The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
All y are x
CHAPTER IV.
INTERPRETATION OF BILITERAL DIAGRAM WHEN MARKED WITH COUNTERS.
The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.
As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.
First, let us suppose that we find a Red Counter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“Some xy exist” = “Some x are y” = “Some y are x”.
Similarly we may interpret a Red Counter, when placed in the North-East, or South-West, or South-East Cell.
Next, let us suppose that we find a Grey Counter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“No xy exist” = “No x are y” = “No y are x”.
Similarly we may interpret a Grey Counter, when placed in the North-East, or South-West, or South-East Cell.
Next, let us suppose that we find a Red Counter placed on the partition which divides the North Half.
We know that this represents the Proposition “Some x exist.”
Similarly we may interpret a Red Counter, when placed on the partition which divides the South, or West, or East Half.
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Next, let us suppose that we find two Red Counters placed in the North Half, one in each Cell.
We know that this represents the Double Proposition “Some x are y and some are y′”.
Similarly we may interpret two Red Counters, when placed in the South, or West, or East Half.
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Next, let us suppose that we find two Grey Counters placed in the North Half, one in each Cell.
We know that this represents the Proposition “No x exist”.
Similarly we may interpret two Grey Counters, when placed in the South, or West, or East Half.
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Lastly, let us suppose that we find a Red and a Grey Counter placed in the North Half, the Red in the North-West Cell, and the Grey in the North-East Cell.
We know that this represents the Proposition, “All x are y”.
[Note that the Half, occupied by the two Counters, settles what is to be the Subject of the Proposition, and that the Cell, occupied by the Red Counter, settles what is to be its Predicate.]
Similarly we may interpret a Red and a Grey counter, when placed in any one of the seven similar positions
Red in North-East, Grey in North-West;
Red in South-West, Grey in South-East;
Red in South-East, Grey in South-West;
Red in North-West, Grey in South-West;
Red in South-West, Grey in North-West;
Red in North-East, Grey in South-East;
Red in South-East, Grey in North-East.