pg025TABLE I.AdjunctsofClasses.Compartments,or Cells,assigned to them.xNorthHalf.x′South〃yWest〃y′East〃xyNorth -WestCell.xy′〃East〃x′ySouth -West〃x′y′〃East〃Q.“Adjunct for West Half?”A.“y.”Q.“Compartment forxy′?”A.“North-East Cell.”Q.“Adjunct for South-West Cell?”A.“x′y.”&c., &c.
pg025TABLE I.AdjunctsofClasses.Compartments,or Cells,assigned to them.xNorthHalf.x′South〃yWest〃y′East〃xyNorth -WestCell.xy′〃East〃x′ySouth -West〃x′y′〃East〃
Q.“Adjunct for West Half?”A.“y.”Q.“Compartment forxy′?”A.“North-East Cell.”Q.“Adjunct for South-West Cell?”A.“x′y.”&c., &c.
After a little practice, he will find himself able to do without the blank Diagram, and will be able to see itmentally(“in my mind’s eye, Horatio!”) while answering the questions of his genial friend. Whenthisresult has been reached, he may safely go on to the next Chapter.
Let us agree that aRedCounter, placed within a Cell, shall mean “This Cell isoccupied” (i.e. “There is at leastoneThing in it”).
Let us also agree that aRedCounter, placed on the partition between two Cells, shall mean “The Compartment, made up of these two Cells, isoccupied; but it is not knownwhereabouts, in it, its occupants are.” Hence it may be understood to mean “At leastoneof these two Cells is occupied: possiblybothare.”
Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mindwhichof two political parties he will join: such a man is said to be “sitting on the fence.” This phrase exactly describes the condition of the Red Counter.
Let us also agree that aGreyCounter, placed within a Cell, shall mean “This Cell isempty” (i.e. “There isnothingin it”).
[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]
[The Reader had better provide himself with 4 Red Counters and 5 Grey ones.]
Henceforwards, in stating such Propositions as “Somex-Things exist” or “Nox-Things arey-Things”, I shall omit the word “Things”, which the Reader can supply for himself, and shall write them as “Somexexist” or “Noxarey”.
[Note that the word “Things” is here used with a special meaning, as explained atp. 23.]
[Note that the word “Things” is here used with a special meaning, as explained atp. 23.]
A Proposition, containing onlyoneof the Letters used as Symbols for Attributes, is said to be ‘Uniliteral’.
[For example, “Somexexist”, “Noy′exist”, &c.]
[For example, “Somexexist”, “Noy′exist”, &c.]
A Proposition, containingtwoLetters, is said to be‘Biliteral’.
[For example, “Somexy′exist”, “Nox′arey”, &c.]
[For example, “Somexy′exist”, “Nox′arey”, &c.]
A Proposition is said to be ‘in terms of’ the Letters it contains, whether with or without accents.
[Thus, “Somexy′exist”, “Nox′arey”, &c., are said to bein terms ofxandy.]
[Thus, “Somexy′exist”, “Nox′arey”, &c., are said to bein terms ofxandy.]
Let us take, first, the Proposition “Somexexist”.
[Note that this Proposition is (as explained atp. 12) equivalent to “Some existing Things arex-Things.”]
[Note that this Proposition is (as explained atp. 12) equivalent to “Some existing Things arex-Things.”]
Diagram representing x exists
This tells us that there is at leastoneThing in the North Half; that is, that the North Half isoccupied. And this we can evidently represent by placing aRedCounter (here represented by adottedcircle) on the partition which divides the North Half.
[In the “books” example, this Proposition would be “Some old books exist”.]
[In the “books” example, this Proposition would be “Some old books exist”.]
Similarly we may represent the three similar Propositions “Somex′exist”, “Someyexist”, and “Somey′exist”.
[The Reader should make out all these for himself. In the “books” example, these Propositions would be “Some new books exist”, &c.]
[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 “Noxexist”.
Diagram representing x does not exist
This tells us that there isnothingin the North Half; that is, that the North Half isempty; that is, that the North-West Cell and the North-East Cell are both of themempty. And this we can represent by placingtwo GreyCounters in the North Half, one in each Cell.
[The Reader may perhaps think that it would be enough to place aGreyCounter on the partition in the North Half, and that, just as aRedCounter, so placed, would mean “This Half isoccupied”, so aGreyone would mean “This Half isempty”.This, however, would be a mistake. We have seen that aRedCounter, so placed, would mean “At leastoneof these two Cells is occupied: possiblybothare.” Hence aGreyone would merely mean “At leastoneof these two Cells is empty: possiblybothare”. But what we have to represent is, that both Cells arecertainlyempty: and this can only be done by placing aGreyCounter ineachof them.In the “books” example, this Proposition would be “No old books exist”.]
[The Reader may perhaps think that it would be enough to place aGreyCounter on the partition in the North Half, and that, just as aRedCounter, so placed, would mean “This Half isoccupied”, so aGreyone would mean “This Half isempty”.
This, however, would be a mistake. We have seen that aRedCounter, so placed, would mean “At leastoneof these two Cells is occupied: possiblybothare.” Hence aGreyone would merely mean “At leastoneof these two Cells is empty: possiblybothare”. But what we have to represent is, that both Cells arecertainlyempty: and this can only be done by placing aGreyCounter ineachof them.
In the “books” example, this Proposition would be “No old books exist”.]
pg029Similarly we may represent the three similar Propositions “Nox′exist”, “Noyexist”, and “Noy′exist”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No new books exist”, &c.]
[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”.
Diagram representing x y exists
This tells us that there is at leastoneThing in the North-West Cell; that is, that the North-West Cell isoccupied. And this we can represent by placing aRedCounter in it.
[In the “books” example, this Proposition would be “Some old English books exist”.]
[In the “books” example, this Proposition would be “Some old English books exist”.]
Similarly we may represent the three similar Propositions “Somexy′exist”, “Somex′yexist”, and “Somex′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.]
[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 “Noxyexist”.
Diagram representing x y does not exist
This tells us that there isnothingin the North-West Cell; that is, that the North-West Cell isempty. And this we can represent by placing aGreyCounter in it.
[In the “books” example, this Proposition would be “No old English books exist”.]
[In the “books” example, this Proposition would be “No old English books exist”.]
Similarly we may represent the three similar Propositions “Noxy′exist”, “Nox′yexist”, and “Nox′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.]
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old foreign books exist”, &c.]
pg030Diagram representing x does not exist
We have seen that the Proposition “Noxexist” may be represented by placingtwo GreyCounters in the North Half, one in each Cell.
We have also seen that these twoGreyCounters, takenseparately, represent the two Propositions “Noxyexist” and “Noxy′exist”.
Hence we see that the Proposition “Noxexist” is aDoubleProposition, and is equivalent to thetwoPropositions “Noxyexist” and “Noxy′exist”.
[In the “books” example, this Proposition would be “No old books exist”.Hence this is aDoubleProposition, and is equivalent to thetwoPropositions “No oldEnglishbooks exist” and “No oldforeignbooks exist”.]
[In the “books” example, this Proposition would be “No old books exist”.
Hence this is aDoubleProposition, and is equivalent to thetwoPropositions “No oldEnglishbooks exist” and “No oldforeignbooks exist”.]
Let us take, first, the Proposition “Somexarey”.
Diagram representing x y exists
This tells us that at leastoneThing, in theNorthHalf, is also in theWestHalf. Hence it must be in the spacecommonto them, that is, in theNorth-West Cell. Hence the North-West Cell isoccupied. And this we can represent by placing aRedCounter in it.
[Note that theSubjectof the Proposition settles whichHalfwe are to use; and that thePredicatesettles in whichportionof it we are to place the Red Counter.In the “books” example, this Proposition would be “Some old books are English”.]
[Note that theSubjectof the Proposition settles whichHalfwe are to use; and that thePredicatesettles in whichportionof 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 “Somexarey′”, “Somex′arey”, and “Somex′arey′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old books are foreign”, &c.]
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some old books are foreign”, &c.]
pg031Let us take, next, the Proposition “Someyarex”.
Diagram representing x y exists
This tells us that at leastoneThing, in theWestHalf, is also in theNorthHalf. Hence it must be in the spacecommonto them, that is, in theNorth-West Cell. Hence the North-West Cell isoccupied. And this we can represent by placing aRedCounter in it.
[In the “books” example, this Proposition would be “Some English books are old”.]
[In the “books” example, this Proposition would be “Some English books are old”.]
Similarly we may represent the three similar Propositions “Someyarex′”, “Somey′arex”, and “Somey′arex′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some English books are new”, &c.]
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “Some English books are new”, &c.]
Diagram representing x y exists
We see that thisoneDiagram has now served to represent no less thanthreePropositions, viz.
(1)“Somexyexist;(2)Somexarey;(3)Someyarex”.
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”.]
[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, “Somexarey” and “Someyarex”, 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”.]
[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”.]
pg032Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set offourTrios being as follows:—
(1)“Somexyexist” = “Somexarey” = “Someyarex”.(2)“Somexy′exist” = “Somexarey′” = “Somey′arex”.(3)“Somex′yexist” = “Somex′arey” = “Someyarex′”.(4)“Somex′y′exist” = “Somex′arey′” = “Somey′arex′”.
Let us take, next, the Proposition “Noxarey”.
Diagram representing x y does not exist
This tell us that no Thing, in theNorthHalf, is also in theWestHalf. Hence there isnothingin the spacecommonto them, that is, in theNorth-West Cell. Hence the North-West Cell isempty. And this we can represent by placing aGreyCounter in it.
[In the “books” example, this Proposition would be “No old books are English”.]
[In the “books” example, this Proposition would be “No old books are English”.]
Similarly we may represent the three similar Propositions “Noxarey′”, and “Nox′arey”, and “Nox′arey′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No old books are foreign”, &c.]
[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 “Noyarex”.
Diagram representing x y does not exist
This tells us that no Thing, in theWestHalf, is also in theNorthHalf. Hence there isnothingin the spacecommonto them, that is, in theNorth-West Cell. That is, the North-West Cell isempty. And this we can represent by placing aGreyCounter in it.
[In the “books” example, this Proposition would be “No English books are old”.]
[In the “books” example, this Proposition would be “No English books are old”.]
Similarly we may represent the three similar Propositions “Noyarex′”, “Noy′arex”, and “Noy′arex′”.
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No English books are new”, &c.]
[The Reader should make out all these for himself. In the “books” example, these three Propositions would be “No English books are new”, &c.]
pg033Diagram representing x y does not exist
We see that thisoneDiagram has now served to present no less thanthreePropositions, viz.
(1)“Noxyexist;(2)Noxarey;(3)Noyarex.”
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”.]
[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, “Noxarey” and “Noyarex”, 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”.]
[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 offourTrios being as follows:—
(1)“Noxyexist” = “Noxarey” = “Noyarex”.(2)“Noxy′exist” = “Noxarey′” = “Noy′arex”.(3)“Nox′y exist” = “Nox′arey” = “Noyarex′”.(4)“Nox′y′ exist” = “Nox′arey′” = “Noy′arex′”.
Diagram representing all x are y
Let us take, next, the Proposition “Allxarey”.
We know (seep. 17) that this is aDoubleProposition, and equivalent to thetwoPropositions “Somexarey” and “Noxarey′”, each of which we already know how to represent.
[Note that theSubjectof the given Proposition settles whichHalfwe are to use; and that itsPredicatesettles in whichportionof that Half we are to place the Red Counter.]
[Note that theSubjectof the given Proposition settles whichHalfwe are to use; and that itsPredicatesettles in whichportionof that Half we are to place the Red Counter.]
pg034TABLE II.SomexexistDiagram representing x existsNoxexistDiagram representing x does not existSomex′existDiagram representing x prime existsNox′existDiagram representing x prime does not existSomeyexistDiagram representing y existsNoyexistDiagram representing y does not existSomey′existDiagram representing y prime existsNoy′existDiagram representing y prime does not exist
pg034TABLE II.SomexexistDiagram representing x existsNoxexistDiagram representing x does not existSomex′existDiagram representing x prime existsNox′existDiagram representing x prime does not existSomeyexistDiagram representing y existsNoyexistDiagram representing y does not existSomey′existDiagram representing y prime existsNoy′existDiagram representing y prime does not exist
Similarly we may represent the seven similar Propositions “Allxarey′”, “Allx′arey”, “Allx′arey′”, “Allyarex”, “Allyarex′”, “Ally′arex”, and “Ally′arex′”.
Diagram representing x exists
Let us take, lastly, the Double Proposition “Somexareyand some arey′”, each part of which we already know how to represent.
Similarly we may represent the three similar Propositions, “Somex′areyand some arey′”, “Someyarexand some arex′”, “Somey′arexand some arex′”.
The Reader should now get his genial friend to question him, severely, on these two Tables. TheInquisitorshould have the Tables before him: but theVictimshould have nothing but a blank Diagram, and the Counters with which he is to represent the various Propositions named by his friend, e.g. “Someyexist”, “Noy′arex”, “Allxarey”, &c. &c.
pg035TABLE III.Somexyexist= Somexarey= SomeyarexDiagram representing x y existsAllxareyDiagram representing all x are ySomexy′exist= Somexarey′= Somey′arexDiagram representing x y existsAllxarey′Diagram representing all x are y primeSomex′yexist= Somex′arey= Someyarex′Diagram representing x y existsAllx′areyDiagram representing all x prime are ySomex′y′exist= Somex′arey′= Somey′arex′Diagram representing x prime y prime existsAllx′arey′Diagram representing all x prime are y primeNoxyexist= Noxarey= NoyarexDiagram representing x y does not existAllyarexDiagram representing all y are xNoxy′exist= Noxarey′= Noy′arexDiagram representing x y prime does not existAllyarex′Diagram representing all y are x primeNox′yexist= Nox′arey= Noyarex′Diagram representing x prime y does not existAlly′arexDiagram representing all y prime are xNox′y′exist= Nox′arey′= Noy′arex′Diagram representing x prime y prime does not existAlly′arex′Diagram representing all y prime are x primeSomexarey,and some arey′Diagram representing x exists with and without ySomeyarexand some arex′Diagram representing y exists with and without xSomex′arey,and some arey′Diagram representing x prime exists with and without ySomey′arexand some arex′Diagram representing y prime exists with and without x
pg035TABLE III.Somexyexist= Somexarey= SomeyarexDiagram representing x y existsAllxareyDiagram representing all x are ySomexy′exist= Somexarey′= Somey′arexDiagram representing x y existsAllxarey′Diagram representing all x are y primeSomex′yexist= Somex′arey= Someyarex′Diagram representing x y existsAllx′areyDiagram representing all x prime are ySomex′y′exist= Somex′arey′= Somey′arex′Diagram representing x prime y prime existsAllx′arey′Diagram representing all x prime are y primeNoxyexist= Noxarey= NoyarexDiagram representing x y does not existAllyarexDiagram representing all y are xNoxy′exist= Noxarey′= Noy′arexDiagram representing x y prime does not existAllyarex′Diagram representing all y are x primeNox′yexist= Nox′arey= Noyarex′Diagram representing x prime y does not existAlly′arexDiagram representing all y prime are xNox′y′exist= Nox′arey′= Noy′arex′Diagram representing x prime y prime does not existAlly′arex′Diagram representing all y prime are x primeSomexarey,and some arey′Diagram representing x exists with and without ySomeyarexand some arex′Diagram representing y exists with and without xSomex′arey,and some arey′Diagram representing x prime exists with and without ySomey′arexand some arex′Diagram representing y prime exists with and without x
The Diagram is supposed to be set before us, with certain Counters placed upon it; and the problem is to find out what Proposition, or Propositions, the Counters represent.
As the process is simply the reverse of that discussed in the previous Chapter, we can avail ourselves of the results there obtained, as far as they go.
Diagram representing x y exists
First, let us suppose that we find aRedCounter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“Somexyexist” = “Somexarey” = “Someyarex”.
Similarly we may interpret aRedCounter, when placed in the North-East, or South-West, or South-East Cell.
Diagram representing x y does not exist
Next, let us suppose that we find aGreyCounter placed in the North-West Cell.
We know that this represents each of the Trio of equivalent Propositions
“Noxyexist” = “Noxarey” = “Noyarex”.
Similarly we may interpret aGreyCounter, when placed in the North-East, or South-West, or South-East Cell.
pg037Diagram representing x exists
Next, let us suppose that we find aRedCounter placed on the partition which divides the North Half.
We know that this represents the Proposition “Somexexist.”
Similarly we may interpret aRedCounter, when placed on the partition which divides the South, or West, or East Half.
Diagram representing x exists with and without y
Next, let us suppose that we findtwo RedCounters placed in the North Half, one in each Cell.
We know that this represents theDoubleProposition “Somexareyand some arey′”.
Similarly we may interprettwo RedCounters, when placed in the South, or West, or East Half.
Diagram representing x does not exist
Next, let us suppose that we findtwo GreyCounters placed in the North Half, one in each Cell.
We know that this represents the Proposition “Noxexist”.
Similarly we may interprettwo GreyCounters, when placed in the South, or West, or East Half.
Diagram representing all x are y
Lastly, let us suppose that we find aRedand aGreyCounter placed in the North Half, theRedin the North-WestCell, and theGreyin the North-EastCell.
We know that this represents the Proposition, “Allxarey”.
[Note that theHalf, occupied by the two Counters, settles what is to be theSubjectof the Proposition, and that theCell, occupied by theRedCounter, settles what is to be itsPredicate.]
[Note that theHalf, occupied by the two Counters, settles what is to be theSubjectof the Proposition, and that theCell, occupied by theRedCounter, settles what is to be itsPredicate.]
pg038Similarly we may interpret aRedand aGreycounter, when placed in any one of the seven similar positions
Red in North-East, Grey in North-West;Red in South-West, Grey in South-East;Red in South-East, Grey in South-West;Red in North-West, Grey in South-West;Red in South-West, Grey in North-West;Red in North-East, Grey in South-East;Red in South-East, Grey in North-East.
Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not onlyrepresentPropositions, but alsointerpretDiagrams when marked with Counters.
The Questions and Answers should be like this:—
Q.Represent “Nox′arey′.”A.Grey Counter in S.E. Cell.Q.Interpret Red Counter on E. partition.A.“Somey′exist.”Q.Represent “Ally′arex.”A.Red in N.E. Cell; Grey in S.E.Q.Interpret Grey Counter in S.W. Cell.A.“Nox′yexist” = “Nox′arey” = “Noyarex′”.&c., &c.
At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.
[Work Examples §1, 5–8 (p. 97).]
[Work Examples §1, 5–8 (p. 97).]
An annotated biliteral diagramAn annotated triliteral diagram
First, let us suppose that the aboveleft-hand Diagram is the Biliteral Diagram that we have been using in Book III., and that we change it into aTriliteralDiagram by drawing anInner Square, so as to divide each of its 4 Cells into 2 portions, thus making 8 Cells altogether. Theright-hand Diagram shows the result.
[The Reader is strongly advised, in reading this Chapter,notto refer to the above Diagrams, but to make a large copy of the right-hand one for himself,without any letters, and to have it by him while he reads, and keep his finger on that particularpartof it, about which he is reading.]
[The Reader is strongly advised, in reading this Chapter,notto refer to the above Diagrams, but to make a large copy of the right-hand one for himself,without any letters, and to have it by him while he reads, and keep his finger on that particularpartof it, about which he is reading.]
pg040Secondly, let us suppose that we have selected a certain Adjunct, which we may call “m”, and have subdivided thexy-Class into the two Classes whose Differentiæ aremandm′,and that we have assigned the N.W.InnerCell to the one (which we may call “the Class ofxym-Things”, or “thexym-Class”), and the N.W.OuterCell to the other (which we may call “the Class ofxym′-Things”, or “thexym′-Class”).
[Thus, in the “books” example, we might say “Letmmean ‘bound’, so thatm′will mean ‘unbound’”, and we might suppose that we had subdivided the Class “old English books” into the two Classes, “old English bound books” and “old English unbound books”, and had assigned the N.W.InnerCell to the one, and the N.W.OuterCell to the other.]
[Thus, in the “books” example, we might say “Letmmean ‘bound’, so thatm′will mean ‘unbound’”, and we might suppose that we had subdivided the Class “old English books” into the two Classes, “old English bound books” and “old English unbound books”, and had assigned the N.W.InnerCell to the one, and the N.W.OuterCell to the other.]
Thirdly, let us suppose that we have subdivided thexy′-Class, thex′y-Class, and thex′y′-Class in the same manner,and have, in each case, assigned theInnerCell to the Class possessing the Attributem, and theOuterCell to the Class possessing the Attributem′.
[Thus, in the “books” example, we might suppose that we had subdivided the “new English books” into the two Classes, “new English bound books” and “new English unbound books”, and had assigned the S.W.InnerCell to the one, and the S.W.OuterCell to the other.]
[Thus, in the “books” example, we might suppose that we had subdivided the “new English books” into the two Classes, “new English bound books” and “new English unbound books”, and had assigned the S.W.InnerCell to the one, and the S.W.OuterCell to the other.]
It is evident that we have now assigned theInner Squareto them-Class, and theOuter Borderto them′-Class.
[Thus, in the “books” example, we have assigned theInner Squareto “bound books” and theOuter Borderto “unbound books”.]
[Thus, in the “books” example, we have assigned theInner Squareto “bound books” and theOuter Borderto “unbound books”.]
When the Reader has made himself familiar with this Diagram, he ought to be able to find, in a moment, the Compartment assigned to a particularpairof Attributes, or the Cell assigned to a particulartrioof Attributes.The following Rules will help him in doing this:—
(1)Arrange the Attributes in the orderx,y,m.pg041(2)Take thefirstof them and find the Compartment assigned to it.(3)Then take thesecond, and find whatportionof that compartment is assigned to it.(4)Treat thethird, if there is one, in the same way.
[For example, suppose we have to find the Compartment assigned toym. We say to ourselves “yhas theWestHalf; andmhas theInnerportion of that West Half.”Again, suppose we have to find the Cell assigned tox′ym′. We say to ourselves “x′has theSouthHalf;yhas theWestportion of that South Half, i.e. has theSouth-West Quarter; andm′has theOuterportion of that South-West Quarter.”]
[For example, suppose we have to find the Compartment assigned toym. We say to ourselves “yhas theWestHalf; andmhas theInnerportion of that West Half.”
Again, suppose we have to find the Cell assigned tox′ym′. We say to ourselves “x′has theSouthHalf;yhas theWestportion of that South Half, i.e. has theSouth-West Quarter; andm′has theOuterportion of that South-West Quarter.”]
The Reader should now get his genial friend to question him on the Table given on the next page, in the style of the following specimen-Dialogue.
Q.Adjunct for South Half, Inner Portion?A.x′m.Q.Compartment form′?A.The Outer Border.Q.Adjunct for North-East Quarter, Outer Portion?A.xy′m′.Q.Compartment forym?A.West Half, Inner Portion.Q.Adjunct for South Half?A.x′.Q.Compartment forx′y′m?A.South-East Quarter, Inner Portion.&c. &c.pg042TABLE IV.AdjunctofClasses.Compartments,or Cells,assigned to them.xNorthHalf.x′South〃yWest〃y′East〃mInnerSquare.m′OuterBorder.xyNorth-WestQuarter.xy′〃East〃x′ySouth-West〃x′y′〃East〃xmNorthHalf,InnerPortion.xm′〃〃Outer〃x′mSouth〃Inner〃x′m′〃〃Outer〃ymWest〃Inner〃ym′〃〃Outer〃y′mEast〃Inner〃y′m′〃〃Outer〃xymNorth-WestQuarter,InnerPortion.xym′〃〃〃Outer〃xy′m〃East〃Inner〃xy′m′〃〃〃Outer〃x′ymSouth-West〃Inner〃x′ym′〃〃〃Outer〃x′y′m〃East〃Inner〃x′y′m′〃〃〃Outer〃
Q.Adjunct for South Half, Inner Portion?A.x′m.Q.Compartment form′?A.The Outer Border.Q.Adjunct for North-East Quarter, Outer Portion?A.xy′m′.Q.Compartment forym?A.West Half, Inner Portion.Q.Adjunct for South Half?A.x′.Q.Compartment forx′y′m?A.South-East Quarter, Inner Portion.&c. &c.
pg042TABLE IV.AdjunctofClasses.Compartments,or Cells,assigned to them.xNorthHalf.x′South〃yWest〃y′East〃mInnerSquare.m′OuterBorder.xyNorth-WestQuarter.xy′〃East〃x′ySouth-West〃x′y′〃East〃xmNorthHalf,InnerPortion.xm′〃〃Outer〃x′mSouth〃Inner〃x′m′〃〃Outer〃ymWest〃Inner〃ym′〃〃Outer〃y′mEast〃Inner〃y′m′〃〃Outer〃xymNorth-WestQuarter,InnerPortion.xym′〃〃〃Outer〃xy′m〃East〃Inner〃xy′m′〃〃〃Outer〃x′ymSouth-West〃Inner〃x′ym′〃〃〃Outer〃x′y′m〃East〃Inner〃x′y′m′〃〃〃Outer〃
Diagram representing x m exists
Let us take, first, the Proposition “Somexmexist”.
[Note that thefullmeaning of this Proposition is (as explained atp. 12) “Some existing Things arexm-Things”.]
[Note that thefullmeaning of this Proposition is (as explained atp. 12) “Some existing Things arexm-Things”.]
This tells us that there is at leastoneThing in the Inner portion of the North Half; that is, that this Compartment isoccupied. And this we can evidently represent by placing aRedCounter on the partition which divides it.
[In the “books” example, this Proposition would mean “Some old bound books exist” (or “There are some old bound books”).]
[In the “books” example, this Proposition would mean “Some old bound books exist” (or “There are some old bound books”).]
Similarly we may represent the seven similar Propositions, “Somexm′exist”, “Somex′mexist”, “Somex′m′exist”, “Someymexist”, “Someym′exist”, “Somey′mexist”, and “Somey′m′exist”.
Diagram representing x m does not exist
pg044Let us take, next, the Proposition “Noxmexist”.
This tells us that there isnothingin the Inner portion of the North Half; that is, that this Compartment isempty. And this we can represent by placingtwo GreyCounters in it, one in each Cell.
Similarly we may represent the seven similar Propositions, in terms ofxandm, or ofyandm, viz. “Noxm′exist”, “Nox′mexist”, &c.
These sixteen Propositions of Existence are the only ones that we shall have to represent on this Diagram.
Diagram representing x m exists
Let us take, first, the Pair of Converse Propositions
“Somexarem” = “Somemarex.”
We know that each of these is equivalent to the Proposition of Existence “Somexmexist”, which we already know how to represent.
Similarly for the seven similar Pairs, in terms ofxandm, or ofyandm.
Diagram representing x m does not exist
Let us take, next, the Pair of Converse Propositions
“Noxarem” = “Nomarex.”
We know that each of these is equivalent to the Proposition of Existence “Noxmexist”, which we already know how to represent.
Similarly for the seven similar Pairs, in terms ofxandm, or ofyandm.
Diagram representing all x are m
pg045Let us take, next, the Proposition “Allxarem.”
We know (seep. 18) that this is aDoubleProposition, and equivalent to thetwoPropositions “Somexarem” and “Noxarem′”, each of which we already know how to represent.
Similarly for the fifteen similar Propositions, in terms ofxandm, or ofyandm.
These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.
The Reader should now get his genial friend to question him on the following four Tables.
The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor,e.g.“Noy′arem”, “Somexm′exist”, &c., &c.