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Message-id: <9205121557.AA00900@cs.umn.edu> Reply-To: cg@cs.umn.edu Date: Tue, 12 May 92 11:52:12 EDT From: sowa@watson.ibm.com To: cg@cs.umn.edu, interlingua@isi.edu Cc: franconi@irst.it Subject: Sets, plurals, and mereology

Last year, there was a lot of discussion of sets, plurals, and mereology on CG list. I recently received another question about the subject, to which I sent a brief response. Following is an elaboration of that response: 1. I believe that mereology has a more natural application to plurals in ordinary language than set theory. 2. However, logicians are more familiar with set theory than mereology. (Most nonlogicians are blissfully unaware of either one.) 3. Therefore, in my publications, I usually define the CG plural constructions in terms of sets, but I avoid the constructions that mereology does not represent. 4. In particular, mereology and natural languages never distinguish collections whose contents are the same, but with different levels of nesting; e.g. the person John, the set {John}, and the sets {{John}}, {{{John}}}, {{{{John}}}}, etc., would all be treated as exactly the same. 5. If you really want to distinguish a collection from its contents in natural language or in conceptual graphs, you must introduce a new concept type: e.g. "ships" vs. "a convoy of ships". [SHIP: {*}] represents "some ships" [CONVOY]->(MEMB)->[SHIP: {*}] represents "a convoy whose members are some ships" or simply "a convoy of ships". This seems like a more natural approach: you only make that distinction if it is significant, not as in set theory, where it keeps coming up whether you need it or not. 6. The basic difference between set theory and mereology is in the primitive operators: set theory has two primitives, which may be called element and subset; mereology has only one primitive, which may be called part-of. (What you call these operators is irrelevant -- the basic distinction is whether you have one or two primitive operators.) 7. Having two primitive operators in set theory leads to paradoxes, such as the set of all sets that are not members of themselves. In mereology, such paradoxes do not arise, since every collection is part of itself, and there cannot be a collection that is not part of itself. 8. Since conceptual graphs can express anything that could be expressed in natural language, it should be possible to express paradoxes as well. But you must do so by introducing new concept types, such as SET, CONVOY, etc. Then "cats" is represented [CAT: {*}], and "a set of cats" is represented [SET]->(MEMB)->[CAT: {*}]. 9. It is possible to translate any statement about sets, however paradoxical, into either English or conceptual graphs. But to do so, you have to introduce new concept types, such as SET. The paradoxes do not arise from the basic structure of the system, as they do in set theory. 10. Some people are puzzled by the distinction between [CAT: {*}] and [SET]->(MEMB)->[CAT: {*}]. I would answer by giving the translation to and from English, conceptual graphs, and predicate calculus: "cats" <=> [CAT: {*}] <=> (Es)(set(s) & count(s,n) & n>0 & (Ax)(element(x,s) -> cat(x))). "set of cats" <=> [SET]->(MEMB)->[CAT: Col{*}] <=> (Eu)(Ev)(SET(u) & set(v) & count(v,n) & n>0 & (Ax)(element(x,v) -> (cat(x) & MEMB(x,u))). The reason for the confusion is the distinction between the predicates SET(u) vs. set(v) and element(x,v) vs. MEMB(x,u). In lower case, the predicates set and element are being used in the target language of the formula operator phi (i.e. predicate calculus). In upper case, the predicates SET and MEMB arise from the source language conceptual graphs, which is being used to represent the original source language English. To distinguish these two more clearly, let's assume that the target language (predicate calculus) has been enriched with some axioms that define a particular version of set theory, say ZF. To be explicit, we could use the predicates zfset(v) and zfelement(x,v). The source language (English or conceptual graphs) has no built-in axioms for SET and MEMB until we state them. When we introduce the concept type SET and the relation type MEMB, they are nothing but uninterpreted symbols. In order to give meaning to those symbols, we have to state particular axioms for the version of set theory that we prefer; and those axioms could be stated in English or in conceptual graphs. In particular, we could use one version of set theory, say ZF, in predicate calculus and use it to model a different version of set theory, whose axioms would be stated in conceptual graphs. However, as I have said many times before, I prefer mereology as my underlying model for the {*} notation in conceptual graphs. Therefore, I would prefer to use a target language consisting of predicate calculus enriched by the axioms of mereology and with the primitives collection(v) and part-of(x,v). Then English statements about sets would be modeled in terms of mereology: "cats" <=> [CAT: {*}] <=> (Es)(collection(s) & nonempty(s) & (Ax)(part-of(x,s) -> cat(x))). "set of cats" <=> [SET]->(MEMB)->[CAT: Col{*}] <=> (Eu)(Ev)(SET(u) & collection(v) & nonempty(v) & (Ax)(part-of(x,v) -> (cat(x) & MEMB(x,u))). With this approach, we could use mereology as the primitive system, and state the axioms for ZF set theory (or any other set theory) in the source language, which could be either English or conceptual graphs. Although I advocate mereology as a more natural basis for NL semantics, I am not suggesting that the ANSI X3H4 committee or the DARPA-sponsored Knowledge Sharing Effort adopt mereology. Instead, what I suggest for the standards efforts is a subset of the CG theory that can be mapped directly to and from KIF and whose semantics would be based on the constructions used for KIF. For research on NL semantics, people could use any underlying model they found appropriate, but only a subset of it might be translatable to other knowledge bases via KIF. John Sowa