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## Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysi

 To: "[ontolog-forum]" "John F. Sowa" Fri, 16 Mar 2007 14:42:14 -0500 <45FAF316.5070105@xxxxxxxxxxx>
 ```Chris P, Chris M, and Pat,    (01) I just wanted to add a few words to support Pat's comment:    (02) CP> I find large tracts of the reference data are irredeemably > higher-order....    (03) PH> Irredeemably?? I am quite unconvinced that universals really > need true semantic higher-order predication. That would be a > very strong claim to defend. Are you sure that the cardinality > of the set of universals is really uncountable?    (04) The fact that you can quantify over predicates without going beyond FOL can be shown by using a construction defined by Quine (1953):    (05) Quine, Williard Van Orman (1953) "Reduction to a dyadic predicate," reprinted in Quine (1966), pp. 224-226.    (06) Quine, Willard Van Orman (1966) Selected Logic Papers, Enlarged Edition, Harvard University Press, Cambridge, MA, 1995.    (07) What Quine does is to show that any theory that has any set of predicates that take any number of arguments can be replaced by an equivalent theory in which there is only one dyadic predicate F.    (08) Instead of using Quine's notation, I'll assume the "apply" operator of LISP, which is slightly more perspicuous. Let apply be a dyadic operator, whose first argument is any N-adic predicate P and whose second argument is any list of N items,    (09) L = (a1, a2, ..., aN).    (010) Then define    (011) P(a1, a2, ..., aN) = apply(P,L).    (012) This apply operator applies P to the list of arguments in L. If you adapt this approach to predicate calculus, you get a formalism with apply as the only predicate, which takes one of the predicates of the original theory as its first argument and a list of N items as its second argument.    (013) Now quantifiers can freely range over P or the elements of L without making the theory higher order. This is exactly what CL does, but it eliminates the pretense of using the "apply" operator.    (014) This construction illustrates a crucial point: what creates problems in traditional higer-order logic is *not* the syntax, but the semantics. With Quine's construction, the apply operator, or the CL syntax, the crucial feature that keeps the logic within the realm of FOL is the semantics -- i.e., that there is a single domain D over which all the quantifiers range.    (015) What makes HOL complicated is the assumption of a hierarchy of domains:    (016) 1. Ordinary variables range over a set D0 of individuals.    (017) 2. Variables that represent first-order predicates range over D1, whose cardinality is 2^card(D0).    (018) 3. Variables that represent second-order predicates range over D2, whose cardinality is 2^card(D1).    (019) 4......    (020) Notice that the cardinality of these domains grows exponentially (and if the starting domain is countably infinite, the Ds grow very fast indeed).    (021) If you just assume a single domain D, in which everything of interest is a member, you never go beyond FOL.    (022) John    (023) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: 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 Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (024) ```
 Current Thread Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, (continued) Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, Christopher Menzel Re: [ontolog-forum] The Relation Between Logic and Ontology inMetaphysics, Chris Partridge Re: [ontolog-forum] The Relation Between Logic and Ontology inMetaphysics, Christopher Menzel Re: [ontolog-forum] The Relation Between Logic and Ontology inMetaphysics, Duane Nickull Re: [ontolog-forum] The Relation Between Logic and Ontology inMetaphysics, Christopher Menzel Re: [ontolog-forum] The Relation Between Logic and OntologyinMetaphysics, Chris Partridge Re: [ontolog-forum] The Relation Between Logic and OntologyinMetaphysics, Chris Menzel Re: [ontolog-forum] The Relation Between Logic and OntologyinMetaphysics, Christopher Menzel Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, Pat Hayes Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, Chris Partridge Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, John F. Sowa <= Re: [ontolog-forum] The Relation Between Logic and Ontology in Metaphysics, Pat Hayes Re: [ontolog-forum] Ontological Assumptions of FOL, Kathryn Blackmond Laskey Re: [ontolog-forum] Ontological Assumptions of FOL, John F. Sowa Re: [ontolog-forum] Ontological Assumptions of FOL, Pat Hayes Re: [ontolog-forum] Ontological Assumptions of FOL, John F. Sowa Re: [ontolog-forum] Ontological Assumptions of FOL, Christopher Menzel Re: [ontolog-forum] Ontological Assumptions of FOL, Christopher Menzel Re: [ontolog-forum] Ontological Assumptions of FOL, Chris Menzel Re: [ontolog-forum] Ontological Assumptions of FOL, paola . dimaio Re: [ontolog-forum] Ontological Assumptions of FOL, Christopher Menzel