Re: Dick Lipton & Ken Regan • (1) • (2)
We continue with the differential analysis of the proposition in Example 1.
Example 1
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(1) |
A proposition defined on one universe of discourse has natural extensions to larger universes of discourse. As a matter of course in a given context of discussion, some of these extensions come to be taken for granted as the most natural extensions to make in passing from one universe to the next and they tend to be assumed automatically, by default, in the absence of explicit notice to the contrary. These are the tacit extensions that apply in that context.
Differential logic, at the first order of analysis, treats extensions from boolean spaces of type to enlarged boolean spaces of type
In this setting
but we use different letters merely to distinguish base and differential features.
In our present example, the tacit extension of
is the boolean function
defined by the following equation:
The boolean expansion of takes the following form:
In other words, is simply
on the base variables
extended by a tautology — commonly known as a “Don’t Care” condition — on the differential variables
To be continued …
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