“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
Re: Dick Lipton & Ken Regan • (1) • (2)
We continue with the differential analysis of the proposition in Example 1.
Example 1
 |
(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 enlargement map 
computed in Post 5 and graphed in Post 6, is shown again here:
The tacit extension 
computed in Post 7 and graphed in Post 8, is shown again here:
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 …
Resources
Like this:
Like Loading...
Related
![\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bsmallmatrix%7D++%26+p+q+r+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+p+q+%5Ctexttt%7B%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dr++%26+%2B+%26+p+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+r+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+p+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dq+%5C%2C+%5Cmathrm%7Bd%7Dr++%5C%5C%5B4pt%5D++%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+q+r+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+q+%5Ctexttt%7B%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D+%5Cmathrm%7Bd%7Dr++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%28%7D+q+%5Ctexttt%7B%29%7D+r+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5C%2C+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26+%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%28%7D+q+%5Ctexttt%7B%29%28%7D+r+%5Ctexttt%7B%29%7D+%5C%2C%5Ccdot%5C%2C++%5Cmathrm%7Bd%7Dp+%5C%2C+%5Cmathrm%7Bd%7Dq+%5C%2C+%5Cmathrm%7Bd%7Dr++%5Cend%7Bsmallmatrix%7D&bg=ffffff&fg=000000&s=2 "\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}")
![\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%2A%7B8%7D%7Bl%7D%7D++%26++p+q+r+%7E++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%7E%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%7E%7D++%5C%5C%5B4pt%5D++%2B+%26++p+q+r+%7E++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%7E%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%29%7D++%26+%2B+%26++p+q+r+%7E++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dr+%5Ctexttt%7B%7E%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0 "\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}")
Pingback: Frankl, My Dear : 10 | Inquiry Into Inquiry
Pingback: Survey of Differential Logic • 1 | Inquiry Into Inquiry