Complete Works of Lewis Carroll (144 page)

[The Reader should make out all these for himself.
In the “books” example, these three Propositions would be “No English books are new”, &c.]

We see that this
one
Diagram has now served to present no less than
three
Propositions, viz.

(1) “No
xy
exist;

(2)   No
x
are
y
;

(3)   No
y
are
x
.”

Hence these three Propositions are equivalent.

[In the “books” example, these Propositions would be

(1) “No old English books exist;

(2)   No old books are English;

(3)   No English books are old”.]

The two equivalent Propositions, “No
x
are
y
” and “No
y
are
x
”, are said to be ‘Converse’ to each other.

[For example, if we were told to convert the Proposition

“No porcupines are talkative”,

we should first choose our Univ.
(say “animals”), and then complete the Proposition, by supplying the Substantive “animals” in the Predicate, so that it would be

“No porcupines are talkative animals”, and we should then convert it, by interchanging its Terms, so that it would be

“No talkative animals are porcupines”.]

Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of
four
Trios being as follows:—

(1) “No
xy
exist” = “No
x
are
y
” = “No
y
are
x
”.

(2) “No
xy
′
exist” = “No
x
are
y
′
” = “No
y
′
are
x
”.

(3) “No
x
′
y exist” = “No
x
′
are
y
” = “No
y
are
x
′
”.

(4) “No
x
′
y′ exist” = “No
x
′
are
y
′
” = “No
y
′
are
x
′
”.

Let us take, next, the Proposition “All
x
are
y
”.

We know (see p.
17) that this is a
Double
Proposition, and equivalent to the
two
Propositions “Some
x
are
y
” and “No
x
are
y
′
”, each of which we already know how to represent.

[Note that the
Subject
of the given Proposition settles which
Half
we are to use; and that its
Predicate
settles in which
portion
of that Half we are to place the Red Counter.]

TABLE II.

  Some
x
exist  

  No
x
exist  

  Some
x
′
exist  

  No
x
′
exist  

  Some
y
exist  

  No
y
exist  

  Some
y
′
exist  

  No
y
′
exist