[SCL] alternate notions for vocabulary and interpretation

[Pat Hayes]

>I have been thinking about the example '(((f a) a) a)' discussed in the SCL
>Core Semantics and have been playing around with the following alternate
>definition of vocabulary and interpretation.

Yes, I also thought of this, ie to extend the interpretation mappings 
to terms as well as names, and use them to define the 'foldings'. 
This might be a good way to present the idea and has the merit of 
making the connection to the syntax quite evident. I rejected it in 
the current draft as I wanted to keep the exposition as close as 
possible to a conventional FO model theory.

>An additional motivation is to
>define an abstract (text-independent) notion of vocabulary and
>interpretation.

Well, it is to the extent that you consider 'position' to be text-independent.

Pat


>
>The vocabulary of any SCL text T is a sextuple
>(ON, RN, FN, OT, RT, FT)
>where ON is the set of names of T in object position,
>where RN is the set of names of T in relation position,
>where FN is the set of names of T in function position,
>where OT is the set of terms of T in object position,
>where RT is the set of terms of T in relation position, and
>where FT is the set of terms of T in function position,
>satisfying
>any elementary term in OT is in ON,
>any elementary term in RT is in RN, and
>any elementary term in FT is in FN
>and
>for any composite term (t, t1, ... tn) in OT or RT or FT
>t is in FT and ti is in OT for 1 <= i <= n.
>
>A bare interpretation of this vocabulary has
>a nonempty set U called the universe and three maps
>int : ON --> U,
>rel : RN --> rel(U), and
>fun : FN --> fun(U).
>
>A folded interpretation is a bare interpretation with three extended maps
>int* : OT --> U,
>rel* : RT --> rel(U), and
>fun* : FT --> fun(U),
>and two folds
>relation : RU --> rel(U) with RU a subset of U and
>function : FU --> fun(U) with FU a subset of U,
>where
>if x is in both ON and RN
>then int(x) is in RU and relation(int(x)) = rel(x),
>if x is in both ON and FN
>then int(x) is in FU and function(int(x)) = fun(x),
>and where
>int* restricts to int for elementary terms in OT,
>rel* restricts to rel for elementary terms in RT,
>fun* restricts to fun for elementary terms in FT,
>and
>for any composite term (t, seq) in OT
>the value of fun*[t] applied to the sequential extension int*[seq]
>is int*[(t, seq)],
>for any composite term (t, seq) in RT
>the value of fun*[t] applied to the sequential extension int*[seq]
>is some z in RU and rel*[(t, seq)] = relation[z], and
>for any composite term (t, seq) in FT
>the value of fun*[t] applied to the sequential extension int*[seq]
>is some y in FU and fun*[(t, seq)] = function[y].
>
>Robert E. Kent
>rekent at ontologos.org
>
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