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## Re: [ontolog-forum] Universal Basic Semantic Structures

 To: "[ontolog-forum]" William Frank Sun, 30 Sep 2012 09:51:32 -0400
 On Sun, Sep 30, 2012 at 1:36 AM, Chris Menzel wrote: On Sat, Sep 29, 2012 at 8:05 PM, William Frank wrote: On Sat, Sep 29, 2012 at 7:03 PM, Chris Menzel wrote: On Fri, Sep 28, 2012 at 11:54 AM, John F Sowa wrote: I would emphasize that set theory has two operators (subset and isIn), but mereology has only one operator (partOf).John, virtually every text on set theory presents the axioms with just a single binary predicate "∈" for membership. The subset relation is always defined; using "isin" instead of "∈": What does this imply to you?It doesn't imply anything to me. It is just a simple fact about how the subset relation is introduced into set theory.  arithmetic has the operations plus and minus and times and successor.  Of course, most texts define all of them from sucessor.  You are quite mistaken. I hope I would be as unlikely to be '*quite* mistaken' as if someone would make clear assertions about operations in set theory and yet not know that subset could be defined in terms of membership.   If I knew that little about the subject of axiomatized arithmetic, I hope I would not speak about it.  It is impossible to define addition in terms of successor and multiplication in terms of addition and successor. You appear to be mistaking the usual recursive axioms for those operators for definitions. They are not. Full arithmetic requires all three operations. (Interestingly, exponentiation can be defined in terms of addition and multiplication.)I think what you meant was,  'in a first order axiomatization of arithmetic'. Otherwise, for a start Defintion of '+'for every f,  f = + means (or iff)for every x,  f(x,0) = xand for every x, y, z  if  f(x,y) = z.then s(fx,y)) = s(z). Then, if you don't like this form of a definition, with the right circumlocution, it can be made to suit whatever form you want a definition to have. I seem to recall that one just has to replace the function with a property, and then show that the for every first two arguments of the property there is exactly one third argument, and then use the iota operator to say + is *the* property P such that .....  As, long, of course, as one does not restrict oneself from quantifying over properties.   I seem to recall I read a paper at a meeting of the Association of Symbolic Logic on this once. Interesting and profound, but it remains the case that all three are part of arithmetic.   Not just the ones you choose in your axoimatization to be primitive.     Instead, they can all be defined from +  and 1, if one chose.  There was a time before successor was discovered.  When it was, did the others go away?  In propositional logic, we can use all the natural deduction operators, and after defining the inference rules for them all, prove many equivelences, or instead, take our pick of a pair such as not and or, or use nor or nand only as primitives.    I have found that some people, depending on what course they happen to have taken, believe that if a then b "really is" not a or b, etc. I have no idea what you are talking about.Sorry.  I read in to your reason for saying that subset can be defined in terms of membership, in almost every axiomatization.  In doing this, I was quite mistaken, not just in misreading, but in reading in instead of just asking, about a subject I do know very little (other people's intentions).  Wm  -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    You are quite mistaken. It is impossible to define addition in terms of successor and multiplication in terms of addition and successor. You appear to be mistaking the usual recursive axioms for those operators for definitions. They are not. Full arithmetic requires all three operations. (Interestingly, exponentiation can be defined in terms of addition and multiplication.) Interesting and profound, but it remains the case that all three are part of arithmetic.   Not just the ones you choose in your axoimatization to be primitive.     Instead, they can all be defined from +  and 1, if one chose.  There was a time before successor was discovered.  When it was, did the others go away?  In propositional logic, we can use all the natural deduction operators, and after defining the inference rules for them all, prove many equivelences, or instead, take our pick of a pair such as not and or, or use nor or nand only as primitives.    I have found that some people, depending on what course they happen to have taken, believe that if a then b "really is" not a or b, etc. I have no idea what you are talking about.-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  -- William Frank413/376-8167This email is confidential and proprietary, intended for its addressees only.It may not be distributed to non-addressees, nor its contents divulged, without the permission of the sender. ``` _________________________________________________________________ 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    (01) ```
 Current Thread Re: [ontolog-forum] Universal Basic Semantic Structures, (continued) Re: [ontolog-forum] Universal Basic Semantic Structures, Avril Styrman Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, Matthew West Re: [ontolog-forum] Universal Basic Semantic Structures, John F Sowa Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, John F Sowa Re: [ontolog-forum] Universal Basic Semantic Structures, Avril Styrman Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, William Frank Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, William Frank <= Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, William Frank Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, William Frank Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, John F Sowa Re: [ontolog-forum] Universal Basic Semantic Structures, Chris Menzel Re: [ontolog-forum] Universal Basic Semantic Structures, doug foxvog Re: [ontolog-forum] Universal Basic Semantic Structures, John Bottoms Re: [ontolog-forum] Universal Basic Semantic Structures, Ed Barkmeyer