The Yablo Paradox
An Essay on Circularity
Roy T. Cook
From Our Blog
The Yablo Paradox (due to Stephen Yablo and Albert Visser) consists of an infinite sequence of sentences of the following form: S1: For all m > 1, Sm is false. S2: For all m > 2, Sm is false. S3: For all m > 3, Sm is false. : : : Sn: For all m > n, Sm is false. Sn+1: For all m > n+1, Sm is false. Hence, the nth sentence in the list 'says' that all of the sentences below it are false.
Posted on November 5, 2017
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The Liar paradox arises via considering the Liar sentence: L: L is not true. and then reasoning in accordance with the: T-schema: Φ is true if and only if what Φ says is the case. Along similar lines, we obtain the Montague paradox (or the paradox of the knower) by considering the following sentence: M: M is not knowable. and then reasoning in accordance with the following two claims: Factivity: If Φ is knowable then what Φ says is the case.
Posted on September 3, 2017
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We are often told that we should be open-minded. In other words, we should be open to the idea that even our most cherished, most certain, most secure, most well-justified beliefs might be wrong. But this is, in one sense, puzzling.
Posted on May 20, 2017
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As regular readers know, I understand paradoxes to be a particular type of argument.
Posted on April 1, 2017
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A directed graph is a pair
where N is any collection or set of objects (the nodes of the graph) and E is a relation on N (the edges). Intuitively speaking, we can think of a directed graph in terms of a dot-and-arrow diagram, where the nodes are represented as dots, and the edges are represented as arrows.
Posted on February 19, 2017
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What is the biggest whole number that you can write down or describe uniquely? Well, there isn't one, if we allow ourselves to idealize a bit. Just write down '1', then '2', then'¦ you'll never find a last one.
Posted on January 22, 2017
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In fiction, an unreliable narrator is a narrator whose credibility is in doubt ' in other words, a proper reading of a narrative with an unreliable narrator requires that the audience question the accuracy of the narrator's representation of the story, and take seriously the idea that what actually happens in the story ' what is fictionally true in the narrative ' is different from what is being said or shown to them.
Posted on December 18, 2016
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The idea that many, if not most, people exhibit physical signs ' tells ' when they lie is an old idea ' one that has been extensively studied by psychologists, and is of obvious practical interest to fields as otherwise disparate as gambling and law enforcement. Some of the tells that indicate someone is lying include:
Posted on November 13, 2016
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Before looking at the person-less variant of the Bernedete paradox, lets review the original: Imagine that Alice is walking towards a point ' call it A ' and will continue walking past A unless something prevents her from progressing further.
Posted on October 16, 2016
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Let us say that a sentence is periphrastic if and only if there is a single word in that sentence such that we can remove the word and the result (i) is grammatical, and (ii) has the same truth value as the original sentence.
Posted on September 18, 2016
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For many months now this column has been examining logical/mathematical paradoxes. Strictly speaking, a paradox is a kind of argument. In literary theory, some sentences are also called paradoxes, but the meaning of the term is significantly different.
Posted on August 14, 2016
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Imagine that we have a black and white monitor, a black and white camera, and a computer. We hook up the camera and monitor to the computer, and we write a program where, for some medium-ish shade of grey G.
Posted on June 12, 2016
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One of the most famous, and most widely discussed, paradoxes is the Liar paradox. The Liar sentence is true if and only if it is false, and thus can be neither (unless it can be both). The variants of the Liar that I want to consider in this instalment arise by taking the implicit temporal aspect of the word 'is' in the Liar paradox seriously.
Posted on April 17, 2016
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A theory is inconsistent if we can prove a contradiction using basic logic and the principles of that theory. Consistency is a much weaker condition that truth: if a theory T is true, then T consistent, since a true theory only allows us to prove true claims, and contradictions are not true. There are, however, infinitely many different consistent theories that we can construct.
Posted on March 13, 2016
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Imagine that you are an extremely talented, and extremely ambitious, shepherd, and an equally talented and equally ambitious carpenter. You decide that you want to explore what enclosures, or fences, you can build, and which groups of objects, or flocks, you can shepherd around so that they are collected together inside one of these fences.
Posted on February 14, 2016
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The Liar paradox is often informally described in terms of someone uttering the sentence: I am lying right now. If we equate lying with merely uttering a falsehood, then this is (roughly speaking) equivalent to a somewhat more formal, more precise version of the paradox that arises by considering a sentence like: "This sentence is false".
Posted on January 17, 2016
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While most of you probably don't believe in Santa Claus (and some of you of course never did!), you might not be aware that Santa Claus isn't just imaginary, he is impossible! In order to show that the very concept of Santa Claus is riddled with incoherence, we first need to consult the canonical sources to determine what properties and powers this mystical man in red is supposed to have.
Posted on December 13, 2015
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According to philosophical lore many sentences are self-evident. A self-evident sentence wears its semantic status on its sleeve: a self-evident truth is a true sentence whose truth strikes us immediately, without the need for any argument or evidence, once we understand what the sentence means.
Posted on November 8, 2015
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In a 1929 lecture, Martin Heidegger argued that the following claim is true: Nothing nothings. In German: 'Das Nichts nichtet'. Years later Rudolph Carnap ridiculed this statement as the worst sort of meaningless metaphysical nonsense in an essay titled 'Overcoming of Metaphysics Through Logical Analysis of Language'. But is this positivistic attitude reasonable?
Posted on October 10, 2015
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Imagine that, on a Tuesday night, shortly before going to bed one night, your roommate says 'I promise to only utter truths tomorrow.' The next day, your roommate spends the entire day uttering unproblematic truths like: 1 + 1 = 2.
Posted on September 13, 2015
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A 'Liar cycle' is a finite sequence of sentences where each sentence in the sequence except the last says that the next sentence is false, and where the final sentence in the sequence says that the first sentence is false.
Posted on August 9, 2015
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Here I want to present a novel version of a paradox first formulated by Jos© Bernardete in the 1960s ' one that makes its connections to the Yablo paradox explicit by building in the latter puzzle as a 'part'. This is not the first time connections between Yablo's and Bernardete's puzzles have been noted (in fact, Yablo himself has discussed such links). But the version given here makes these connections particularly explicit.
Posted on July 12, 2015
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The Liar paradox arises when we consider the following declarative sentence: This sentence is false. Given some initially intuitive platitudes about truth, the Liar sentence is true if and only if it is false. Thus, the Liar sentence can't be true, and can't be false, violating out intuition that all declarative sentences are either true or false (and not both). There are many variants of the Liar paradox. For example, we can formulate relatively straightforward examples of interrogative Liar paradoxes, such as the following Liar question: Is the answer to this question 'no'?
Posted on June 14, 2015
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The collection of infinite Yabloesque sequences that contain both infinitely many Y-all sentence and infinitely many Y-exists sentences, however, is a much larger collection. It is what is called continuum-sized, and a collection of this size is not only infinite, but strictly larger than any countably infinite collection. Thus, although the simplest cases of Yabloesque sequence ' the Yablo Paradox itself and its Dual ' are paradoxical, the vast majority of mixed Yabloesque sequences are not!
Posted on April 5, 2015
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A generalization is a claim of the form: (1) All A's are B's. A generalization about generalizations is thus a claim of the form: (2) All generalizations are B. Some generalizations about generalizations are true. For example: (3) All generalizations are generalizations. And some generalizations about generalizations are false. For example: (4) All generalizations are false. In order to see that (4) is false, we could just note that (3) is a counterexample to (4).
Posted on March 8, 2015
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Imagine that Banksy, (or J.S.G. Boggs, or some other artist whose name starts with 'B', and who is known for making fake money) creates a perfectly accurate counterfeit dollar bill ' that is, he creates a piece of paper that is indistinguishable from actual dollar bills visually, chemically, and in every other relevant physical way. Imagine, further, that our artist looks at his creation and realizes that he has succeeded in creating a perfect forgery. There doesn't seem to be anything mysterious about such a scenario at first glance ' creating a perfect forgery.
Posted on February 8, 2015
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One of the central tasks when reading a mystery novel (or sitting on a jury, etc.) is figuring out which of the characters are trustworthy. Someone guilty will of course say they aren't guilty, just like the innocent ' the real question in these situations is whether we believe them. The guilty party ' let's call her Annette ' can try to convince us of her trustworthiness by only saying things that are true, insofar as such truthfulness doesn't incriminate her.
Posted on January 11, 2015
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Early twentieth century packaging for Quaker Oats depicted the eponymous Quaker holding a package of the oats, which, in turn, depicted the Quaker holding a package of the oats, which itself depicted the Quaker holding a package of the oats, ad infinitum. It inspired a generation of philosophers.
Posted on November 30, 2014
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Why study paradoxes? The easiest way to answer this question is with a story: In 2002 I was attending a conference on self-reference in Copenhagen, Denmark. During one of the breaks I spoke with Raymond Smullyan; a mathematical logician and renowned author of 'Knights and Knaves' (K&K) puzzles.
Posted on September 7, 2014
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