Chapter 38
_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. pg037 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.
* * * * *
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.
* * * * *
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.
* * * * *
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_.] pg038 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.
Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not only _represent_ Propositions, but also _interpret_ Diagrams when marked with Counters.
The Questions and Answers should be like this:--
Q. Represent "No x' are y'." A. Grey Counter in S.E. Cell. Q. Interpret Red Counter on E. partition. A. "Some y' exist." Q. Represent "All y' are x." A. Red in N.E. Cell; Grey in S.E. Q. Interpret Grey Counter in S.W. Cell. A. "No x'y exist" = "No x' are y" = "No y are x'". &c., &c.
At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.
[Work Examples § =1=, 5-8 (p. 97).]
pg039