Complete Works of Lewis Carroll (155 page)

  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
′
.”

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”. 

(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.

(1)

“No son of mine is dishonest;

  People always treat an honest man with respect.”

Univ.
“men”;
m
 = honest;
x
 = my sons;
y
 = treated with respect.

“No
x
are
m
′
;

 All
m
are
y
.”

∴
“No
x
are
y
′
.”

i.e.
“No son of mine ever fails to be treated with respect.”

(2)

“All cats understand French;

  Some chickens are cats”.

Univ.
“creatures”;
m
 = cats;
x
 = understanding French;
y
 = chickens.

“All
m
are
x
;

 Some
y
are
m
.”

∴
“Some
y
are
x
.”

i.e.
“Some chickens understand French.”

(3)

“All diligent students are successful;

  All ignorant students are unsuccessful”.

Univ.
“students”;
m
 = successful;
x
 = diligent;
y
 = ignorant.