CHAPTER III.. _representation of Propositions._

REPRESENTATION OF PROPOSITIONS.

§ 1.

Introductory.

Henceforwards, in stating such Propositions as "Some x-Things exist" or "No x-Things are y-Things", I shall omit the word "Things", which the Reader can supply for himself, and shall write them as "Some x exist" or "No x are y".

A Proposition, containing only one of the Letters used as Symbols for Attributes, is said to be '=Uniliteral='.

A Proposition, containing two Letters, is said to be ='Biliteral'=.

A Proposition is said to be '=in terms of=' the Letters it contains, whether with or without accents.

pg028 § 2.

Representation of Propositions of Existence.

Let us take, first, the Proposition "Some x exist".

This tells us that there is at least one Thing in the North Half; that is, that the North Half is occupied. And this we can evidently represent by placing a Red Counter (here represented by a dotted circle) on the partition which divides the North Half.

·-------· | (.) | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "Some x' exist", "Some y exist", and "Some y' exist".

Let us take, next, the Proposition "No x exist".

This tells us that there is nothing in the North Half; that is, that the North Half is empty; that is, that the North-West Cell and the North-East Cell are both of them empty. And this we can represent by placing two Grey Counters in the North Half, one in each Cell.

·-------· |( )|( )| |---|---| | | | ·-------·

This, however, would be a mistake. We have seen that a Red Counter, so placed, would mean "At least one of these two Cells is occupied: possibly both are." Hence a Grey one would merely mean "At least one of these two Cells is empty: possibly both are". But what we have to represent is, that both Cells are certainly empty: and this can only be done by placing a Grey Counter in each of them.

In the "books" example, this Proposition would be "No old books exist".] pg029 Similarly we may represent the three similar Propositions "No x' exist", "No y exist", and "No y' exist".

Let us take, next, the Proposition "Some xy exist".

This tells us that there is at least one Thing in the North-West Cell; that is, that the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

·-------· |(.)| | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "Some xy' exist", "Some x'y exist", and "Some x'y' exist".

Let us take, next, the Proposition "No xy exist".

This tells us that there is nothing in the North-West Cell; that is, that the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

·-------· |( )| | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "No xy' exist", "No x'y exist", and "No x'y' exist".

·-------· |( )|( )| |---|---| | | | ·-------·

We have also seen that these two Grey Counters, taken separately, represent the two Propositions "No xy exist" and "No xy' exist".

Hence we see that the Proposition "No x exist" is a Double Proposition, and is equivalent to the two Propositions "No xy exist" and "No xy' exist".

Hence this is a Double Proposition, and is equivalent to the two Propositions "No old English books exist" and "No old foreign books exist".]

§ 3.

Representation of Propositions of Relation.

Let us take, first, the Proposition "Some x are y".

This tells us that at least one Thing, in the North Half, is also in the West Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

·-------· |(.)| | |---|---| | | | ·-------·

In the "books" example, this Proposition would be "Some old books are English".]

Similarly we may represent the three similar Propositions "Some x are y'", "Some x' are y", and "Some x' are y'".

This tells us that at least one Thing, in the West Half, is also in the North Half. Hence it must be in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is occupied. And this we can represent by placing a Red Counter in it.

·-------· |(.)| | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "Some y are x'", "Some y' are x", and "Some y' are x'".

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

(1) "Some xy exist; (2) Some x are y; (3) Some y are x".

·-------· |(.)| | |---|---| | | | ·-------·

Hence these three Propositions are equivalent.

(1) "Some old English books exist; (2) Some old books are English; (3) Some English books are old".]

The two equivalent Propositions, "Some x are y" and "Some y are x", are said to be '=Converse=' to each other; and the Process, of changing one into the other, is called '=Converting=', or '=Conversion='.

"Some apples are not ripe,"

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

"Some apples are not-ripe fruit";

and we should then convert it by interchanging its Terms, so that it would be

"Some not-ripe fruit are apples".] pg032 Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of four Trios being as follows:--

(1) "Some xy exist" = "Some x are y" = "Some y are x". (2) "Some xy' exist" = "Some x are y'" = "Some y' are x". (3) "Some x'y exist" = "Some x' are y" = "Some y are x'". (4) "Some x'y' exist" = "Some x' are y'" = "Some y' are x'".

Let us take, next, the Proposition "No x are y".

This tell us that no Thing, in the North Half, is also in the West Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. Hence the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

·-------· |( )| | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "No x are y'", and "No x' are y", and "No x' are y'".

Let us take, next, the Proposition "No y are x".

This tells us that no Thing, in the West Half, is also in the North Half. Hence there is nothing in the space common to them, that is, in the North-West Cell. That is, the North-West Cell is empty. And this we can represent by placing a Grey Counter in it.

·-------· |( )| | |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions "No y are x'", "No y' are x", and "No y' are x'".

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.

(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.

"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.

·-------· |(.)|( )| |---|---| | | | ·-------·

pg034 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 | |---|---| | | | | | | | | | |( )| | | | ·-------· | | ·-------· | ·-----------------------------------------------------·

Similarly we may represent the seven similar Propositions "All x are y'", "All x' are y", "All x' are y'", "All y are x", "All y are x'", "All y' are x", and "All y' are x'".

Let us take, lastly, the Double Proposition "Some x are y and some are y'", each part of which we already know how to represent.

·-------· |(.)|(.)| |---|---| | | | ·-------·

Similarly we may represent the three similar Propositions, "Some x' are y and some are y'", "Some y are x and some are x'", "Some y' are x and some are x'".

The Reader should now get his genial friend to question him, severely, on these two Tables. The Inquisitor should have the Tables before him: but the Victim should have nothing but a blank Diagram, and the Counters with which he is to represent the various Propositions named by his friend, e.g. "Some y exist", "No y' are x", "All x are y", &c. &c.

pg035 TABLE III.

·-------------------------------------------------------------· | | ·-------· | | ·-------· | | Some xy exist | |(.)| | | | |(.)|( )| | | = Some x are y | |---|---| | All x are y | |---|---| | | = Some y are x | | | | | | | | | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | Some xy' exist | | |(.)| | | |( )|(.)| | | = Some x are y' | |---|---| | All x are y' | |---|---| | | = Some y' are x | | | | | | | | | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | Some x'y exist | | | | | | | | | | | = Some x' are y | |---|---| | All x' are y | |---|---| | | = Some y are x' | |(.)| | | | |(.)|( )| | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | Some x'y' exist | | | | | | | | | | | = Some x' are y'| |---|---| | All x' are y' | |---|---| | | = Some y' are x'| | |(.)| | | |( )|(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------·

·-------------------------------------------------------------· | | ·-------· | | ·-------· | | No xy exist | |( )| | | | |(.)| | | | = No x are y | |---|---| | All y are x | |---|---| | | = No y are x | | | | | | |( )| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | No xy' exist | | |( )| | | |( )| | | | = No x are y' | |---|---| | All y are x' | |---|---| | | = No y' are x | | | | | | |(.)| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | No x'y exist | | | | | | | |(.)| | | = No x' are y | |---|---| | All y' are x | |---|---| | | = No y are x' | |( )| | | | | |( )| | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | No x'y' exist | | | | | | | |( )| | | = No x' are y' | |---|---| | All y' are x' | |---|---| | | = No y' are x' | | |( )| | | | |(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------·

·-------------------------------------------------------------· | | ·-------· | | ·-------· | | | |(.)|(.)| | | |(.)| | | | Some x are y, | |---|---| | Some y are x | |---|---| | | and some are y' | | | | | and some are x' | |(.)| | | | | ·-------· | | ·-------· | |------------------|-----------|------------------|-----------| | | ·-------· | | ·-------· | | | | | | | | | |(.)| | | Some x' are y, | |---|---| | Some y' are x | |---|---| | | and some are y' | |(.)|(.)| | and some are x' | | |(.)| | | | ·-------· | | ·-------· | ·-------------------------------------------------------------·

pg036