[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INTRO.  Note 33

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3.2.2.  Umpire Operators

| Nota Bene
|
| For this production, the part of the upper-case
| Greek character Upsilon will be played by "!Y!".
|
| In addition, I am going to experiment with marking
| Cactus Language or Ref Log expressions by means of
| single underscore marks at their beginning and end.

In order to get a handle on the space of higher order propositions and
eventually to carry out a functional approach to quantification theory,
it serves to construct some specialized tools.  Specifically, I define
a higher order operator !Y!, called the "umpire operator", which takes
up to three propositions as arguments and returns a single truth value
as the result.  Formally, this so-called "multi-grade" property of !Y!
can be expressed as a union of function types, in the following manner:

   !Y! : |_|^(m = 1, 2, 3) ((B^k -> B)^m -> B).

In contexts of application the intended sense can be discerned by
the number of arguments that actually appear in the argument list.
Often, the first and last arguments appear as indices, the one in
the middle being treated as the main argument while the other two
arguments serve to modify the sense of the operation in question.
Thus, we have the following forms:

   !Y!_p^r q  =  !Y!(p, q, r)

   !Y!_p^r : (B^k -> B) -> B

The intention of this operator is that we evaluate the proposition q
on each model of the proposition p and combine the results according
to the method indicated by the connective parameter r.  In principle,
the index r might specify any connective on as many as 2^k arguments,
but usually we have in mind a much simpler form of combination, most
often either collective products or collective sums.  By convention,
each of the accessory indices p, r is assigned a default value that
is understood to be in force when the corresponding argument place
is left blank, specifically, the constant proposition 1 : B^k -> B
for the lower index p, and the continued conjunction or continued
product operation ]¯[ for the upper index r. `Taking the upper
default value gives license to the following readings:

   1.  !Y!_p q  =  !Y!(p, q)  =  !Y!(p, q, product).

   2.  !Y!_p    =  !Y!(p, -, product) : (B^k -> B) -> B.

This means that !Y!_p q = 1 if and only if q holds for all models of p.
In propositional terms, this is tantamount to the assertion that p => q,
or that _(p (q))_ = 1.

Throwing in the lower default value permits the following abbreviations:

   3.  !Y!q  =  !Y!(q)  =  !Y!_1 q  =  !Y!(1, q, product).

   4.  !Y!   =  !Y!(1, -, product) : (B^k -> B) -> B.

This means that !Y!q = 1 if and only if q holds for the whole
universe of discourse in question, that is, if and only q is the
constantly true proposition 1 : B^k -> B.  The ambiguities of this
usage are not a problem so long as we distinguish the context of
definition from the context of application and restrict all
shorthand notations to the latter.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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