Revenge of the Liar: New Essays on the Paradox
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My profile My library Metrics Alerts. Sign in. Get my own profile Cited by View all All Since Citations h-index 29 22 iindex 61 David Ripley Monash University Verified email at monash. Thomas Forster University of Cambridge Verified email at dpmms. Verified email at uconn. Articles Cited by Co-authors. Australasian journal of philosophy 78 4 , , Journal of philosophical logic 41 3 , , Logical Consequences.
Kluwer Academic Publishers, to appear , She trained them in the art of espionage, tested their skills in weaponry, surveillance, and sabotage, and sharpened their minds with nerve-wracking psychological games. As they grew older they came to question her motives, her methods—and her sanity. Now twenty-six years old, the twins are trying to lead normal lives.
A twisted trail leads from the CIA, to the KGB, to an underground network of global assassins where hunters become the hunted. But the radical poisonof this paradox is not eluc i-dated till this essay comes. Liars' Paradox. Click Download or Read Online button to get logic language and the liar paradox book now. The contingent versions of the Liar Paradox are going to be troublesome because, if the production of the paradox does not depend only on something intrinsic to the sentence but also depends on what circumstances occur in the world, then there needs to be a detailed description of when those circumstances are troublesome and when they are not, and why.
It would be ideal if we had a solution to both the Liar Paradox and Curry's Paradox, another paradox that turns on self-reference. Haskell Curry's very interesting paradox concerns sentences such as C which contains themselves:. The sentence C above contains C. This can lead to a paradoxes because one instance of Tarski's Convention T is the equivalence. Now let's begin to construct a multi-step Conditional Proof.
Assume that C is true. Hence, by Conditional Proof, we have established that. So, we have proved a contradiction. The outcome is a self-referential paradox that does not rely on negation, as the Liar Paradox does. Ask yourself whether the first person's sentence in the sequence is true or false. To produce the paradox it is crucial that the line of speakers be infinite. Notice that no sentence overtly refers to itself. There is controversy in the literature about whether the paradox actually contains a hidden appeal to self-reference or circularity.
See Beall for more discussion. To summarize, an important goal for the best solution, or solutions, to the Liar Paradox is to offer us a deeper understanding of how our semantic concepts and principles worked to produce the Paradox in the first place, especially if a solution to the Paradox requires changing them. We want to understand the concepts of truth, reference, and negation that are involved in the Liar Paradox.
In addition to these, there are the subsidiary principles and related notions of denial, definability, naming, meaning, predicate, property, presupposition, antecedent, and operating on prior sentences to form newer meaningful ones rather than merely newer grammatical ones.
We would like to know what limits there are on all these notions and mechanisms, and how one impacts another. What are the important differences among the candidates for bearers of truth? The leading candidates are sentences, propositions, statements, claims, and utterances. Is one primary, while the others are secondary or derivative?
Ideally, we would like to know a great deal more about truth, but also falsehood and the related notions of fact, situation and state of affairs. We want to better understand what a language is and what the relationship is between an interpreted formal language and a natural language, relative to different purposes. Finally, it would be instructive to learn how the Liar Paradoxes are related to all the other paradoxes.
That may be quite a lot to ask, but then our civilization does have some time to investigate all this before the Sun expands and vaporizes our little planet. An important question regarding the Liar Paradox is: What is the relationship between a solution to the Paradox for interpreted formal languages and a solution to the Paradox for natural languages?
There is significant disagreement on this issue. Is appeal to a formal language a turn away from the original problem, and so just changing the subject? Can one say we are still on the subject when employing a formal language because a natural language contains implicitly within it some formal language structure? Or should we be in the business of building an ideal language to replace natural language for the purpose of philosophical study?
Is our natural language, for example, English, a semantically closed language? Does English have one or more logics? Should we conclude from the Liar Paradox that the logic of English cannot be standard logic but must be one that restricts the explosion that occurs due to our permitting the deduction of anything whatsoever from a contradiction? Should we say English really has truth gaps or perhaps occasional truth gluts sentences that are both true and false?
So many questions. Or instead can a formal language be defended on the ground that natural language is inconsistent and the formal language is showing the best that can be done rigorously? Can sense even be made of the claim that a natural language is inconsistent, for is not consistency a property only of languages with a rigorous structure, namely formal languages and not natural languages?
Should we say people can reason inconsistently in natural language without declaring the natural language itself to be inconsistent? This article raises, but will not resolve, these questions, although some are easier to answer than others.
Many of the most important ways out of the Liar Paradox recommend revising classical formal logic. Classical logic is the formal logic known to introductory logic students as "predicate logic" in which, among other things, i all sentences of the formal language have exactly one of two possible truth values TRUE, FALSE , ii the rules of inference allow one to deduce any sentence from an inconsistent set of assumptions, iii all predicates are totally defined on the range of the variables, and iv the formal semantics is the one invented by Tarski that provided the first precise definition of truth for a formal language in its metalanguage.
A few philosophers of logic argue against any revision of classical logic by saying classical logic is the incumbent formalism that should be accepted unless an alternative is required probably it is believed to be incumbent because of its remarkable success in expressing most of modern mathematical inference. Still, most other philosophers argue that classical logic is not the incumbent which must remain in office unless an opponent can dislodge it. Instead, the office has always been vacant.
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In the decades since Tarski's treatment of the Liar Paradox, there have been many new approaches that reject his classical, extensional logic in favor of alternative logics that do not require that his T-sentences be theorems of the metalanguage. One critic of classical formal logic, Hartley Slater, says the usual formal languages fail at the crucial point of properly treating indexicals, words whose reference changes with context:. Slater , p. Some philosophers object to revising classical logic if the purpose in doing so is merely to find a way out of the Paradox.
There are more pressing problems in the philosophy of logic and language than finding a solution to the Paradox, so any treatment of it should wait until these problems have a solution. From the future resulting theory which solves those problems, one could hope to deduce a solution to the Liar Paradox. However, for those who believe the Paradox is not a minor problem but is one deserving of immediate attention, there can be no waiting around until the other problems of language are solved.
Perhaps the investigation of the Liar Paradox will even affect the solutions to those other problems. There have been many systematic proposals for ways out of the Liar Paradox. Below is a representative sample of five of the main ways out. Bertrand Russell said natural language is incoherent, but its underlying sensible part is an ideal formal language such as the applied predicate logic of Principia Mathematica.
Russell's way out was to rule out self-referential sentences as being ungrammatical or not well-formed in his ideal language. This is a formal language involving an infinite hierarchy of, among other things, orders of propositions:. If we now revert to the contradictions, we see at once that some of them are solved by the theory of types. This solves the liar. This theory is one of his formalizations of the Vicious-Circle Principle: Whatever involves all of a collection must not be one of the collection.
Russell believed that violations of this principle are the root of all the logical paradoxes. His solution to the Liar Paradox has the drawback that it places so many subscript restrictions on what can refer to what. The type theory also rules out explicitly saying within his formalism that legitimate terms must have a unique type, or saying that properties have the property of belonging to exactly one category in the hierarchy of types, which, if we step outside the theory of types, seems to be true about the theory of types.
Bothered by this, Tarski took a different approach to the Liar Paradox. Because of the vagueness of natural language, Tarski quit trying to find the paradox-free structure within natural languages and concentrated on developing formal languages that did not allow the deduction of a contradiction, but which diverge from natural language "as little as possible. One virtue of Tarski's way out of the Paradox is that it does permit the concept of truth to be applied to sentences that involve the concept of truth, provided we apply level subscripts to the concept of truth and follow the semantic rule that any subscript inside a pair of quotation marks is smaller than the subscript outside but still within the sentence; any violation of this rule produces a meaningless, ungrammatical formal sentence, but not a false one.
The language of level 1 is the meta-language of the object language in level 0. Level 0 sentences don't contain truth or similar terms, but would contain, say, "Paris is the capital of France. It would be: "'Paris is the capital of France' is true 0. The rule for subscripts stops the formation of both the Classical Liar Sentence and the Strengthened Liar Sentence anywhere within the hierarchy. The subscripting also stops paradoxical chains that start as follows:.
Another virtue of the Tarski way out is that it provides a way out of the Yablo Paradox. Tarski allows some self-reference, but not the self-reference involved in the Liar Paradox. Intuitively, a more global truth predicate should be expressible in the language it applies to. The Tarski way out does not allow us even to say that in all languages of the hierarchy, some sentences are true. To use Wittgenstein's phrase from his Tractatus , the character of the hierarchy can be shown but not said. Despite these restrictions and despite the unintuitive and awkward hierarchy, Quine defends Tarski's way out as the best of the ways.
Here is Quine's defense:. Revision of a conceptual scheme is not unprecedented. It happens in a small way with each advance in science, and it happens in a big way with the big advances, such as the Copernican revolution and the shift from Newtonian mechanics to Einstein's theory of relativity. We can hope in time even to get used to the biggest such changes and to find the new schemes natural. There was a time when the doctrine that the earth revolves around the sun was called the Copernican paradox, even by the men who accepted it.
And perhaps a time will come when truth locutions without implicit subscripts, or like safeguards, will really sound as nonsensical as the antinomies show them to be. Quine The languages either the formalized languages or—what is more frequently the case—the portions of everyday language which are used in scientific discourse do not have to be semantically closed.
Tarski, One criticism of Quine is that he is asking us to be patient and not to be so bothered by the complexity of the hierarchy, but he is giving no other justification for the hierarchy. That is, Tarski's solution does not provide a way to specify the circumstances in which a sentence leads to a paradox and the other circumstances in which that same sentence is paradox-free.
But neither can it be a regimented language, for no regimented language can make semantic generalizations about itself or about languages on a higher level than itself. Putnam , Tarski's Undefinability Theorem does not apply to languages having sentences that are neither true nor false. So, it can be argued that Kripke successfully shows that a semantically coherent formal language can contain its own global truth predicate. Kripke trades Russell's and Tarski's infinite syntactic complexity of languages for infinite semantic complexity of a single formal language. He rejects Tarski's infinite hierarchy of meta-languages in favor of one formal language having an infinite hierarchy of partial interpretations.
Consider a single formal language capable of expressing elementary number theory and containing a predicate T for truth that is, for truth in an interpretation. Kripke assigns to T an elaborate interpretation, namely its extension the set of sentences it is true of , its anti-extension the set of sentences it is false of , and its undecideds the set of sentences it is neither true nor false of. No sentence is allowed to be a member of both the extension and anti-extension of any predicate. Kripke allows the interpretation of T to change throughout the hierarchy.
The basic predicates except the T predicate must have their interpretations already fixed in this base level. In the base level of the hierarchy, the predicate T is given a special extension and anti-extension.
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As we ascend the hierarchy, distancing ourselves from the basic level, more and more complex sentences involving the symbol 'T' get added into the extension and anti-extension of the intended truth predicate T. As we go up a level we add into the extension of T all the true sentences containing T from the lower level. Ditto for the anti-extension. It will become true at the next higher level. And so goes the hierarchy of interpretations as it attributes truth to more and more sentences involving the concept of truth itself.
The extension of T, that is, the class of names of sentences that satisfy T, grows but never contracts as we move up the hierarchy, and it grows by calling more true sentences true. Similarly the anti-extension of T grows but never contracts as more false sentence involving T are correctly said to be false.
At this fixed point, the formal equivalent of the Liar Sentence still is neither true nor false, and so falls into the truth gap, just as Kripke set out to show. In this way, the Liar Paradox is solved, the formal language has a global truth predicate, the formal semantics is coherent, and many of our intuitions about semantics are preserved.
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However, there are difficulties with Kripke's way out. His treatment of the Classical Liar stumbles on the Strengthened Liar and reveals why that paradox deserves its name. For a discussion of why, see Kirkham , pp. Other philosophers say this is not a fair criticism of Kripke's theory since Tarski's Convention T, or some other intuitive feature of our concept of truth, must be restricted in some way if we are going to have a formal treatment of truth.
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What can more easily be agreed upon by the critics is that Kripke's candidate for the Liar sentence falls into the truth gap in Kripke's theory at all levels of his hierarchy, so it is not true in his theory. Therefore, Kripke's truth-gap theory cannot state its own thesis. Robert Martin and Peter Woodruff created the same way out as Kripke, though a few months earlier and in less depth. Another way out says the Liar Sentence is meaningful and is true or else false, but one special step of the argument in the Liar Paradox is incorrect. Arthur Prior, following the informal suggestions of Jean Buridan and C.
Peirce, takes this way out and concludes that the Liar Sentence is simply false. So do Jon Barwise and John Etchemendy, but they go on to present a detailed, formal treatment of the Paradox that depends crucially upon using propositions rather than sentences. The details of their treatment will not be sketched here. Their treatment says the Liar Proposition is simply false on one interpretation but simply true on another interpretation, and that the argument of the Paradox improperly exploits this ambiguity. The key ambiguity is to conflate the Liar Proposition's negating itself with its denying itself.
Similarly, in ordinary language we are not careful to distinguish asserting that a proposition is false from denying that it is true. Three positive features of the Barwise-Etchemendy solution are that i it applies to the Strengthened Liar, ii its propositions are always true or false, but never both, and iii it shows the way out of paradox both for natural language and interpreted formal language. Yet there is a price to pay. No proposition in their system can be about the whole world, and this restriction is there for no independent reason but only because otherwise we would get a paradox.
A more radical way out of the Paradox is to argue that the Liar Sentence is both true and false. This solution, a version of dialethism, embraces the contradiction, then tries to limit the damage that is ordinarily a consequence of that embrace. This way out was initially promoted primarily by Graham Priest in It succeeds in avoiding semantic incoherence while offering a formal, detailed treatment of the Paradox. One noteworthy feature of Priest's "truth-glut" semantics is that it is the same as Kleene's strong three-valued semantics with truth-gaps if we apply this translation scheme:.
A principal virtue of the paraconsistency treatment is that, unlike with Barwise and Etchemendy's treatment, a sentence can be about the whole world.
See the last paragraph of " Paradoxes of Self-Reference ," for more discussion of using paraconsistency as a way out of the Liar Paradox. To summarize, when we treat the Liar Paradox we should provide two things, an informal diagnosis which pinpoints the part of the paradox's argument that has led us astray, and a formalism that prevents the occurrence of the paradox's argument within that formalism. Russell, Tarski, Kripke, Barwise-Etchemendy, and Priest among many others deserve credit for providing a philosophical justification for their proposed solutions while also providing a formal treatment in symbolic logic that shows in detail both the character and implications of their proposed solutions.
Perhaps more work needs to be done in finding the best way, or the best ways , out of the Liar Paradox that will preserve the most important intuitions we have about semantics while avoiding semantic incoherence. In this vein, one can draw a pessimistic conclusion and an optimist conclusion. Taking the pessimistic route, Putnam says:. And if it is meaningful, is it true or false? Does it express a proposition or not? Does it have a truth value or not? And which one? In Putnam ,p. More optimistically, should there really be so much fear and loathing about limitations on our ability to formally express all the theses of our favored theory?
Many fields have learned to live with their limitations.