<div style="font-family: 'Times New Roman'; font-size: 16px;"><br /><span>On 09/08/12, <b class="name">Jay Sulzberger </b> <jays@panix.com> wrote:</span><blockquote cite="mid:Pine.NEB.4.64.1208081639300.5843@panix3.panix.com" class="iwcQuote" style="border-left: 1px solid #00F; padding-left: 13px; margin-left: 0;" type="cite"><div class="mimepart text plain"><br /><br />Here we are close to the distinction between a class of "objects<br />which satisfy a condition" vs "objects with added structure", for<br />which see:<br /><br /> <a href="http://math.ucr.edu/home/baez/qg-spring2004/discussion.html" target="_blank">http://math.ucr.edu/home/baez/qg-spring2004/discussion.html</a><br /> <a href="http://ncatlab.org/nlab/show/stuff,+structure,+property" target="_blank">http://ncatlab.org/nlab/show/stuff,+structure,+property</a><br /><br />oo--JS.<br /></div></blockquote>This seems to be addressing my my question, but I am not sure that I can relate the above ideas to Haskell.<br _moz_dirty="" />Below is my current (naive) understanding and some further question:<br _moz_dirty="" /><br _moz_dirty="" />"objects which satisfy a condition" <br _moz_dirty="" />Could these objects be models that have the same signature (instances in Haskell). <br _moz_dirty="" />Haskell type classes seem to be signature only (no equations, ignoring default methods) so in general they provide an empty theory with no logical consequences.<br _moz_dirty="" /><br _moz_dirty="" />"objects with added structure"<br _moz_dirty="" />I am struggling with this concept both in general and in relation to the hierarchy from my earlier posting.<br _moz_dirty="" />Could this be "model expansion" where a theory describing an existing model is enriched with additional axioms.<br _moz_dirty="" />The enriched theory is then satisfied by models with more structure (operations).<br _moz_dirty="" />I am unsure about the size of this expanded model and the number of potential expanded models. <br _moz_dirty="" />Would a expanded model have less elements?<br _moz_dirty="" />Would there be fewer models for the enriched theory?<br _moz_dirty="" /><br _moz_dirty="" />In relation to Haskell data types also have structure (constructors). <br _moz_dirty="" />The data types can be used to build other data types (is this model expansion?)<br _moz_dirty="" />I am not sure if the model (instance) of a sub-class could be considered as expanded model of its super-class.<br _moz_dirty="" /><br _moz_dirty="" /><br _moz_dirty="" />Your reply was very helpful<br _moz_dirty="" />Thanks,<br _moz_dirty="" />Pat<br /><blockquote cite="mid:Pine.NEB.4.64.1208081639300.5843@panix3.panix.com" class="iwcQuote" style="border-left: 1px solid #00F; padding-left: 13px; margin-left: 0;" type="cite"><div class="mimepart text plain"><br /></div></blockquote></div>
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