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

 To: "[ontolog-forum] " , Rick Murphy Pat Hayes Thu, 18 Aug 2011 17:15:28 -0500
 ``` On Aug 18, 2011, at 4:58 PM, Rick Murphy wrote:    (01) > Hey All: > > Long time no ont. > > I am researching type free logics with higher order quantification.    (02) You need to be achingly precise about what you count as 'higher order'. Common Logic is type-free and (some would say) uses (what certainly looks very like) higher-order quantification, in that it allows one to quantify over names in predicate and function positions, like this:    (03) (forall (f)(iff (transitive f)(forall x y z)(if (and (f x y)(f y z)) (f x z) )) ))    (04) However, CL has a first-order semantics and its metatheory is much closer to FO than HO, so many logicians would *not* count it as a higher-order logic. It has been called FO logic with a HO syntax, or HO logic with a FO semantics.    (05) Here is a test case to hone your intuitions. Consider these axioms:    (06) (R a) (Q b)    (07) And ask yourself whether these entail the following:    (08) (exists (p)(and (p a)(p b) )    (09) i.e. that there is a property that applies both to a and to b.    (010) If you intuitively answer "no", then you are thinking first-order, and would likely find CL congenial. If it seems obviously "yes", then you are thinking in a genuinely higher-order way.    (011) It really does matter which way you choose, as you will be interested in very different logics.    (012) Pat    (013) > I > would like to read some (free) academic research on this topic and > wondered whether anyone could point me at published papers. > > Google only matches six documents on the terms "type free logic" and > "higher order quantification". > > Research here seems thin, probably my ignorance. > > Chris I have read your paper. The references don't seem to point at > direct prior research on these combined subjects. > > -- > Rick > > > _________________________________________________________________ > 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 > >    (014) ------------------------------------------------------------ IHMC (850)434 8903 or (650)494 3973 40 South Alcaniz St. (850)202 4416 office Pensacola (850)202 4440 fax FL 32502 (850)291 0667 mobile phayesAT-SIGNihmc.us http://www.ihmc.us/users/phayes    (015) _________________________________________________________________ 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    (016) ```
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