[SCL] 'moderately' folded semantics

[Pat Hayes]

>Pat, John and other SCL folk,
>
>In another forum today, John discussed the connections between 
>various indexicals, such as pronouns, variables, and names.
>
>John Sowa wrote:
>For the Common Logic project, Pat Hayes suggested, and I quickly 
>agreed, that we need not make any syntactic distinction between 
>variables and proper names. If a character string is introduced by a 
>quantifier, then it is a variable. Otherwise, it is assumed to be a 
>name, which is treated logically in the same way as a variable 
>introduced with an existential quantifier and having global scope 
>(i.e., the entire text in which it occurs).
>
>Linguistically, a proper name (noun) corresponds to an object (a 
>specific individual), whereas a common name (noun) corresponds to a 
>class or a relation. However, in the SCL there is no distinction 
>between proper names and common names -- there is only names, and we 
>can quantify over any name, no matter whether it represents an 
>object, a class or a relation. As an example, the two expressions 
>'married(John, Abigail)' and '(exists (m) (exists (j) (exists (a) (m 
>j a))))' should be equivalent. Note that the term 'married' is in 
>relation position in the first expression, and the term 'm' is in 
>relation position in the second expression.

They are equivalent as long as nothing else is said using the name 
'married'. But in a Webbish open-world context, you never know 
whether something else has been said. In effect you can never 'reach' 
the global scope level except by using names. Notationally, this 
would be like putting brackets around the entire WWWeb.

>The SCL document suggests using as default the 'minimally' folded 
>semantics that when computing contextual signatures, the assumption 
>(for relation position) is that "a name which occurs only in 
>relation position is not a denoting term. Such names are not 
>required to denote individuals (although in the SCL semantics, they 
>may do so)." But John's suggestion that in the SCL an unquantified 
>name "is treated logically in the same way as a variable introduced 
>with an existential quantifier and having global scope" seems to be 
>more in keeping with a 'moderately' folded semantics, which makes 
>the assumption that "a name which occurs in relation position also 
>occurs in object position". This may simplify the SCL semantics. It 
>is not the 'maximally' folded semantics, since we are only assuming 
>that "all names denote individuals". We are not "associating every 
>individual with both a relational and a functional extension".

But maximal folding is vacuous once we have moderate folding, since 
we can simply associate the empty relation extension and a suitable 
constant functional extension to every non-relational individual. 
This does not extend the universe. The 'harm' is done (ie the MT 
fails to correspond to a conventional FO MT) in the case where a 
relation name used purely relationally forces the universe to contain 
entities that it would not otherwise contain, as in Horrock's example:

(forall (x) (= x a))
(P a)
(not (Q a))

which is FO satisfiable, and SCL satisfiable (with an identical 
model) but does not have a satisfying moderately folded 
interpretation.

Pat

>
>Robert E. Kent
><mailto:rekent at ontologos.org>rekent at ontologos.org
>
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