Chapter 37
_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".
[Note that the word "Things" is here used with a special meaning, as explained at p. 23.]
A Proposition, containing only _one_ of the Letters used as Symbols for Attributes, is said to be '=Uniliteral='.
[For example, "Some x exist", "No y' exist", &c.]
A Proposition, containing _two_ Letters, is said to be ='Biliteral'=.
[For example, "Some xy' exist", "No x' are y", &c.]
A Proposition is said to be '=in terms of=' the Letters it contains, whether with or without accents.
[Thus, "Some xy' exist", "No x' are y", &c., are said to be _in terms of_ x and y.]
pg028 § 2.
_Representation of Propositions of Existence._
Let us take, first, the Proposition "Some x exist".
[Note that this Proposition is (as explained at p. 12) equivalent to "Some existing Things are x-Things."]
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.
·-------· | (.) | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "Some old books exist".]
Similarly we may represent the three similar Propositions "Some x' exist", "Some y exist", and "Some y' exist".
[The Reader should make out all these for himself. In the "books" example, these Propositions would be "Some new books exist", &c.]
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.
·-------· |( )|( )| |---|---| | | | ·-------·
[The Reader may perhaps think that it would be enough to place a _Grey_ Counter on the partition in the North Half, and that, just as a _Red_ Counter, so placed, would mean "This Half is _occupied_", so a _Grey_ one would mean "This Half is _empty_".
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".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No new books exist", &c.]
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.
·-------· |(.)| | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "Some old English books exist".]
Similarly we may represent the three similar Propositions "Some xy' exist", "Some x'y exist", and "Some x'y' exist".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "Some old foreign books exist", &c.]
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.
·-------· |( )| | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "No old English books exist".]
Similarly we may represent the three similar Propositions "No xy' exist", "No x'y exist", and "No x'y' exist".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No old foreign books exist", &c.] pg030 We have seen that the Proposition "No x exist" may be represented by placing _two Grey_ Counters in the North Half, one in each Cell.
·-------· |( )|( )| |---|---| | | | ·-------·
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".
[In the "books" example, this Proposition would be "No old books 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.
·-------· |(.)| | |---|---| | | | ·-------·
[Note that the _Subject_ of the Proposition settles which _Half_ we are to use; and that the _Predicate_ settles in which _portion_ of it we are to place the Red Counter.
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'".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "Some old books are foreign", &c.] pg031 Let us take, next, the Proposition "Some y are x".
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.
·-------· |(.)| | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "Some English books are old".]
Similarly we may represent the three similar Propositions "Some y are x'", "Some y' are x", and "Some y' are x'".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "Some English books are new", &c.]
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.
[In the "books" example, these Propositions would be
(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='.
[For example, if we were told to convert the Proposition
"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.
·-------· |( )| | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "No old books are English".]
Similarly we may represent the three similar Propositions "No x are y'", and "No x' are y", and "No x' are y'".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No old books are foreign", &c.]
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.
·-------· |( )| | |---|---| | | | ·-------·
[In the "books" example, this Proposition would be "No English books are old".]
Similarly we may represent the three similar Propositions "No y are x'", "No y' are x", and "No y' are x'".
[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No English books are new", &c.] pg033 ·-------· |( )| | |---|---| | | | ·-------·
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.]
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