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## Re: [ontolog-forum] First-Order Semantics

 To: "[ontolog-forum] " Kathryn Blackmond Laskey Sun, 10 Jun 2007 11:43:11 -0400
 ```>[JS] ...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. > >What makes HOL complicated is the assumption of a hierarchy of domains: > > 1. Ordinary variables range over a set D0 of individuals. > > 2. Variables that represent first-order predicates range over D1, > whose cardinality is 2^card(D0). > > 3. Variables that represent second-order predicates range over D2, > whose cardinality is 2^card(D1). > > 4...... > >Notice that the cardinality of these domains grows exponentially >(and if the starting domain is countably infinite, the Ds grow >very fast indeed). > >If you just assume a single domain D, in which everything of interest >is a member, you never go beyond FOL.    (01) [KBL] A typed logic has quantifiers for each type, ranging over entities of that type. Typed logics can have first-order semantics. CL is untyped, but can represent typed logics. I can use CL to reason about predicates or functions, and write quantified CL expressions in which the variables range only over the predicates and/or functions.    (02) The key assumption, as I understand it, is that in first-order semantics, the domain D is a set, and not a proper class. We can create syntactic conventions that restrict variable substitutions in quantified expressions to subsets of D, but the restricted ranges are subsets of a set, not subclasses of a proper class. This simplifies the semantics.    (03) Is that right?    (04) Kathy    (05) _________________________________________________________________ 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    (06) ```
 Current Thread Re: [ontolog-forum] First-Order Semantics, Kathryn Blackmond Laskey <= Re: [ontolog-forum] First-Order Semantics, Pat Hayes Re: [ontolog-forum] First-Order Semantics, John F. Sowa Re: [ontolog-forum] First-Order Semantics, Pat Hayes Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk Re: [ontolog-forum] First-Order Semantics, John F. Sowa Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Christopher Menzel