretically inconsistent.
First, there is the question of relevant implication. Relevance logicians 8 brand the truth-functional calculus as libertine, charging that some of the inferences it sanctions notably Disjunctive Syllogism introduce informational irrelevancies and are therefore not ones that English speakers would or should make. But however we feel about this as a complaint against the analysis of English or any other natural language in truth-functional terms, it does not apply to my idea of reconstructing paradoxes into Paradoxes; I am here using the propositional calculus and its libertine notion of validity only as a tool for strict truth-preservation in inferential moves between propositions that are already formulated in the truth-functional idiom.
Disjunctive Syllogism may not be a valid principle in the logic of English any more than are Antecedent-Strengthening and Modus Ponens , but it is valid trivially and by definition for the tilde and the vel.
Second, paraconsistent systems have been offered as rivals to standard contradiction-shunning logic, 9 and in particular dialetheists led by Graham Priest , have maintained that there are actually true contradictions. But this view is no opponent of my characterization of a Paradox. It merely takes a refreshing attitude toward Paradoxes once they are identified.
If, following Tarski, we were to try to assign levels of language to this pair of utterances, then how could we do it? The two principles which Martin takes to categorise the Liar we have just seen, namely. Tarski, in these terms, took claim S to be incorrect. But one also might claim that T is incorrect, maybe because there are sentences without a truth value, being meaningless, or lacking in content in some other way, as is held by the theorists mentioned before.
Fourthly it is possible to argue that S and T are correct, but really compatible. Martin sees this happening as a result of some possible ambiguity in the terms used in the two principles. We can isolate a further, fifth option, although Martin does not consider it. That option is to hold that both S and T are incorrect, as is done by the tradition which holds that it is not sentences which are true or false. Following the second world war, because of this sort of thing, it became more common to think of semantical notions as attached not to sentences and words, but to what such sentences and words mean Kneale and Kneale , pf.
But it was shown by Thomason, following work by Montague, that the same sorts of problems can be generated even in this case. We can create self-referential paradoxes to do with statements and propositions which again cannot be obviously escaped Thomason , , And the problems are not just confined to the semantics of truth and falsity, but also arise in just the same way with more general semantical notions like knowledge, belief, and provability.
In recent years, the much larger extent of the problems to do with self-reference has, in this way, become increasingly apparent.
Such representational treatments of the attitudes have found many advocates; and it is probably true that some of their proponents have not been sufficiently alert to the pitfalls of self-reference even after those had been so clearly exposed in Montague … To such happy-go-lucky representationalists, Thomason is a stern warning of the obstacles that a precise elaboration of their proposals would encounter.
Asher and Kamp go on to explain the general method which achieves these results Asher and Kamp , p87 :. What is also true, and even provable in such a system is that, if it is consistent then a a certain specific self-referential formula is not provable in the system, and b the consistency of the system is not provable in the system.
This means the consistency of the system cannot be proved in the system unless it is inconsistent, and it is commonly believed that the appropriate systems are consistent. But if they are consistent then this result shows they are incomplete, that is there are truths which they cannot prove. And that fact has been fed into the very large debate about our differences from, even superiority over mechanisms see e.
Penrose But if we consider the way many people would argue about, for instance,. If that sentence is provable then it is true, since provability entails truth; but that makes it unprovable, which is a contradiction. Hence it must be unprovable. But by this process we seem to have proved that it is unprovable — another contradiction!
So, unless we can extricate ourselves from this impasse, as well as the many others we have looked at, we would not seem to be too bright. Or does this sort of argument show that there is, indeed, no escape?
The intractability of the impasse here, and the failure of many great minds to make headway with it, has lead some theorists to believe that indeed there is no escape. Notable amongst these is Priest compare Priest , who believes we must now learn to accept that some contradictions can be true, and adjust our logic accordingly.
It is thought that, if this traditional rule were removed from logic then, at least, any true contradictions we find, e. But there is the broader, philosophical question, as well, about whether a switch to a different logic does not just change the subject, leaving the original problems unattacked.
It seems we may have just turned our backs on the real difficulty. There have been developments, in the last few years, which have shown that the previous emphasis on paradoxes involving self-reference was to some extent misleading. For a family of paradoxes, with similar levels of intractability, have been discovered, which are not reflexive in this way. It was mentioned before that a form of the Liar paradox could be derived in connection with the pair of statements.
For, if what Socrates is saying is true, then, according to the former, what Plato is saying is false, but then, according to the latter, what Socrates is saying is false. On the other hand, if what Socrates is saying is false then, according to the former, what Plato is saying is true, and then, according to the latter, what Socrates is saying is true.
Yablo asked us to consider an infinite sequence of sentences of which the following is representative Yablo :. Hence for no n is S n true. But that means that S 1 is true, S 2 is true, etc; in fact it means every statement is true, which is another contradiction. So no student is thinking an untruth. But if some student is consequently thinking a truth, then some student behind them is thinking an untruth, which we know to be impossible.
Indeed every supposition seems impossible, and we are in the characteristic impasse. Gaifman has worked up a way of dealing with such more complex paradoxes of the Liar sort, which can end up denying the sentences in such loops, chains, and infinite sequences have any truth value whatever. If, in the course of applying the evaluation procedure, a closed unevaluated loop forms and none of its members can be assigned a standard value by any of the rules, then all of its members are assigned GAP in a single evaluation step.
But the major question with such approaches, as before, is how they deal with The Strengthened Liar. Surely there remain major problems with. For more discussion of the logical paradoxes, see the following articles within this encyclopedia:.
Barry Hartley Slater Email: slaterbh cyllene. Logical Paradoxes A paradox is generally a puzzling conclusion we seem to be driven towards by our reasoning, but which is highly counterintuitive, nevertheless.
Moving to Modern Times Between the classical times of Aristotle and the late nineteenth century when Cantor worked, there was a period in the middle ages when paradoxes of a logical kind were considered intensively. Some Recent Logical Paradoxes It was developments in other parts of mathematics which were integral to the discovery of the next logical paradoxes to be considered. Mackie disagreed with Ramsey to a certain extent, although he was prepared to say Mackie , p : The semantical paradoxes…can thus be solved in a philosophical sense by demonstrating the lack of content of the key items, the fact that various questions and sentences, construed in the intended way, raise no substantial issue.
Specifically he held that statements about all the members of certain collections were nonsense compare Haack , p : Whatever involves all of a collection must not be one of a collection, or, conversely, if, provided a certain collection had a total it would have members only definable in terms of that total, then the said collection has no total.
And one may very well ask, with respect to What I am now saying is false, for instance, whether this has any sense, or involves a substantive issue, as Mackie would have it see also Parsons The Unstrengthened Liar comes in a whole host of variations, for instance: This very sentence is false, or Some sentence in this book is false, if that sentence is the only sentence in a book, say in its preface.
The two principles which Martin takes to categorise the Liar we have just seen, namely S There is a sentence which says of itself only that it is not true, and T Any sentence is true if and only if what it says is the case.
But if we consider the way many people would argue about, for instance, this very sentence is unprovable, then our abilities as humans might not seem to be too great. For many people would argue: If that sentence is provable then it is true, since provability entails truth; but that makes it unprovable, which is a contradiction.
A Contemporary Twist There have been developments, in the last few years, which have shown that the previous emphasis on paradoxes involving self-reference was to some extent misleading. It was mentioned before that a form of the Liar paradox could be derived in connection with the pair of statements What Plato is saying is false, What Socrates is saying is true, when Socrates says the former, and Plato the latter.
References and Further Reading Asher, N. Halpern ed. Asher, N. Chierchia, B. Partee, and R. Turner eds. Properties, Types and Meaning 1. Beall, J. Copi, I. Macmillan, New York. Gaifman, H.
Goldstein, L. Grattan-Guinness, I. Haack, S. Hallett, M. Keefe, R. Kneale, W. Lavine, S. On the other hand, the consistency of the system is ruined by extensionality and this could be counted as an additional paradox! Interestingly, it has been shown that closely related systems have unexpected applications to the characterization of complexity classes Girard , Terui ; on the other hand, the system is computationally complete it can interpret combinatory logic, Cantini Besides tools from algebra and analysis, logical investigations about paradoxes have recently applied graph theory see Cook , Rabern, Rabern and Macauley , Beringer and Schindler , Hsiung : a basic idea is the attempt to grasp in geometric terms the patterns of paradox , their structural features.
It is further conjectured that a solution to the characterization problem for dangerous rfgs amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. This route is independently developed in Rossi by exploring the wide range of semantic behaviours displayed by paradoxical sentences, and providing a unified theory of truth and paradox.
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.
Introduction 2. Paradoxes: early developments — 2. Difficulties involving ordinal and cardinal numbers 2. Paradoxes, predicativism and the doctrine of types: — 3. Logical developments and paradoxes until 4.
Paradoxes: between metamathematics and type-free foundations — 5. A glance at present-day investigations 6. Introduction Between the end of the 19th century and the beginning of the 20th century, the foundations of logic and mathematics were affected by the discovery of a number of difficulties—the so-called paradoxes—involving fundamental notions and basic methods of definition and inference , which were usually accepted as unproblematic.
Difficulties involving ordinal and cardinal numbers The earliest modern paradoxes concerned the notions of ordinal and cardinal number. From each answer, the opposite follows. Likewise, there is no class as a totality of those classes which, each taken as a totality, do not belong to themselves.
From this I conclude that under certain circumstances a definable collection does not form a totality. It contains a new paradox credited to Grelling with a semantical flavor see also the entry self-reference : To each word there corresponds a concept, that the very word designates, and which applies to it or does not apply; in the first case, we call the word autological , else heterological.
Logical developments and paradoxes until In the period until , the problem of paradoxes led naturally to and was subsumed under the investigation of logical calculi its final by-product being the Hilbert-Ackermann textbook of Bibliography Primary Sources: — The items occurring in this list mainly concern the primary literature on paradoxes in the period — Bernstein, F.
Bochvar, D. Bergmann, of the Russian original, which appeared in Mathematicheski Sbornik , 4 46 : — Borel, E. Brouwer, L. Burali-Forti C. Rendiconti del Circolo Matematico di Palermo , , —; English translation in van Heijenoort , — Cantor, G. Zermelo ed. Meschkowski and W. Nilson eds. Carnap, R. Church, A. McCall ed. Curry, H. Finsler, P. Fitch, F. Frege, G. Grelling, K. Hessenberg, G. Hilbert, D. Kleene, S.
Levi, B. Lewis, C. Mirimanoff, D. Peano, G. Quine, W. Richard, J. Rosser, J. Russell, B. Tarski, A.
A source book in mathematical logic — , Cambridge, Mass. Weyl, H. Whitehead, A. Zermelo, E. Zygmunt, J. Recent Sources 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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Fujimoto, J. Galinon, T. Achourioti eds. Cook, R. Coquand, T. Prawitz, B. Skyrms, D. Westerstahl eds. Dean, W. Piecha and P. Schroeder-Heister eds. De Vos, M. Bezhanishvilii eds. Douven I. Eberhard, S. Achourioti, H. Galinon, J. Fernandez, K. Fujimoto eds. Enayat, A. Feferman, S. Boffa and D. Field, H. Flagg, R. Annals of Pure and Applied Logic , 33— Forti, M. Friedman, H. Gaifman, H. Vardi ed. Garciadiego, A. Geach, P. Girard, J. USSR Izv. Grue, K. Anderson and M. Gupta, A. Hajek, P.
Halbach, V. Analysis, 53— Hinnion, R. Holmes, M. Horsten, L. Hsiung, M. Irvine, A. Gabbay and J. Wood eds. Kahle, R. Kaplan, D. Kikuchi, M. Klement, K. Kreisel, G. Kripke, S. Kritchman, S. Kurahashi, T. Kyburg, H. Lawvere, F. Hilton ed. Leigh, G. Leitgeb, H. Libert, T. Link, G. Linsky, B. Calude ed. Malitz, R. Mancosu, P. Mras, P. Weingartner, and B. Ritter eds. Haaparanta ed. Mares, E.
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