Is there anyone who has rigorous understanding of integration
and differentiation in the continuous case who would be able
to explain the discrete case?
I’ve been looking at Pascals traingle and the most striking
feature of this triangle seems to be the fact that it embodies
(aspects of) integration and differentiation in multiple dimensions in
the discrete case.
For any non-negative number of logical variables, the number of
propositions which have X interpretations can be seen as the surface
area of the propositions which have (X+1) interpretations.
The dimensionality is determined by the number of interpretations, while
the ‘resolution’ is determined by the number of propositional variables.
All propositions for 3 logical variables are partitioned (on the number
of interpretations that satisfy them) as follows:
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256

The 8 propositions which have one interpretation are the surface area of
the 28 propositions which have two interpretations. Likewise (except for
the fact that the dimensionality goes from 2 to 3), the 28 propositions
which have three interpretations can be seen as the surface area of
the 56 propositions which have four interpretations.

This link between dimensionality and the triangle of Pascal can be
observed from the following ‘interpretation’:
    1                
   1 1                
  1 2 1              
 1 3 3 1              
1 4 6 4 1            

Every layer indicates how many (spaces), points, lines, faces, volumes,
etc. are involved, given a non-negative number of points.
The first layer, "1", is the case for 0 points.
The second layer, "1 1", is the case for 1 point.
The third layer, "1 2 1", indicates that a line is a
relation between two points.
The fourth layer, "1 3 3 1", indicates that a face is a relation
between three lines which is, in turn, a relation between 3 points.
The fifth layer, "1 4 6 4 1" is simply a tetrahedron.
4 points, 6 lines, 4 faces, 1 volume.

The other (obvious) interpretation is the link between logic and sets
since it demonstrates how propositions can be seen as subsets of
interpretations, while interpretations can be seen as subsets of logical
variables. Every layer in the triangle of Pascal indicates how a powerset
of a given set is partitioned on cardinality. Moreover, since the powerset
is a set itself, certain powersets can be identified with powersets of
powersets. For instance, the 256 propositions in 3 logical variables
can be seen as elements from the powerset of 8 possible interpretations.
Since the powerset of a set of 3 elements has 8 elements, the elements
from the powerset of a set of 8 elements coincide with the elements
from the powerset of (the powerset of a set of 3 elements).
Note that propositions can always be seen as elements from the powerset of
(the powerset of the set of logical variables).
As far as I understand this, I think the powerset operator can be seen as
the way to dualize between propositions and interpretations.
However I don’t understand the exact distinction between interpretations
and propositions although I reckon they can be identified somewhat.

                                                regards, Niek