tag:blogger.com,1999:blog-4987609114415205593.post8866579493076927127..comments2021-06-14T17:02:48.428+01:00Comments on M-Phi: Yablo's ParadoxJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-4987609114415205593.post-73274070825692351152012-07-25T03:54:57.478+01:002012-07-25T03:54:57.478+01:00- BUTLER, Jesse (2005) «Circularity and infinite l...- BUTLER, Jesse (2005) «Circularity and infinite liar-like paradoxes», Master of Arts Thesis, University of Florida.<br />- KENNEDY, Niel (2007) «The definite story on Yablo's paradox», In : Institute for Language Logic and Computation. Paris-Amsterdam Logic Meeting of Young Researchers - 6, Amsterdam, 14-15 Décembre, 2007.<br />- LUNA, Laureano (2009) «Yablo’s paradox and beginningless time», Disputatio 26(3).<br />- ANDRÁS, Ferenc (2010) «The new clothes of paradox». (Available online)<br />- HASSMAN, Benjamin John (2011) «Semantic objects and paradox. A study of Yablo’s omega-liar», PhD Dissertation, University of Iowa.<br />- LEACH-KROUSE, Graham (2011) «Yablo’s paradox and arithmetical incompleteness», arXiv.<br />- FORSTER, Thomas (2012) «Yablo’s paradox and the omitting types theorem for propositional languages». (Available online) <br /><br /><br />http://ferenc.andrasek.hu/index.php?page=/right9.phpAZnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-33014655062142331332011-08-20T18:52:52.112+01:002011-08-20T18:52:52.112+01:00Exactly.Exactly.Rafal Urbaniakhttps://www.blogger.com/profile/10277466578023939272noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-92150374708123727232011-08-20T14:40:59.267+01:002011-08-20T14:40:59.267+01:00Hi Rafal, many thanks!
The main confusion of Bueno...Hi Rafal, many thanks!<br />The main confusion of Bueno & Colyvan 2004 (and their mistaken reply) is to confuse variables with numerals, i.e., confusing a universally quantified formula $\forall x \phi(x)$ with the set $\{\phi(\underline{n}): n \in \omega\}$.<br />In this particular case, the set of instances,<br /><br />$\{\mathbf{True}(\ulcorner Y_n \urcorner) \leftrightarrow Y_n: n \in \omega\}$<br /><br />is confused the universal generalization, the uniform sentence,<br /><br />$\forall x(\mathbf{True}(\ulcorner Y(\dot{x}) \urcorner) \leftrightarrow Y(x))$<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-90181411367266828792011-08-20T12:36:39.324+01:002011-08-20T12:36:39.324+01:00Hi Jeff,
Hi Jeff,
Many thanks for your comments,...Hi Jeff,<br />Hi Jeff,<br /> <br />Many thanks for your comments, a few further remarks.<br />—-<br />QUOTE<br />"Ketland was probably a bit hasty when he said the existence of Yablo sequence is a theorem of logic: for it is conditional upon PA extended with a truth predicate. (Although, this of course hinges on what you mean by mathematical logic)."<br /><br />Right - but I did not say that it is a theorem of "logic"!! <br />I said it is a "theorem of mathematical logic". Mathematical logic is branch of *mathematics*, covered in books such as Enderton, Shoenfield, and other textbooks. <br />—-<br />COMMENT<br />I thought you had that in mind, hence the remark "(Although, this of course hinges on what you mean by mathematical logic.)" Still, the use of term "logic" might have caused some confusion, as witnessed by the case in question.<br />—-<br /><br />—-<br />QUOTE<br />You quote B&C saying,<br />"Still, if the list is a theorem of Peano arithmetic suppelemented with a truth predicate, isn't that enough? Not really." <br /><br />The list is *not* a theorem of PA + truth predicate. How can a list be a theorem? The existence of the list is a theorem of mathematical logic. The universally quantified biconditional is a theorem of PA + truth predicate. The *existence* of this list is a theorem of mathematical logic [...] And the existence of its instances then follows.<br />—-<br />COMMENT<br />Yup, B&C's phrasing struck me as unfortunate, but I charitably assumed they had the existence of the list in mind.<br />—-<br /><br />—-<br />QUOTE<br />You quote B&C as saying:<br />"While it is true that each finite subset of Yablo sentences is not paradoxical, it is not true that each subset is satisfiable."<br /><br />This confuses finite subsets of Yablo *biconditionals* with finite subsets of Yablo sentences. The consistency proof concerns the former. Each finite subset of Yablo *biconditionals* is satisfiable. <br />—-<br />COMMENT<br />Good point, I didn't make this distinction. But still, even if you look at the sentences not biconditionals, their argument that s_1+s_2 is not satisfiable because s_2 is vacuously true fails.<br />—-<br /><br />—-<br />QUOTE<br />Right - but I don't think there is a difference between Yablo sentences and the list of Yablo sentences. <br />There is a difference between <br />- the Yablo sentences (indexed by n∈ω)<br />- what a non-standard model *thinks* is a Yablo sentence.<br />[...]<br /><br />Without these distinctions, I don't see how one can distinguish between local and uniform schemes. This seems to be the central mistake of B&C (original and reply). The local scheme gives the ω-inconsistency, while the uniform one gives a genuine inconsistency.<br />—-<br />COMMENT<br />I completely agree. My point was that by including whatever the non-standard model thinks is a Yablo sentence on the list of Yablo sentences (to which the local disquotation scheme applies) B&C in effect tacitly replaced the local scheme (applying to sentences indexed with natural numbers) with the uniform one.<br />—-<br /><br />cheers,<br />RafalRafal Urbaniakhttps://www.blogger.com/profile/10277466578023939272noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-20920958798296801092011-08-19T23:18:28.136+01:002011-08-19T23:18:28.136+01:00Hi Rafal, (by the way, many thanks for writing the...Hi Rafal, (by the way, many thanks for writing the post on this), here's another comment on your comment ...<br /><br />"Well, Ketland's response is slightly misleading, for it doesn't emphasize the (possible) difference between being on the Yablo list, and being a Yablo sentence. On one interpretation (interpretation A) the Yablo list is the list of Yablo sentences obtained for natural numbers as indices. In another (B) the Yablo list is just the set of all Yablo sentences, ordered by the less-then relation, no matter whether the numbers involved are standard or not."<br /><br />Right - but I don't think there is a difference between Yablo sentences and the list of Yablo sentences. <br />There is a difference between <br />- the Yablo sentences (indexed by $n \in \omega$)<br />- what a non-standard model *thinks* is a Yablo sentence.<br /><br />So, on interpretation (A), this simply is the definition of the Yablo sentences (= the Yablo list), which is:<br /><br />$Y_n$ is the result of substituting the numeral of $n$ for all free occurrences of $x$ in $Y(x)$. <br /><br />And the numeral of $n$ is defined by:<br /><br />The numeral of $0$ is $\underline{0}$.<br />The numeral of $n+1$ is $s\underline{n}$.<br /><br />Let $\mathcal{A} \models PA$ be non-standard. Let $a \in dom(\mathcal{A})$ be a non-standard element. What then is its numeral? More or less by definition, non-standard elements do not have numerals. The concept "numeral of $n$" is defined in the metalanguage. Unless one accepts this, one cannot even define "non-standard"! For "$\mathcal{A} \models PA$ is non-standard" means "$\mathcal{A}$ is not isomorphic to $(\mathbb{N}, 0, S, +, \times)$".<br /><br />Without these distinctions, I don't see how one can distinguish between local and uniform schemes. This seems to be the central mistake of B&C (original and reply). The local scheme gives the $\omega$-inconsistency, while the uniform one gives a genuine inconsistency. Quantification of $\phi(x)$ with respect to a variable $x$ is not always the same as infinitary conjunction of its instances with numerals, for there are non-standard models.<br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-60408285128547771982011-08-19T22:25:51.834+01:002011-08-19T22:25:51.834+01:00Hi Rafal, yes, that's pretty much right.
You ...Hi Rafal, yes, that's pretty much right. <br />You write,<br /><br />"Ketland was probably a bit hasty when he said the existence of Yablo sequence is a theorem of logic: for it is conditional upon PA extended with a truth predicate. (Although, this of course hinges on what you mean by mathematical logic)."<br /><br />Right - but I did not say that it is a theorem of "logic"!! <br />I said it is a "theorem of mathematical logic". Mathematical logic is branch of *mathematics*, covered in books such as Enderton, Shoenfield, and other textbooks. <br /><br />You quote B&C saying,<br />"Still, if the list is a theorem of Peano arithmetic suppelemented with a truth predicate, isn't that enough? Not really." <br /><br />The list is *not* a theorem of PA + truth predicate. How can a list be a theorem? The existence of the list is a theorem of mathematical logic. The universally quantified biconditional is a theorem of PA + truth predicate. The *existence* of this list is a theorem of mathematical logic (if one wanted to be really fussy, I'm pretty sure provable in $I \Sigma_1$, and not even using a truth predicate!). And the existence of its instances then follows.<br /><br />The original paper by B&C claimed that one could, pace Priest, deduce a contradiction using just the *local T-scheme*. The uniform one amounts to using a satisfaction predicate - Priest's point. But I'd noticed that only an omega-inconsistency follows with the local T-scheme (i.e., disquotation applied to Yablo sentences one-by-one), since I'd written some notes for myself on the problem in 2000, I think, and discussed it with Stephen Yablo. So, one needs the stronger uniform scheme - which is equivalent to using satisfaction. (At the time, I didn't know the papers by Hardy 1995 and Leitgeb 2001. Note that B&C are also objecting to Hardy and Leitgeb.)<br /><br />You quote B&C as saying:<br /><br />"While it is true that each finite subset of Yablo sentences is not paradoxical, it is not true that each subset is satisfiable."<br /><br />This confuses finite subsets of Yablo *biconditionals* with finite subsets of Yablo sentences. The consistency proof concerns the former. Each finite subset of Yablo *biconditionals* is satisfiable. <br /><br />So, I didn't mention the unpublished B&C reply in this bibliography because of these apparent mathematical errors. <br /><br />Cheers,<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-69764438875468332492011-08-19T20:55:22.126+01:002011-08-19T20:55:22.126+01:00Hi Jeff, thanks for the mention.:) I think you for...Hi Jeff, thanks for the mention.:) I think you forgot the Bueno&Colyvan response to your paper. It's "Yablo's paradox rides again" (I think it's unpublished, but available online at <br /><br />http://homepage.mac.com/mcolyvan/papers/yra.pdf )<br /><br />Although, I don't think their case is too strong, just sketched a few remarks at Entia et Nomina:<br /><br />http://entiaetnomina.blogspot.com/2011/08/yablos-paradox-bueno-colyvan-vs-ketland.html<br /><br />If you think I misunderstood your position somehow in those comments, please let me know.Rafal Urbaniakhttps://www.blogger.com/profile/10277466578023939272noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-78090726035336207362011-05-08T18:02:51.495+01:002011-05-08T18:02:51.495+01:00Hi Roy, yes - good point! I knew Kripke had briefl...Hi Roy, yes - good point! I knew Kripke had briefly mentioned something in the area, but I'd forgotten when making the list. (Both Gabriel Uzquiano 2004 in Analysis, and Hannes's 2005 Analysis paper have similar ideas.)<br /><br />I'll update the post and, since you're here on the blog too, I'll change the intro bit too.Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-8581378874669379732011-05-08T03:33:46.846+01:002011-05-08T03:33:46.846+01:00You forgot the earliest: Kripke's classic [197...You forgot the earliest: Kripke's classic [1975], where he writes:<br /><br />One surprise to me was the fact that the orthodox approach by no means obviously guarantees groundedness… Even if unrestricted truth definitions are in question, standard theorems easily allow us to construct a descending chain of first-order languages L0, L1, L2,…, such that Li contains a truth predicate for Li+1. I don’t know whether such a chain can engender ungrounded sentences, or even quite how to state the problem here; some substantial technical questions in this area are yet to be resolved. ([1975]: 698)RoyTCookhttps://www.blogger.com/profile/05233569728242084863noreply@blogger.com