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## Re: [ontolog-forum] HOL decidability [Was: using SKOS for controlled val

 To: "[ontolog-forum]" Christopher Menzel Tue, 12 Oct 2010 11:42:48 -0500 <21CB16F0-274D-4DFE-9C25-8EC2CB6D5A35@xxxxxxxx>
 ```On Oct 11, 2010, at 10:21 PM, doug foxvog wrote: > On Mon, October 11, 2010 23:04, Christopher Menzel said: >> On Mon, 2010-10-11 at 12:41 -0700, Rich Cooper wrote: >> ... >> Higher-order languages are indeed more expressive [than FOL]. As a >> consequence, they are not even partially decidable. > > Maybe Higher-order languages as a whole are not even partially decidable, > but large numbers of statements in them are.    (01) Sure, but I'm not sure I see the point. We're talking about the logics themselves.    (02) > It was noted that normal computer languages embodied HOL.    (03) That's a statement that needs a great deal of qualification. I am far from an expert on these matters, so someone with a more authoritative grasp of the subjects might well usefully correct/qualify some of my remarks. But I believe the gist of the following is correct.    (04) The logic that corresponds to most programming languages is the untyped lambda calculus, which is designed to represent recursion. The reason the lambda calculus is often referred to as higher-order is because it quantifies over functions. It is true that higher-order logics can quantify over functions (and properties and relations) and, for this reason, functions/properties/relations are often referred to as "higher-order objects". But simply quantifying over such objects is not in itself what makes a logic higher-order; rather, the function/property/relation space must consist of all possible functions/properties/relations over (in the second-order case) the domain of individuals. Absent that requirement, in the context of basic classical logic, one's logic is still first-order. (I'd recommend Herbert Enderton's article "Second-order and Higher-order Logic" in the Stanford Encyclopedia of Philosophy for more on this -- http://goo.gl/Ubko.)    (05) Now, with its ability to represent recursion as well as basic arithmetic, the lambda calculus certain far exceeds first-order logic in expressive power. But it is not straightforwardly classified as a species of classical higher-order logic either, as standard higher-order logic does not of itself support recursion, in the sense that mechanisms for defining recursive functions and a proof of their existence do not simply fall out of the semantics of higher-order logic; nor does the semantics allow for such constructs as functional self-application, which is possible in the untyped lambda calculus. Because of these features, the untyped lambda calculus does not have a classical higher-order semantics and, indeed, finding a semantics for it proved to be a very difficult problem that was not solved until Scott's discovery of domain theory in the 1960s. Together with Strachey, Scott applied domain theory to the development of denotational semantics for programming languages. Theoretically, this is a very beautiful semantics, but it is much more complex than the simple and straightforward semantics of ordinary higher-order logic.    (06) Bottom line: The sense in which normal computer languages "embody" higher-order logic is not a simple and straightforward matter.    (07) -chris    (08) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (09) ```
 Current Thread [ontolog-forum] Conation as an explanation of human thought processes, (continued) Re: [ontolog-forum] using SKOS for controlled values for controlled vocabulary, Obrst, Leo J.