[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D22

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Tacit Extensions

| I would really like to have slipped imperceptibly into this lecture, as
| into all the others I shall be delivering, perhaps over the years ahead.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]

Strictly speaking, however, there is a subtle distinction in type between
the function f_i : X -> B and the corresponding function g_j : EX -> B,
even though they share the same logical expression.  Being human, we
insist on preserving all of the aesthetic delights afforded by the
abstractly unified form of the "cake" while giving up none of the
diverse contents that its substantive consummation can provide.
In short, we want to maintain the logical equivalence of
expressions that represent the same proposition, while
recognizing the full diversity of its functional and
its typical representations.  Both perspectives,
and all of the levels of abstraction extending
through them, have their uses, as we shall see
in time.

Because this special circumstance points up an important general theme,
it is a good idea to discuss it more carefully.  Whenever there arises
a situation like this, where one alphabet !X! is a subset of another
alphabet !Y!, then we say that any proposition f : <|!X!|> -> B has
a "tacit extension" to a proposition !e!f : <|!Y!|> -> B, and that
the space (<|!X!|> -> B) has an "automatic embedding" within the
space (<|!Y!|> -> B).  The extension is defined in such a way
that !e!f puts the same constraint on the variables of X that
are contained in Y as the proposition f initially did, while
it puts no constraint on the variables of Y outside of X,
in effect, conjoining the two constraints.

If the variables in question are indexed as !X! = {x_1, ..., x_n}
and !Y! = {x_1, ..., x_n, ..., x_n+k}, then the definition of the
tacit extension from !X! to !Y! may be expressed in the form of
an equation:

   !e!f(x_1, ..., x_n, ..., x_n+k)  =  f(x_1, ..., x_n).

On formal occasions, such as the present context of definition,
the tacit extension from !X! to !Y! is explicitly symbolized by
the operator !e! : (<|!X!|> -> B) -> (<|!Y!|> -> B), where the
appropriate alphabets !X! and !Y! are understood from context,
but normally one may leave the "!e!" silent.

Let's explore what this means for the present Example.
Here, !X! = {A} and !Y! = E!X! = {A, dA}.  For each of
the propositions f_i over X, specifically, those whose
expression e_i lies in the collection {0, (A), A, 1},
the tacit extension !e!f of f to EX can be phrased as
a logical conjunction of two factors, f_i = e_i.!t!,
where !t! is a logical tautology that uses all the
variables of !Y! - !X!.  Working in these terms,
the tacit extensions !e!f of f to EX may be
explicated as shown in Table 15.

Table 15.  Tacit Extensions of [A] to [A, dA]
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|                                                                     |
|    0    =      0  . ((dA), dA)        =              0              |
|                                                                     |
|   (A)   =     (A) . ((dA), dA)        =      (A)(dA) + (A) dA       |
|                                                                     |
|    A    =      A  . ((dA), dA)        =       A (dA) +  A  dA       |
|                                                                     |
|    1    =      1  . ((dA), dA)        =              1              |
|                                                                     |
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In its effect on the singular propositions over X, this analysis has an
interesting interpretation.  The tacit extension takes us from thinking
about a particular state, like A or (A), to considering the collection
of outcomes, the outgoing changes or the singular dispositions, that
spring from that state.

Jon Awbrey

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