Jon Awbrey
Registered: Feb 2004
Posts: 557 
Logic In Graphs
LIG. Note 3
I probably ought to explain my remark that reentrant forms are
related to the "differential extension" of propositional calculi,
as this may not be immediately clear from the links that I listed.
The easiest way to see this is to consider the reentrant form "x = (x)".
As long as we stick to classical logic this will remain a false statement,
but we can understand its relation to the assignment statement "x := (x)",
that is, a command that sets the next value of x equal to the negation of
the current value of x, where x is a boolean variable, x in B = {0, 1}.
NB. In this text, I use parentheses for logical negation and
angle brackets for argument lists in functional notation.
A more explicit model of what is happening here may be formulated
by considering the space of functions {f : N > B}, where N is the
set of natural numbers or nonnegative integers N = {0, 1, 2, ...}.
In a typical scenario we may consider N to be a discrete dimension
of time, and x<n> to be the value of the variable x at the time n.
The assignment statement x := (x) is tantamount to a differential
equation that has two solutions in the function space {f : N > B}.
With initial condition x<0> = 0, we get the sequence 0, 1, 0, 1, ...
With initial condition x<0> = 1, we get the sequence 1, 0, 1, 0, ...
Another way to write this differential equation, strictly speaking,
a boolean finite difference equation, is via the statement "dx = 1",
that is, an assertion that the first difference of x is a constant 1.
Moreover, since we are working in a logical domain, it is sufficient
simply to write "dx", reading "dx" as a "differential proposition"
that says that the first difference of x is true for all n in N.
We should note, however, that dx is not the same thing
as an imaginary truth value, since it does not satisfy
the initial equation, as in the expression "dx = (dx)".
The differential proposition dx asserts that x changes
at every step, while the differential proposition (dx)
asserts that dx = 0, in other words, that the value of
x<n> is the same for all values of n.
Jon Awbrey

Last edited by Jon Awbrey on 03082005 at 07:36 PM
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