From LOJBAN%CUVMB.bitnet@YaleVM.YCC.YALE.EDU Sat Mar 6 22:55:06 2010 Received: from YALEVM.YCC.YALE.EDU by MINERVA.CIS.YALE.EDU via SMTP; Mon, 8 Mar 1993 17:53:57 -0500 Received: from CUVMB.CC.COLUMBIA.EDU by YaleVM.YCC.Yale.Edu (IBM VM SMTP V2R2) with BSMTP id 6324; Mon, 08 Mar 93 17:50:12 EST Received: from CUVMB.BITNET by CUVMB.CC.COLUMBIA.EDU (Mailer R2.07) with BSMTP id 6964; Mon, 08 Mar 93 17:53:47 EST Date: Mon, 8 Mar 1993 17:07:39 -0500 Reply-To: John Cowan Sender: Lojban list From: John Cowan Subject: TECH: Properties problems solved? X-To: Lojban List To: Erik Rauch Status: OR X-From-Space-Date: Mon Mar 8 12:07:39 1993 X-From-Space-Address: @YaleVM.YCC.YALE.EDU:LOJBAN@CUVMB.BITNET Message-ID: Having read the comments of Nick, Bob Chassell, lojbab/Nora, Gerald Koenig, and Jim Carter (who is the Mark Hubey of the Loglan Project :-) :-) ), and having reread Quine (>Methods of Logic<, 4th ed.) some light begins to dawn in my fevered brain. There seem to be two main differences in form between standard logic and logic a la Lojban. One has been brought up peripherally, and I think I will knock it off before proceeding to the meat of this posting. In standard logic notation, quantifiers have short scope: in "(Ex) Fx . Gx", the scope of the binding of "x" is "Fx" only. In Lojban, however, the scope extends to the next prenex binding the same variable; or the next cmavo that unbinds quantifiers, either "ni'o" or "da'o"; or the "tu'u" (explicit or implicit) that terminates a prenexed "tu'e...tu'u" group. A consequence of this view is that Lojban is a "fully alphatized" language; that is, there can be no rebinding of variables in inner scopes that have already been used in outer scopes. Every scope starts with the prenex that contains it and extends to the end-of-all-scopes point. Each variable in such a set of nested scopes must have a distinct name. The other, more important, difference is that in standard logic variables are not implicitly quantified. In Lojban logic, "Fx" means the same as "(Ex) Fx"; that is, "da prenu" is the same as "su'o da zo'u da prenu". What, then, is the meaning of "Fx" in standard logic? It is what Quine calls an "open sentence", and others have called a "propositional function". It is neither true nor false, but rather true >of< something. Thus, if "F" is short for "is a baseball", then "Fx" is true if x is a baseball. This is precisely what I was groping for when I talked of "certain abstract objects, namely those which can be applied to objects to produce truth values." I therefore am now in a position to propose a resolution of the property problem by convention. Let us hold that any da-series variables which first appear within a "ka...kei" property abstractions are taken to be free variables in the logician's sense; they are not quantified, but serve as mere placeholders. Thus, the translation of "Fx" is now "le ka da broda", or "de" if "da" is already in use, etc. Now we can distinguish clearly between: 1) la djan. zmadu la djordj. le ka da prami mi John exceeds George in-the property-of (x loves me). John loves me more than George loves me. and 2) la djan. zmadu la djordj. le ka mi prami da John exceeds George in-the property-of (I love x). I love John more than I love George. Since "da" can be ellipsized like any other sumti, it is legal to omit it and in these contexts will probably cause no confusion. However, oddities like: 3) le ka [zo'e] binxo lo bitmu da the property-of ( [something] becomes a-wall under-conditions x) the property of being the conditions under which something not specified, but understood from context, becomes a wall can be clearly and precisely distinguished from the more normal 4) le ka [da] binxo lo bitmu [zo'e] the property-of ( x becomes a wall [under-some-conditions] ) the property of becoming a wall (under some unspecified conditions). I believe that using this convention eliminates any need for a "kau" suffix here, leaving "kau" to its original role for indicating indirect questions; it seems clear that this also satisfies Nora's desire for a conservative solution, being a mere change of interpretation in (presumably rare) sentences with explicit unbound "da/de/di" within "ka...nei". I think, but I am not sure, that "ka...nei" without unbound "da" continues to provide what JCB, Bob Chassell, Gerald Koenig (insofar as I understand him), and Jim Carter want as regards "essentialism". And of course, Lojban remains an elliptical rather than an auto-replicating language, where the values of missing sumti must be glorked from context. -- John Cowan cowan@snark.thyrsus.com ...!uunet!lock60!snark!cowan e'osai ko sarji la lojban.