[Inquiry] Re: Logic In Graphs

[Jon Awbrey]

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LIG.  Note 3

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I probably ought to explain my remark that re-entrant 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 re-entrant 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 non-negative 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

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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