“It doesn’t matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn’t make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.
Robert Musil • The Man Without Qualities
We continue with the differential analysis of the proposition in Example 1.
The difference operator is defined as the difference between the enlargement operator and the tacit extension operator
The difference map is the result of applying the difference operator to the function When the sense is clear, we may refer to simply as the difference of
In boolean spaces there is no difference between the sum and the difference so the difference operator is equally well expressed as the exclusive disjunction or symmetric difference In this case the difference map can be computed according to the formula
The action of on our present example, can be computed from the data on hand according to the following prescription.
The difference map is the sum of the enlargement map and the tacit extension
Here we adopt a paradigm of computation for that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.
The terms of the enlargement map are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.
The terms of the tacit extension are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.
Finally, the terms of the difference map are obtained by overlaying the displays for and and taking their boolean sum entry by entry.
Notice that the “loop” or “no change” term cancels out, leaving 14 terms in the end.
To be continued …
- Differential Logic : Introduction
- Differential Propositional Calculus
- Differential Logic and Dynamic Systems