tag:blogger.com,1999:blog-4987609114415205593.post2277964894498274356..comments2024-03-28T13:40:26.497+00:00Comments on M-Phi: Interpreting a StructureJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-4987609114415205593.post-77884713500609375712011-07-21T13:59:16.121+01:002011-07-21T13:59:16.121+01:00Tristan, "(i) One may think of the intended i...Tristan, "(i) One may think of the intended interpretation as just another kind of structure - albeit one which might contain extra-mathematical things, as in standard first-order model theory."<br /><br />Right - the interpretation $I$ isn't itself a structure but it does determine one, namely $M^{I} = (D_I, \{R^{I}\})$. Then $M$ is correct under $I$ iff $f_I : M \rightarrow M^{I}$ is an isomomorphism.Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-11071179725346857772011-07-21T06:27:42.248+01:002011-07-21T06:27:42.248+01:00Also: 'One could even give a *further* structu...Also: 'One could even give a *further* structure which contains a function which maps the 'denotation function' to the idea of denotation, for example - or the denotation relation itself, in some non-extensional sense.'<br /><br />On second thoughts, there's no particular reason why the denotation relation would have to be regarded non-extensionally.Tristan Hazehttp://sprachlogik.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-34238368385472459632011-07-21T06:21:04.765+01:002011-07-21T06:21:04.765+01:00Sorry, ignore the 'in' in 'In another ...Sorry, ignore the 'in' in 'In another use, an in interpretation'. (And don't try to read that correction out loud.)Tristan Hazehttp://sprachlogik.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-80496061156350695212011-07-21T06:16:50.976+01:002011-07-21T06:16:50.976+01:00Interesting post! Some thoughts:
In discussions a...Interesting post! Some thoughts:<br /><br />In discussions about formal theories and their 'intended interpretations' this latter notion seems curiously indefinite between a couple of quite different things:<br /><br />(i) One may think of the intended interpretation as just another kind of structure - albeit one which might contain extra-mathematical things, as in standard first-order model theory. Or,<br /><br />(ii) One might think of this as some "thing" of a completely different kind. The understanding of a structure, rather than a structure itself.<br /><br />This puts me in mind of Wittgenstein's discussions of interpretation and understanding. In various places, he speaks of 'interpretation' being used to mean two quite different things; on one use, an interpretation of some signs consists in more signs. In another use, an in interpretation is something like the application of some signs.<br /><br />In this connection, note that one could talk about the intended interpretation of I itself: this involves, for example, that the function in I is to be regarded assigning denotations. One could even give a *further* structure which contains a function which maps the 'denotation function' to the idea of denotation, for example - or the denotation relation itself, in some non-extensional sense.<br /><br />One basic moral to draw from this is that, when talking and thinking about interpretations of formal structures, we should take care to know what we are talking about (i.e. another structure - which may contain extra-mathematical objects - or something of a quite different kind).Tristan Hazehttp://sprachlogik.blogspot.com/noreply@blogger.com