ontolog-forum
[Top] [All Lists]

## Re: [ontolog-forum] type free logic and higher order quantification

 To: "[ontolog-forum]" Rick Murphy Fri, 19 Aug 2011 18:53:22 -0400 <1313794402.4456.127.camel@metho-laptop>
 ```Chris, thank you. The explanation below is very helpful in answering my question regarding predicativity in logics with FO semantics and HO syntax.    (01) I enjoyed your paper [1] and recommend it highly for anyone who wants to understand the design goals of CL.    (02) -- Rick    (03) [1] http://cmenzel.org/Papers/Menzel-KRTheWWWAndTheEvolutionOfLogic.pdf    (04) On Fri, 2011-08-19 at 14:56 -0500, Christopher Menzel wrote:    (05) > > In CL, the statement equivalent to Russell's paradox > > I'd say: The sentence that *generates* Russell's paradox... > > > has a stable truth value: false. No paradox. > > Right. In more detail: the chief culprit in Russell's paradox is the > so-called "naive" comprehension principle that, for any formula φ there > is a set containing (or a property true of) all the things that are φ. > In the CL dialect CLIF, this principle is expressed schematically as: > > NC (exists (p) (forall (q) (iff (p x) φ))). > > In CL, where self-predication is permitted, an instance of NC is: > > R (exists (p) (forall (q) (iff (p x) (not q q)))). > > A contradiction follows from R immediately. For let r be one of the > things p said to exist by R: > > (1) (forall (q) (iff (r x) (not q q)))). > > But CL is type-free, so r itself is among the things being quantified > over; so by universal instantiation: > > (2) (iff (r r) (not r r)), > > contradiction. So, as John notes, in CL, sentence R is false; more > exactly, it is *logically* false; its negation is a theorem of CL. > > -chris > > > > > > > > _________________________________________________________________ > 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 >    (06) _________________________________________________________________ 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    (07) ```
 Current Thread Re: [ontolog-forum] type free logic and higher order quantification, (continued) Re: [ontolog-forum] type free logic and higher order quantification, Pat Hayes Re: [ontolog-forum] type free logic and higher order quantification, Christopher Menzel Re: [ontolog-forum] type free logic and higher order quantification, Rick Murphy Re: [ontolog-forum] type free logic and higher order quantification, John F. Sowa Re: [ontolog-forum] type free logic and higher order quantification, Pat Hayes Re: [ontolog-forum] type free logic and higher order quantification, John F. Sowa Re: [ontolog-forum] type free logic and higher order quantification, Christopher Menzel Re: [ontolog-forum] type free logic and higher order quantification, Pat Hayes Re: [ontolog-forum] type free logic and higher order quantification, Christopher Menzel Re: [ontolog-forum] type free logic and higher order quantification, Rick Murphy <= Re: [ontolog-forum] type free logic and higher order quantification, k Goodier Re: [ontolog-forum] type free logic and higher order quantification, Christopher Menzel Re: [ontolog-forum] type free logic and higher order quantification, Rick Murphy Re: [ontolog-forum] type free logic and higher order quantification, John F. Sowa