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

 To: "[ontolog-forum] " Christopher Menzel Thu, 18 Aug 2011 17:52:49 -0500 <66116D87-6E0D-4D06-8FD0-CC7D6275925C@xxxxxxxx>
 ```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)    (01) Pat, you obviously meant to write:    (02) (R a) (R b)    (03) > 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.    (04) 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    (05) (exists (p) (forall (x1 ... xn) (iff (p x1 ... xn) φ)))    (06) 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.    (07) This, of course, only strengthens your initial point to Rick that "you need to be achingly precise about what you count as 'higher order'".    (08) -chris    (09) _________________________________________________________________ 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    (010) ```
 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, Rick Murphy