Example 1. A Square Rigging (concl.)
If we eliminate from view the regions of 
What we have achieved in this example is to give a differential description of a simple dynamic process. We did this by embedding a directed graph, representing the state transitions of a finite automaton, in the share of a boolean lattice or n‑cube cut out by nullifying all the regions the dynamics outlaws.
With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description. In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of n‑cubes without being forced to actualize all their points.
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is one‑dimensional we can easily fit the second order extension 
within the compass of a single venn diagram, charting the pair of converging trajectories as shown in the following Figure.
suppose we are given the initial condition 
and the second order differential law 
Since the equation 
we may infer two possible trajectories, as shown in the following Table.
is a stable attractor or terminal condition for both starting points.
are changed or unchanged in the next moment. To know that one would have to determine 
and so on, pursuing an infinite regress. In order to rest with a finitely determinate system it is necessary to make an infinite assumption, for example, that 
for all 
greater than some fixed value 
Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.
is 
If the feature 
may be taken as an attribute of the same object or state which tells it is changing significantly with respect to the property 
In practice, differential features acquire their meaning through a class of differential inference rules.
will be true in the next moment of observation. Taken all together we have the fourfold scheme of inference shown below.
be a logical basis containing one boolean variable or logical feature 
Corresponding to the basis 
which serves whenever we need to make explicit mention of the symbols used in our formulas and representations.
of points (cells, vectors, interpretations) has cardinality 
and is isomorphic to 
Moreover, 
may be identified with the set of singular propositions 
is algebraically dual to 
Here, 
is interpreted as denoting the constant function 
amounting to the linear proposition of rank 
while 
of rank 
and 
is understood as denoting the constant function 
propositions in the universe of discourse ![\mathcal{X}^\bullet = [\mathcal{X}],](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BX%7D%5E%5Cbullet+%3D+%5B%5Cmathcal%7BX%7D%5D%2C&bg=ffffff&fg=000000&s=0&c=20201002)
collectively forming the set 
![[\mathrm{E}\mathcal{A}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathrm%7BE%7D%5Cmathcal%7BA%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
makes the task of solving differential propositions relatively straightforward. The solution set of the differential proposition 
is the set of models 
in 
Finding all models of 
the extended interpretations in 
which satisfy 
of a universe of discourse ![[\mathcal{A}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathcal%7BA%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is formed by taking the initial basis 
together with the differential basis 
Thus we have the following formula.
![\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BE%7DA%5E%5Cbullet+%3D+%5B%5Cmathrm%7BE%7D%5Cmathcal%7BA%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is the full collection of points and functions, or interpretations and propositions, based on the extended set of features 
a fact summed up in the following notation.![\mathrm{E}A^\bullet ~=~ [\mathrm{E}\mathcal{A}] ~=~ [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BE%7DA%5E%5Cbullet+%7E%3D%7E+%5B%5Cmathrm%7BE%7D%5Cmathcal%7BA%7D%5D+%7E%3D%7E+%5Ba_1%2C+%5Cldots%2C+a_n%2C+%5Cmathrm%7Bd%7Da_1%2C+%5Cldots%2C+%5Cmathrm%7Bd%7Da_n%5D&bg=ffffff&fg=000000&s=0&c=20201002)
the following type.
we arrive at the launchpad of our space explorations.
taken by itself. In like fashion, the adjective extended or the substantive bundle is systematically attached to any construct associated with the full complement of 
features.
sometimes written 
takes the form 
Strictly speaking, the name cotangent space is probably more correct for this construction but since we take up spaces and their duals in pairs to form our universes of discourse it allows our language to be pliable here.
is a set consisting of two differential propositions, 
where 
is a proposition with the logical value of 
Each component 
operating under the ordered correspondence 
A measure of clarity is achieved, however, by acknowledging the differential usage with a superficially distinct type 
whose sense may be indicated as follows.
and 
may appear to be identical sets of binary vectors, but taking a view at that level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
Inquiry Into Inquiry 