(2) Construct a Dictionary, making m and m (or m and m′) represent the pair of codivisional Classes, and x (or x′) and y (or y′) the other two.
(3) Translate the proposed Premisses into abstract form.
(4) Represent them, together, on a Triliteral Diagram.
(5) Ascertain what Proposition, if any, in terms of x and y, is also represented on it.
(6) Translate this into concrete form.
It is evident that, if the proposed Premisses were true, this other Proposition would also be true. Hence it is a Conclusion consequent from the proposed Premisses.
(1)
“No son of mine is dishonest;
People always treat an honest man with respect”.Taking “men” as Univ., we may write these as follows:—
“No sons of mine are dishonest men;
All honest men are men treated with respect”.We can now construct our Dictionary, viz. m = honest; x = sons of mine; y = treated with respect.
(Note that the expression “x = sons of mine” is an abbreviated form of “x = the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)
The next thing is to translate the proposed Premisses into abstract form, as follows:—
“No x are m′;
All m are y”.
[pg061]Next, by the process described at [p. 50], we represent these on a Triliteral Diagram, thus:—
Next, by the process described at [p. 53], we transfer to a Biliteral Diagram all the information we can.
The result we read as “No x are y′” or as “No y′ are x,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose
“No x are y′”,
which, translated into concrete form, is
“No son of mine fails to be treated with respect”.
(2)
“All cats understand French;
Some chickens are cats”.Taking “creatures” as Univ., we write these as follows:—
“All cats are creatures understanding French;
Some chickens are cats”.We can now construct our Dictionary, viz. m = cats; x = understanding French; y = chickens.
The proposed Premisses, translated into abstract form, are
“All m are x;
Some y are m”.In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get the three Propositions
(1) “Some m are x;
(2) No m are x′;
(3) Some y are m”.
The Rule, given at [p. 50], would make us take these in the order 2, 1, 3.
This, however, would produce the result
[pg062]So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition “Some m are x” is already represented on the Diagram.
Transferring our information to a Biliteral Diagram, we get
This result we can read either as “Some x are y” or “Some y are x”.
After consulting our Dictionary, we choose
“Some y are x”,
which, translated into concrete form, is
“Some chickens understand French.”
(3)
“All diligent students are successful;
All ignorant students are unsuccessful”.Let Univ. be “students”; m = successful; x = diligent; y = ignorant.
These Premisses, in abstract form, are
“All x are m;
All y are m′”.These, broken up, give us the four Propositions
(1) “Some x are m;
(2) No x are m′;
(3) Some y are m′;
(4) No y are m”.
which we will take in the order 2, 4, 1, 3.
Representing these on a Triliteral Diagram, we get
And this information, transferred to a Biliteral Diagram, is
Here we get two Conclusions, viz.
“All x are y′;
All y are x′.”[pg063]And these, translated into concrete form, are
“All diligent students are (not-ignorant, i.e.) learned;
All ignorant students are (not-diligent, i.e.) idle”. (See [p. 4].)(4)
“Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict ‘guilty’ was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour”.Let Univ. be “the prisoners who were put on their trial at the last Assizes”; m = who were sentenced to imprisonment; x = against whom the verdict ‘guilty’ was returned; y = who were sentenced to hard labour.
The Premisses, translated into abstract form, are
“All x are m;
Some m are y”.Breaking up the first, we get the three
(1) “Some x are m;
(2) No x are m′;
(3) Some m are y”.
Representing these, in the order 2, 1, 3, on a Triliteral Diagram, we get
Here we get no Conclusion at all.
You would very likely have guessed, if you had seen only the Premisses, that the Conclusion would be
“Some, against whom the verdict ‘guilty’ was returned,
were sentenced to hard labour”.But this Conclusion is not even true, with regard to the Assizes I have here invented.
“Not true!” you exclaim. “Then who were they, who were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict ‘guilty’ returned against them, or how could they be sentenced?”
Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded ‘guilty’. So no verdict was returned at all; and they were sentenced at once.]
I will now work out, in their briefest form, as models for the Reader to imitate in working examples, the above four concrete Problems.
[pg064](1) [see [p. 60]]
“No son of mine is dishonest;
People always treat an honest man with respect.”