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Re: [ontolog-forum] First-Order Semantics

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Kathryn Blackmond Laskey <klaskey@xxxxxxx>
Date: Sun, 10 Jun 2007 11:43:11 -0400
Message-id: <p06110403c291c8846d9b@[]>
>[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)

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