Transformations of Discourse
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well‑spring of the times, the fons et origo of an unfathomable transformation.
— Robert Musil • The Man Without Qualities
Here we take up the general study of logical transformations, or maps relating one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
The goal is to answer a single question: What is a propositional tangent functor? In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
As a first step we examine the types of transformations we already know as extensions and projections and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.
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‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit.
where 
may be read as a temporal parameter indicating the present time of the state and where 
is the decimal equivalent of the binary numeral 
with a subscript 
equal to the numerator of its rational index, taking for granted the constant denominator of 
In that way the temporal succession of states can be reckoned by a parallel round‑up rule. Namely, if 
is any pair of adjacent digits in the state index 
then the value of 
in the next state is 
with the dyadic rationals (or the binary fractions) in the half-open interval 
Formally and canonically, a state 
is indexed by a fraction 
whose denominator is the power of two 
and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.
are just the values 
for 
where 
is defined as being identical to 
To form the binary index 
of the state 
Expressed by way of algebraic formulas, the rational index 
Those are the trajectories generated subject to the dynamic law 
where it’s understood all higher order differences are equal to 
and all higher differences 
are fixed, the state vectors vary only with respect to their projections as points of 
Thus there is just enough space in a planar venn diagram to plot all the orbits and to show how they partition the points of 
It turns out there are exactly two possible orbits, of eight points each, as shown in the following Figure.
The discussion to follow considers a class of trajectories having the property that 
for all 
greater than a fixed value 
and indulges in the use of a picturesque vocabulary to describe salient classes of those curves.
exhibited at each point of any trajectory one may consider. With respect to any point of the corresponding curve let us call that highest order differential feature 
‑gear curves.
and 
has a canonical expression 
in the set 
The tacit extension 
may then be expressed as a logical conjunction 
where 
is a logical tautology using all the variables in 
The following Table shows how the tacit extensions 
of the propositions 
may be expressed in terms of the extended basis 
![\text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]](https://s0.wp.com/latex.php?latex=%5Ctext%7BTacit+Extension+of%7D%7E+%5BA%5D+%7E%5Ctext%7Bto%7D%7E+%5BA%2C+%5Cmathrm%7Bd%7DA%5D&bg=ffffff&fg=000000&s=1&c=20201002)
![Tacit Extension of [A] to [A, dA]](https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-tacit-extensions.png)
the above analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like 
or 
to considering the collection of outcomes, the outgoing changes or singular dispositions springing or stemming from that state.
even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions representing the same proposition while appreciating the full diversity of a proposition’s functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
is a subset of another basis 
we say any proposition 
has a tacit extension to a proposition 
and we say the space 
has an automatic embedding within the space 
is defined in such a way that 
puts the same constraint on the variables of 
as the proposition 
initially put on 
while it puts no constraint on the variables of 
and 
the tacit extension from 
where the bases 
may be silent.![[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7BE%7D%5Cmathcal%7BX%7D%5D+%3D+%5BA%2C+%5Cmathrm%7Bd%7DA%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
Over the extended alphabet 
of cardinality 
we generate the set of points 
of cardinality 
which bears the following chain of equivalent descriptions.![\begin{array}{lll} \mathrm{E}X & = & \langle A, \mathrm{d}A \rangle \\[4pt] & = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \} \\[4pt] & = & \{ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~ \texttt{(} A \texttt{)} \mathrm{d}A,~ A \texttt{(} \mathrm{d}A \texttt{)},~ A ~ \mathrm{d}A \}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++%5Cmathrm%7BE%7DX+%26+%3D+%26+%5Clangle+A%2C+%5Cmathrm%7Bd%7DA+%5Crangle++%5C%5C%5B4pt%5D++%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+A+%5Ctexttt%7B%29%7D%2C+A+%5C%7D+%7E%5Ctimes%7E+%5C%7B+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7DA+%5Ctexttt%7B%29%7D%2C+%5Cmathrm%7Bd%7DA+%5C%7D++%5C%5C%5B4pt%5D++%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+A+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7DA+%5Ctexttt%7B%29%7D%2C%7E++%5Ctexttt%7B%28%7D+A+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7DA%2C%7E++A+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7DA+%5Ctexttt%7B%29%7D%2C%7E++A+%7E+%5Cmathrm%7Bd%7DA+%5C%7D.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
at root isomorphic to 
An element of ![\mathrm{E}X^\bullet = [A, \mathrm{d}A]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BE%7DX%5E%5Cbullet+%3D+%5BA%2C+%5Cmathrm%7Bd%7DA%5D&bg=ffffff&fg=000000&s=0&c=20201002)
the basic dispositions in 
each of type 
There are 
propositions in 
as detailed in the following Table.
Thus the first set of propositions 
is automatically embedded in the present set 
and the corresponding inclusions are indicated at the far left margin of the Table.
Inquiry Into Inquiry 