Timeline for What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?
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| May 20, 2016 at 11:35 | comment | added | Thomas Benjamin | (cont.) cannot be a predicate). Note also that he states, in his letter to Frege, the source of the problem (You state (p. 17 [pg. 23 in the van Heijenoort translation of the Begriffsschrift--mine and Bauer-Mengelberg's comment]) that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful because of the following contradiction."). Frege's restriction to propositions tout court is consistent, as shown by Antonelli and May's paper "Frege's Other Program" (Notre Dame Journal of Formal Logic, Vol. 26, No 1, 2005). | |
| May 20, 2016 at 11:13 | comment | added | Thomas Benjamin | (cont.) for which an extension does exist-- those extensions that satisfy the Axiom of Foundation (the collection $V$). Can $V$ be correctly said to be an "extension" (if so, the contradiction follows)? Frege, in the Grundgesetze (as you rightly pointed out), said (in essence) that the Russell sentence does not have "sharp boundaries" and thus in his sense, is "not a concept at all" (note that Russell said, in his letter to Frege, that the 'predicate', "to be a predicate that cannot be predicated of itself", because of the contradiction that follows when applied to itself, | |
| May 20, 2016 at 10:47 | comment | added | Thomas Benjamin | @MauroALLEGRANZA: though $\lambda$$x$$\exists$$X$($x$=$\epsilon$$X$ $\land$ $\lnot$$X$($x$)) is a wff of Frege' s Grundgesetze, at the point it is applied to it's 'extension' (because of the contradiction), can it be said that it rightly has either an "extension", "sense", or "reference" (its assignment to either the True or the False)? The well- formedness of formulas is syntactical--does the grammatical sentence "Colorless green ideas sleep furiously" have an "extension", "sense", or "reference" in Frege's sense? Consider also the cases of the aforementioned sentence | |
| May 19, 2016 at 14:18 | comment | added | Mauro ALLEGRANZA | Regarding your last comment, I think that your suggested "way out" from the paradox is not viable by the fact that (Frege's symbolic equivalent) of Russell's "contradictory" property is a well-formed formula in F's system... In Appendix to vol 2 of Grundgesetze, F tried to fix his system "limiting" BLV; it is interesting to note F's claim that, with this modification, he can now prove (the symbolic equivalent of): $\vdash \lnot (x \in x)$. | |
| May 19, 2016 at 13:38 | comment | added | Mauro ALLEGRANZA | For a (very) different point of view on the "correct" way to read Frege's logical symbolism, see: Gregory Landini, Frege's Notations: What They Are and How They Mean (2012) (but I've haven't read it yet). | |
| May 17, 2016 at 12:43 | comment | added | Thomas Benjamin | @MauroALLEGRANZA: Very interesting--Thanks! Since for Frege, a 'concept without sharp boundaries' is, in his "sense", "not a concept at all", what forbids him to state this to Russell as a rejoinder to Russell's claim of an inconsistency in the Grundegesetze? | |
| May 17, 2016 at 9:06 | comment | added | Mauro ALLEGRANZA | ... I here use the capital Greek letters as if they were names referring to something, without stating their reference." 2/2 | |
| May 17, 2016 at 9:06 | comment | added | Mauro ALLEGRANZA | Comment to bedeuting; see Grungesetze (new Engl.transl.), footnote 3 to §5 (page 9), regarding the "horizontal": "Evidently, the sign "$\Delta$" must not be without reference, but it has to refer to an object. Names without reference must not occur in concept-script. The stipulation is made such that under all circumstances "$-- \Delta$" refers to something, provided only that "$\Delta$" refers to something. Otherwise $-- \xi$ would not be a concept with sharp boundaries, thus in our sense not be a concept at all. ... 1/2 | |
| May 15, 2016 at 11:39 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
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| May 15, 2016 at 10:14 | history | answered | Thomas Benjamin | CC BY-SA 3.0 |