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

 To: "[ontolog-forum]" Pat Hayes Thu, 18 Aug 2011 19:53:15 -0500 <34C4A73D-75B9-482D-8CE2-BFC0BF946216@xxxxxxx>
 ``` On Aug 18, 2011, at 5:52 PM, Christopher Menzel wrote:    (01) > On Aug 18, 2011, at 5:15 PM, Pat Hayes wrote: >> ... >> Here is a test case to hone your intuitions. Consider these axioms: >> >> (R a) >> (Q b) > > Pat, you obviously meant to write: > > (R a) > (R b) > >> And ask yourself whether these entail the following: >> >> (exists (p)(and (p a)(p b) ) >> >> i.e. that there is a property that applies both to a and to b. >> >> 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. > > I agree that's a good place to start for testing your intuitions, but >answering "yes" here does not necessarily mean that you are thinking in a >genuinely higher-order way, i.e., in a way that commits you to full >second-order logic. For we can in fact add all of the comprehension axioms > > (exists (p) (forall (x1 ... xn) (iff (p x1 ... xn) φ))) > > to the usual axioms for second-order quantification and the resulting logic >can still be given a Henkin-style general model theory that renders the logic >semantically first-order. The comprehension axiom with φ = "(R x)" and some >simple logic give us the implication in your example.    (02) Well, I would say that this is best stated by saying that my two axioms *together with the comprehension axioms* entail the conclusion. Of course, we can haver back and forth as to whether these comprehension axioms are part of a theory expressed in the logic or are part of the logic itself :-)    (03) Pat    (04) > > This, of course, only strengthens your initial point to Rick that "you need >to be achingly precise about what you count as 'higher order'". > > -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    (05) ------------------------------------------------------------ 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    (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 [ontolog-forum] type free logic and higher order quantification, Rick Murphy 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, Pat Hayes <= Re: [ontolog-forum] type free logic and higher order quantification, Rick Murphy