pg056BOOK V.

pg046TABLE V.Diagram representing x m existsSomexmexist= Somexarem= SomemarexDiagram representing x m does not existNoxmexist= Noxarem= NomarexDiagram representing x m prime existsSomexm′exist= Somexarem′= Somem′arexDiagram representing x m prime does not existNoxm′exist= Noxarem′= Nom′arexDiagram representing x prime m existsSomex′mexist= Somex′arem= Somemarex′Diagram representing x prime m does not existNox′mexist= Nox′arem= Nomarex′Diagram representing x prime m prime existsSomex′m′exist= Somex′arem′= Somem′arex′Diagram representing x prime m prime does not existNox′m′exist= Nox′arem′= Nom′arex′

pg047TABLE VI.Diagram representing y m existsSomeymexist= Someyarem= SomemareyDiagram representing y m does not existNoymexist= Noyarem= NomareyDiagram representing y m prime existsSomeym′exist= Someyarem′= Somem′areyDiagram representing y m prime does not existNoym′exist= Noyarem′= Nom′areyDiagram representing y prime m existsSomey′mexist= Somey′arem= Somemarey′Diagram representing y prime m does not existNoy′mexist= Noy′arem= Nomarey′Diagram representing y prime m prime existsSomey′m′exist= Somey′arem′= Somem′arey′Diagram representing y prime m prime does not existNoy′m′exist= Noy′arem′= Nom′arey′

pg048TABLE VII.Diagram representing all x are mAllxaremDiagram representing all x are m primeAllxarem′Diagram representing all x prime are mAllx′aremDiagram representing all x prime are m primeAllx′arem′Diagram representing all m are xAllmarexDiagram representing all m are x primeAllmarex′Diagram representing all m prime are xAllm′arexDiagram representing all m prime are x primeAllm′arex′

pg049TABLE VIII.Diagram representing all y are mAllyaremDiagram representing all y are m primeAllyarem′Diagram representing all y prime are mAlly′aremDiagram representing all y prime are m primeAlly′arem′Diagram representing all m are yAllmareyDiagram representing all m are y primeAllmarey′Diagram representing all m prime are yAllm′areyDiagram representing all m prime are y primeAllm′arey′

The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits “I” and “O”, instead of using the Board and Counters: he may put a “I” to represent aRedCounter (this may be interpreted to mean “There is at leastoneThing here”), and a “O” to represent aGreyCounter (this may be interpreted to mean “There isnothinghere”).

The Pair of Propositions, that we shall have to represent, will always be, one in terms ofxandm, and the other in terms ofyandm.

When we have to represent a Proposition beginning with “All”, we break it up into thetwoPropositions to which it is equivalent.

When we have to represent, on the same Diagram, Propositions, of which some begin with “Some” and others with “No”, we represent thenegativeonesfirst. This will sometimes save us from having to put a “I” “on a fence” and afterwards having to shift it into a Cell.

[Let us work a few examples.(1)“Noxarem′;Noy′arem”.Let us first represent “Noxarem′”. This gives us Diagrama.Then, representing “Noy′arem” on the same Diagram, we get Diagramb.pg051abDiagram representing x m prime does not existDiagram representing x m prime and y prime m do not exist(2)“Somemarex;Nomarey”.If, neglecting the Rule, we were begin with “Somemarex”, we should get Diagrama.And if we were then to take “Nomarey”, which tells us that the Inner N.W. Cell isempty, we should be obliged to take the “I” off the fence (as it no longer has the choice oftwoCells), and to put it into the Inner N.E. Cell, as in Diagramc.This trouble may be saved by beginning with “Nomarey”, as in Diagramb.Andnow, when we take “Somemarex”, there is no fence to sit on! The “I” has to go, at once, into the N.E. Cell, as in Diagramc.abcDiagram a representing x m existsDiagram b representing y m does not existDiagram c representing x m exists and y m does not exist(3)“Nox′arem′;Allmarey”.Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we havethreePropositions to represent, viz.—(1)“Nox′arem′;(2)Somemarey;(3)Nomarey′”.These we will take in the order 1, 3, 2.First we take No. (1), viz. “Nox′arem′”. This gives us Diagrama.pg052Adding to this, No. (3), viz. “Nomarey′”, we get Diagramb.This time the “I”, representing No. (2), viz. “Somemarey,” has to sit on the fence, as there is no “O” to order it off! This gives us Diagramc.abcDiagram a representing x prime m prime does not existDiagram b representing x prime m prime and y prime m do not existDiagram c representing x prime m prime and y prime m do not exist and y m does exist(4)“Allmarex;Allyarem”.Here we break upbothPropositions, and thus getfourto represent, viz.—(1)“Somemarex;(2)Nomarex′;(3)Someyarem;(4)Noyarem′”.These we will take in the order 2, 4, 1, 3.First we take No. (2), viz. “Nomarex′”. This gives us Diagrama.To this we add No. (4), viz. “Noyarem′”, and thus get Diagramb.If we were to add to this No. (1), viz. “Somemarex”, we should have to put the “I” on a fence: so let us try No. (3) instead, viz. “Someyarem”. This gives us Diagramc.And now there is no need to trouble about No. (1), as it would not add anything to our information to put a “I” on the fence. The Diagramalreadytells us that “Somemarex”.]

[Let us work a few examples.

“Noxarem′;Noy′arem”.

Let us first represent “Noxarem′”. This gives us Diagrama.

Then, representing “Noy′arem” on the same Diagram, we get Diagramb.

pg051abDiagram representing x m prime does not existDiagram representing x m prime and y prime m do not exist

“Somemarex;Nomarey”.

If, neglecting the Rule, we were begin with “Somemarex”, we should get Diagrama.

And if we were then to take “Nomarey”, which tells us that the Inner N.W. Cell isempty, we should be obliged to take the “I” off the fence (as it no longer has the choice oftwoCells), and to put it into the Inner N.E. Cell, as in Diagramc.

This trouble may be saved by beginning with “Nomarey”, as in Diagramb.

Andnow, when we take “Somemarex”, there is no fence to sit on! The “I” has to go, at once, into the N.E. Cell, as in Diagramc.

abcDiagram a representing x m existsDiagram b representing y m does not existDiagram c representing x m exists and y m does not exist

“Nox′arem′;Allmarey”.

Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we havethreePropositions to represent, viz.—

(1)“Nox′arem′;(2)Somemarey;(3)Nomarey′”.

These we will take in the order 1, 3, 2.

First we take No. (1), viz. “Nox′arem′”. This gives us Diagrama.

pg052Adding to this, No. (3), viz. “Nomarey′”, we get Diagramb.

This time the “I”, representing No. (2), viz. “Somemarey,” has to sit on the fence, as there is no “O” to order it off! This gives us Diagramc.

abcDiagram a representing x prime m prime does not existDiagram b representing x prime m prime and y prime m do not existDiagram c representing x prime m prime and y prime m do not exist and y m does exist

“Allmarex;Allyarem”.

Here we break upbothPropositions, and thus getfourto represent, viz.—

(1)“Somemarex;(2)Nomarex′;(3)Someyarem;(4)Noyarem′”.

These we will take in the order 2, 4, 1, 3.

First we take No. (2), viz. “Nomarex′”. This gives us Diagrama.

To this we add No. (4), viz. “Noyarem′”, and thus get Diagramb.

If we were to add to this No. (1), viz. “Somemarex”, we should have to put the “I” on a fence: so let us try No. (3) instead, viz. “Someyarem”. This gives us Diagramc.

And now there is no need to trouble about No. (1), as it would not add anything to our information to put a “I” on the fence. The Diagramalreadytells us that “Somemarex”.]

abcDiagram a representing x prime m does not existDiagram b representing x prime m prime and y m prime do not existDiagram c representing x prime m prime and y prime m do not exist and y m does exist

[Work Examples §1, 9–12 (p. 97); §2, 1–20 (p. 98).]

The problem before us is, given a marked Triliteral Diagram, to ascertainwhatPropositions of Relation, in terms ofxandy, are represented on it.

The best plan, for abeginner, is to draw aBiliteralDiagram alongside of it, and to transfer, from the one to the other, all the information he can. He can then read off, from the Biliteral Diagram, the required Propositions. After a little practice, he will be able to dispense with the Biliteral Diagram, and to read off the result from the Triliteral Diagram itself.

Totransferthe information, observe the following Rules:—

(1)Examine the N.W. Quarter of the Triliteral Diagram.(2)If it contains a “I”, ineitherCell, it is certainlyoccupied, and you may mark the N.W. Quarter of the Biliteral Diagram with a “I”.(3)If it containstwo“O”s, one ineachCell, it is certainlyempty, and you may mark the N.W. Quarter of the Biliteral Diagram with a “O”.pg054(4)Deal in the same way with the N.E., the S.W., and the S.E. Quarter.

[Let us take, as examples, the results of the four Examples worked in the previous Chapters.(1)Diagram representing example 1In the N.W. Quarter, onlyoneof the two Cells is marked asempty: so we do not know whether the N.W. Quarter of the Biliteral Diagram isoccupiedorempty: so we cannot mark it.Diagram representing conclusion of example 1In the N.E. Quarter, we findtwo“O”s: sothisQuarter is certainlyempty; and we mark it so on the Biliteral Diagram.In the S.W. Quarter, we have no informationat all.In the S.E. Quarter, we have not enough to use.We may read off the result as “Noxarey′”, or “Noy′arex,” whichever we prefer.(2)Diagram representing example 2In the N.W. Quarter, we have not enough information to use.Diagram representing conclusion of example 2In the N.E. Quarter, we find a “I”. This shows us that it isoccupied: so we may mark the N.E. Quarter on the Biliteral Diagram with a “I”.In the S.W. Quarter, we have not enough information to use.In the S.E. Quarter, we have none at all.We may read off the result as “Somexarey′”, or “Somey′arex”, whichever we prefer.pg055(3)Diagram representing example 3In the N.W. Quarter, we havenoinformation. (The “I”, sitting on the fence, is of no use to us until we know onwhichside he means to jump down!)Diagram representing conclusion of example 2In the N.E. Quarter, we have not enough information to use.Neither have we in the S.W. Quarter.The S.E. Quarter is the only one that yields enough information to use. It is certainlyempty: so we mark it as such on the Biliteral Diagram.We may read off the results as “Nox′arey′”, or “Noy′arex′”, whichever we prefer.(4)Diagram representing example 4Diagram representing partial conclusion of example 4The N.W. Quarter isoccupied, in spite of the “O” in the Outer Cell. So we mark it with a “I” on the Biliteral Diagram.The N.E. Quarter yields no information.Diagram representing complete conclusion of example 4The S.W. Quarter is certainlyempty. So we mark it as such on the Biliteral Diagram.The S.E. Quarter does not yield enough information to use.We read off the result as “Allyarex.”]

[Let us take, as examples, the results of the four Examples worked in the previous Chapters.

(1)Diagram representing example 1

In the N.W. Quarter, onlyoneof the two Cells is marked asempty: so we do not know whether the N.W. Quarter of the Biliteral Diagram isoccupiedorempty: so we cannot mark it.

Diagram representing conclusion of example 1

In the N.E. Quarter, we findtwo“O”s: sothisQuarter is certainlyempty; and we mark it so on the Biliteral Diagram.

In the S.W. Quarter, we have no informationat all.

In the S.E. Quarter, we have not enough to use.

We may read off the result as “Noxarey′”, or “Noy′arex,” whichever we prefer.

(2)Diagram representing example 2

In the N.W. Quarter, we have not enough information to use.

Diagram representing conclusion of example 2

In the N.E. Quarter, we find a “I”. This shows us that it isoccupied: so we may mark the N.E. Quarter on the Biliteral Diagram with a “I”.

In the S.W. Quarter, we have not enough information to use.

In the S.E. Quarter, we have none at all.

We may read off the result as “Somexarey′”, or “Somey′arex”, whichever we prefer.

pg055(3)Diagram representing example 3

In the N.W. Quarter, we havenoinformation. (The “I”, sitting on the fence, is of no use to us until we know onwhichside he means to jump down!)

Diagram representing conclusion of example 2

In the N.E. Quarter, we have not enough information to use.

Neither have we in the S.W. Quarter.

The S.E. Quarter is the only one that yields enough information to use. It is certainlyempty: so we mark it as such on the Biliteral Diagram.

We may read off the results as “Nox′arey′”, or “Noy′arex′”, whichever we prefer.

(4)Diagram representing example 4

Diagram representing partial conclusion of example 4

The N.W. Quarter isoccupied, in spite of the “O” in the Outer Cell. So we mark it with a “I” on the Biliteral Diagram.

The N.E. Quarter yields no information.

Diagram representing complete conclusion of example 4

The S.W. Quarter is certainlyempty. So we mark it as such on the Biliteral Diagram.

The S.E. Quarter does not yield enough information to use.

We read off the result as “Allyarex.”]

[Review Tables V, VI (pp. 46,47). Work Examples §1, 13–16 (p. 97); §2, 21–32 (p. 98); §3, 1–20 (p. 99).]

When a Trio of Biliteral Propositions of Relation is such that

(1)all their six Terms are Species of the same Genus,(2)every two of them contain between them a Pair of codivisional Classes,(3)the three Propositions are so related that, if the first two were true, the third would be true,

the Trio is called a ‘Syllogism’; the Genus, of which each of the six Terms is a Species, is called its‘Universe of Discourse’, or, more briefly, its ‘Univ.’;the first two Propositions are called its ‘Premisses’,and the third its ‘Conclusion’;also the Pair of codivisional Terms in the Premisses are called its ‘Eliminands’,and the other two its ‘Retinends’.

The Conclusion of a Syllogism is said to be ‘consequent’ from its Premisses:hence it is usual to prefix to it the word “Therefore” (or theSymbol “∴”).

pg057[Note that the ‘Eliminands’ are so called because they areeliminated, and do not appear in the Conclusion; and that the ‘Retinends’ are so called because they areretained, anddoappear in the Conclusion.Note also that the question, whether the Conclusion is or is notconsequentfrom the Premisses, is not affected by theactualtruth or falsity of any of the Trio, but depends entirely on theirrelationship to each other.As a specimen-Syllogism, let us take the Trio“Nox-Things arem-Things;Noy-Things arem′-Things.Nox-Things arey-Things.”which we may write, as explained atp. 26, thus:—“Noxarem;Noyarem′.Noxarey”.Here the first and second contain the Pair of codivisional Classesmandm′; the first and third contain the Pairxandx; and the second and third contain the Pairyandy.Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.Hence the Trio is aSyllogism; the two Propositions, “Noxarem” and “Noyarem′”, are itsPremisses; the Proposition “Noxarey” is itsConclusion; the Termsmandm′are itsEliminands; and the Termsxandyare itsRetinends.Hence we may write it thus:—“Noxarem;Noyarem′.∴ Noxarey”.As a second specimen, let us take the Trio“All cats understand French;Some chickens are cats.Some chickens understand French”.These, put into normal form, are“All cats are creatures understanding French;Some chickens are cats.Some chickens are creatures understanding French”.Here all the six Terms are Species of the Genus “creatures.”Also the first and second Propositions contain the Pair of codivisional Classes “cats” and “cats”; the first and third contain the Pair “creatures understanding French” and “creatures understanding French”; and the second and third contain the Pair “chickens” and “chickens”.pg058Also the three Propositions are (as we shall see atp. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens,notstrictly true inourplanet. But there is nothing to hinder them from being true in someotherplanet, sayMarsorJupiter—in which case the third wouldalsobe true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singularcontingentprivilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)Hence the Trio is aSyllogism; the Genus “creatures” is its ‘Univ.’; the two Propositions, “All cats understand French“ and ”Some chickens are cats”, are itsPremisses, the Proposition “Some chickens understand French” is itsConclusion; the Terms “cats” and “cats” are itsEliminands; and the Terms, “creatures understanding French” and “chickens”, are itsRetinends.Hence we may write it thus:—“All cats understand French;Some chickens are cats;∴ Some chickens understand French”.]

Note also that the question, whether the Conclusion is or is notconsequentfrom the Premisses, is not affected by theactualtruth or falsity of any of the Trio, but depends entirely on theirrelationship to each other.

As a specimen-Syllogism, let us take the Trio

“Nox-Things arem-Things;Noy-Things arem′-Things.Nox-Things arey-Things.”

which we may write, as explained atp. 26, thus:—

“Noxarem;Noyarem′.Noxarey”.

Here the first and second contain the Pair of codivisional Classesmandm′; the first and third contain the Pairxandx; and the second and third contain the Pairyandy.

Also the three Propositions are (as we shall see hereafter) so related that, if the first two were true, the third would also be true.

Hence the Trio is aSyllogism; the two Propositions, “Noxarem” and “Noyarem′”, are itsPremisses; the Proposition “Noxarey” is itsConclusion; the Termsmandm′are itsEliminands; and the Termsxandyare itsRetinends.

Hence we may write it thus:—

“Noxarem;Noyarem′.∴ Noxarey”.

As a second specimen, let us take the Trio

“All cats understand French;Some chickens are cats.Some chickens understand French”.

These, put into normal form, are

“All cats are creatures understanding French;Some chickens are cats.Some chickens are creatures understanding French”.

Here all the six Terms are Species of the Genus “creatures.”

Also the first and second Propositions contain the Pair of codivisional Classes “cats” and “cats”; the first and third contain the Pair “creatures understanding French” and “creatures understanding French”; and the second and third contain the Pair “chickens” and “chickens”.

pg058Also the three Propositions are (as we shall see atp. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens,notstrictly true inourplanet. But there is nothing to hinder them from being true in someotherplanet, sayMarsorJupiter—in which case the third wouldalsobe true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singularcontingentprivilege, unknown in England, namely, that they would be able, at any time when provisions ran short, to utilise the nursery-governess for the nursery-dinner!)

Hence the Trio is aSyllogism; the Genus “creatures” is its ‘Univ.’; the two Propositions, “All cats understand French“ and ”Some chickens are cats”, are itsPremisses, the Proposition “Some chickens understand French” is itsConclusion; the Terms “cats” and “cats” are itsEliminands; and the Terms, “creatures understanding French” and “chickens”, are itsRetinends.

Hence we may write it thus:—

“All cats understand French;Some chickens are cats;∴ Some chickens understand French”.]

When the Terms of a Proposition are represented bywords, it is said to be ‘concrete’; when byletters, ‘abstract.’

To translate a Proposition from concrete into abstract form, we fix on a Univ., and regard each Term as aSpeciesof it, and we choose a letter to represent itsDifferentia.

[For example, suppose we wish to translate “Some soldiers are brave” into abstract form. We may take “men” as Univ., and regard “soldiers” and “brave men” asSpeciesof theGenus“men”; and we may choosexto represent the peculiar Attribute (say “military”) of “soldiers,” andyto represent “brave.” Then the Proposition may be written “Some military men are brave men”;i.e.“Somex-men arey-men”;i.e.(omitting “men,” as explained atp. 26) “Somexarey.”In practice, we should merely say “Let Univ. be “men”,x= soldiers,y= brave”, and at once translate “Some soldiers are brave” into “Somexarey.”]

In practice, we should merely say “Let Univ. be “men”,x= soldiers,y= brave”, and at once translate “Some soldiers are brave” into “Somexarey.”]

The Problems we shall have to solve are of two kinds, viz.

(1) “Given a Pair of Propositions of Relation, which contain between them a pair of codivisional Classes, and which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.”

(2) “Given a Trio of Propositions of Relation, of which every two contain a pair of codivisional Classes, and which are proposed as a Syllogism: to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it iscomplete.”

These Problems we will discuss separately.

The Rules, for doing this, are as follows:—

(1) Determine the ‘Universe of Discourse’.

(2) Construct a Dictionary, makingmandm(ormandm′) represent the pair of codivisional Classes, andx(orx′) andy(ory′) the other two.

(3) Translate the proposed Premisses into abstract form.

(4) Represent them, together, on a Triliteral Diagram.

(5) Ascertain what Proposition, if any, in terms ofxandy, isalsorepresented on it.

(6) Translate this into concrete form.

It is evident that, if the proposed Premisses were true, this other Proposition wouldalsobe true. Hence it is aConclusionconsequent from the proposed Premisses.

[Let us work some examples.(1)“No son of mine is dishonest;People always treat an honest man with respect”.Taking “men” as Univ., we may write these as follows:—“No sons of mine are dishonest men;All honest men are men treated with respect”.We can nowconstructour Dictionary, viz.m= honest;x= sons of mine;y= treated with respect.(Note that the expression “x= sons of mine” is an abbreviated form of “x= the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)The next thing is to translate the proposed Premisses into abstract form, as follows:—“Noxarem′;Allmarey”.Diagram representing x m prime does not exist and all m are ypg061Next, by the process described atp. 50, we represent these on a Triliteral Diagram, thus:—Diagram representing x y prime does not existNext, by the process described atp. 53, we transfer to a Biliteral Diagram all the information we can.The result we read as “Noxarey′” or as “Noy′arex,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose“Noxarey′”,which, translated into concrete form, is“No son of mine fails to be treated with respect”.(2)“All cats understand French;Some chickens are cats”.Taking “creatures” as Univ., we write these as follows:—“All cats are creatures understanding French;Some chickens are cats”.We can now construct our Dictionary, viz.m= cats;x= understanding French;y= chickens.The proposed Premisses, translated into abstract form, are“Allmarex;Someyarem”.In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get thethreePropositions(1)“Somemarex;(2)Nomarex′;(3)Someyarem”.Diagram representing x m and y m exist and x prime m does not existThe Rule, given atp. 50, would make us take these in the order 2, 1, 3.This, however, would produce the resultAlternative diagram representing x m and y m exist and x prime m does not existpg062So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition “Somemarex” isalreadyrepresented on the Diagram.Diagram representing x y existsTransferring our information to a Biliteral Diagram, we getThis result we can read either as “Somexarey” or “Someyarex”.After consulting our Dictionary, we choose“Someyarex”,which, translated into concrete form, is“Some chickens understand French.”(3)“All diligent students are successful;All ignorant students are unsuccessful”.Let Univ. be “students”;m= successful;x= diligent;y= ignorant.These Premisses, in abstract form, are“Allxarem;Allyarem′”.These, broken up, give us the four Propositions(1)“Somexarem;(2)Noxarem′;(3)Someyarem′;(4)Noyarem”.Diagram representing four propositionswhich we will take in the order 2, 4, 1, 3.Representing these on a Triliteral Diagram, we getDiagram representing all x are y prime and all y are x primeAnd this information, transferred to a Biliteral Diagram, isHere we gettwoConclusions, viz.“Allxarey′;Allyarex′.”pg063And these, translated into concrete form, are“All diligent students are (not-ignorant, i.e.) learned;All ignorant students are (not-diligent, i.e.) idle”. (Seep. 4.)(4)“Of the prisoners who were put on their trial at the lastAssizes, all, against whom the verdict ‘guilty’ wasreturned, were sentenced to imprisonment;Some, who were sentenced to imprisonment, were alsosentenced 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“Allxarem;Somemarey”.Breaking up the first, we get the three(1)“Somexarem;(2)Noxarem′;(3)Somemarey”.Diagram representing x m and y m exist and x m does not existRepresenting these, in the order 2, 1, 3, on a Triliteral Diagram, we getHere we get no Conclusion at all.You would very likely have guessed, if you had seenonlythe Premisses, that the Conclusion would be“Some, against whom the verdict ‘guilty’ was returned,were sentenced to hard labour”.But this Conclusion is not eventrue, with regard to the Assizes I have here invented.“Nottrue!” you exclaim. “Then whowerethey, who were sentenced to imprisonment and were also sentenced to hard labour? Theymusthave had the verdict ‘guilty’ returned against them, or how could they be sentenced?”Well, it happened likethis, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, theypleaded‘guilty’. So noverdictwas returned at all; and they were sentenced at once.]

[Let us work some examples.

“No son of mine is dishonest;People always treat an honest man with respect”.

Taking “men” as Univ., we may write these as follows:—

“No sons of mine are dishonest men;All honest men are men treated with respect”.

We can nowconstructour Dictionary, viz.m= honest;x= sons of mine;y= treated with respect.

(Note that the expression “x= sons of mine” is an abbreviated form of “x= the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)

The next thing is to translate the proposed Premisses into abstract form, as follows:—

“Noxarem′;Allmarey”.

Diagram representing x m prime does not exist and all m are y

pg061Next, by the process described atp. 50, we represent these on a Triliteral Diagram, thus:—

Diagram representing x y prime does not exist

Next, by the process described atp. 53, we transfer to a Biliteral Diagram all the information we can.

The result we read as “Noxarey′” or as “Noy′arex,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose

“Noxarey′”,

which, translated into concrete form, is

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

“All cats understand French;Some chickens are cats”.

Taking “creatures” as Univ., we write these as follows:—

“All cats are creatures understanding French;Some chickens are cats”.

We can now construct our Dictionary, viz.m= cats;x= understanding French;y= chickens.

The proposed Premisses, translated into abstract form, are

“Allmarex;Someyarem”.

In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get thethreePropositions

(1)“Somemarex;(2)Nomarex′;(3)Someyarem”.

Diagram representing x m and y m exist and x prime m does not exist

The Rule, given atp. 50, would make us take these in the order 2, 1, 3.

This, however, would produce the result

Alternative diagram representing x m and y m exist and x prime m does not exist

pg062So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition “Somemarex” isalreadyrepresented on the Diagram.

Diagram representing x y exists

Transferring our information to a Biliteral Diagram, we get

This result we can read either as “Somexarey” or “Someyarex”.

After consulting our Dictionary, we choose

“Someyarex”,

which, translated into concrete form, is

“Some chickens understand French.”

“All diligent students are successful;All ignorant students are unsuccessful”.

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

These Premisses, in abstract form, are

“Allxarem;Allyarem′”.

These, broken up, give us the four Propositions

(1)“Somexarem;(2)Noxarem′;(3)Someyarem′;(4)Noyarem”.

Diagram representing four propositions

which we will take in the order 2, 4, 1, 3.

Representing these on a Triliteral Diagram, we get

Diagram representing all x are y prime and all y are x prime

And this information, transferred to a Biliteral Diagram, is

Here we gettwoConclusions, viz.

“Allxarey′;Allyarex′.”

pg063And these, translated into concrete form, are

“All diligent students are (not-ignorant, i.e.) learned;All ignorant students are (not-diligent, i.e.) idle”. (Seep. 4.)

“Of the prisoners who were put on their trial at the lastAssizes, all, against whom the verdict ‘guilty’ wasreturned, were sentenced to imprisonment;Some, who were sentenced to imprisonment, were alsosentenced 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

“Allxarem;Somemarey”.

Breaking up the first, we get the three

(1)“Somexarem;(2)Noxarem′;(3)Somemarey”.

Diagram representing x m and y m exist and x m does not exist

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 seenonlythe Premisses, that the Conclusion would be

“Some, against whom the verdict ‘guilty’ was returned,were sentenced to hard labour”.

But this Conclusion is not eventrue, with regard to the Assizes I have here invented.

“Nottrue!” you exclaim. “Then whowerethey, who were sentenced to imprisonment and were also sentenced to hard labour? Theymusthave had the verdict ‘guilty’ returned against them, or how could they be sentenced?”

Well, it happened likethis, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, theypleaded‘guilty’. So noverdictwas 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.

“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.

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

“All cats understand French;Some chickens are cats”.

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

i.e. “Some chickens understand French.”

“All diligent students are successful;All ignorant students are unsuccessful”.

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

i.e. “All diligent students are learned; and all ignorant students are idle”.

“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”.

Univ. “prisoners who were put on their trial at the last Assizes”,m= sentenced to imprisonment;x= against whom the verdict ‘guilty’ was returned;y= sentenced to hard labour.

[Review Tables VII, VIII (pp. 48,49). Work Examples §1, 17–21 (p. 97); §4, 1–6 (p. 100); §5, 1–6 (p. 101).]

The Rules, for doing this, are as follows:—

(1)Take the proposed Premisses, and ascertain, by the process described atp. 60, what Conclusion, if any, is consequent from them.

(2)If there benoConclusion, say so.

(3)If therebea Conclusion, compare it with the proposed Conclusion, and pronounce accordingly.

I will now work out, in their briefest form, as models for the Reader to imitate in working examples, six Problems.

“All soldiers are strong;All soldiers are brave.Some strong men are brave.”

Univ. “men”;m= soldiers;x= strong;y= brave.

Hence proposed Conclusion is right.

“I admire these pictures;When I admire anything I wish to examine it thoroughly.I wish to examine some of these pictures thoroughly.”

Univ. “things”;m= admired by me;x= these pictures;y= things which I wish to examine thoroughly.

Hence proposed Conclusion isincomplete, thecompleteone being “I wish to examineallthese pictures thoroughly”.

“None but the brave deserve the fair;Some braggarts are cowards.Some braggarts do not deserve the fair.”

Univ. “persons”;m= brave;x= deserving of the fair;y= braggarts.

Hence proposed Conclusion is right.

“All soldiers can march;Some babies are not soldiers.Some babies cannot march”.

Univ. “persons”;m= soldiers;x= able to march;y= babies.

“All selfish men are unpopular;All obliging men are popular.All obliging men are unselfish”.

Univ. “men”;m= popular;x= selfish;y= obliging.

Hence proposed Conclusion isincomplete, thecompleteone containing, in addition, “All selfish men are disobliging”.

”No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;

This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.

This party of tourists need not run.”

Univ. “persons meaning to go by the train, and unable to get a conveyance”;m= having enough time to walk to the station;x= needing to run;y= these tourists.

[Here isanotheropportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you itisn’t, don’t believe it! You don’t mean to tell me those touristsneedto run? IfIwere one of them, and knew thePremissesto be true, I should bequiteclear that Ineedn’trun—and Ishould walk!”Andyouwill reply “But suppose there was a mad bull behind you?”And then your innocent friend will say “Hum! Ha! I must think that over a bit!”You may then explain to him, as a convenienttestof the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of thePremisses, would make theConclusionfalse, the Syllogismmustbe unsound.]

[Here isanotheropportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.

He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you itisn’t, don’t believe it! You don’t mean to tell me those touristsneedto run? IfIwere one of them, and knew thePremissesto be true, I should bequiteclear that Ineedn’trun—and Ishould walk!”

Andyouwill reply “But suppose there was a mad bull behind you?”

And then your innocent friend will say “Hum! Ha! I must think that over a bit!”

You may then explain to him, as a convenienttestof the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of thePremisses, would make theConclusionfalse, the Syllogismmustbe unsound.]

[Review Tables V–VIII (pp. 46–49). Work Examples §4, 7–12 (p. 100); §5, 7–12 (p. 101); §6, 1–10 (p. 106); §7, 1–6 (pp. 107,108).]

Let us agree that “x1” shall mean “Some existing Things have the Attributex”, i.e. (more briefly) “Somexexist”; also that “xy1” shall mean “Somexyexist”, and so on.Such a Proposition may be called an ‘Entity.’

[Note that, when there aretwoletters in the expression, it does not in the least matter which standsfirst: “xy1” and “yx1” mean exactly the same.]

Also that “x0” shall mean “No existing Things have the Attributex”, i.e. (more briefly) “Noxexist”; also that “xy0” shall mean “Noxyexist”, and so on.Such a Proposition may be called a ‘Nullity’.

Also that “†” shall mean “and”.

[Thus “ab1†cd0” means “Someabexist and nocdexist”.]

Also that “¶” shall mean “would, if true, prove”.

[Thus, “x0¶xy0” means “The Proposition ‘Noxexist’ would, if true, prove the Proposition ‘Noxyexist’”.]

When two Letters are both of them accented, or bothnotaccented, they are said to have ‘Like Signs’, or to be ‘Like’: when one is accented, and the other not, they are said to have ‘Unlike Signs’, or to be ‘Unlike’.

Let us take, first, the Proposition “Somexarey”.

This, we know, is equivalent to the Proposition of Existence “Somexyexist”. (Seep. 31.) Hence it may be represented by the expression “xy1”.

The Converse Proposition “Someyarex” may of course be represented by thesameexpression, viz. “xy1”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

“Somexarey′” = “Somey′arex”,“Somex′arey” = “Someyarex′”,“Somex′arey′” = “Somey′arex′”.

Let us take, next, the Proposition “Noxarey”.

This, we know, is equivalent to the Proposition of Existence “Noxyexist”. (Seep. 33.) Hence it may be represented by the expression “xy0”.

The Converse Proposition “Noyarex” may of course be represented by thesameexpression, viz. “xy0”.

Similarly we may represent the three similar Pairs of Converse Propositions, viz.—

“Noxarey′” = “Noy′arex”,“Nox′arey” = “Noyarex′”,“Nox′arey′” = “Noy′arex′”.

pg072Let us take, next, the Proposition “Allxarey”.

Now it is evident that the Double Proposition of Existence “Somexexist and noxy′exist” tells us thatsomex-Things exist, but thatnoneof them have the Attributey′: that is, it tells us thatallof them have the Attributey: that is, it tells us that “Allxarey”.

Also it is evident that the expression “x1†xy′0” represents this Double Proposition.

Hence it also represents the Proposition “Allxarey”.

[The Reader will perhaps be puzzled by the statement that the Proposition “Allxarey” is equivalent to the Double Proposition “Somexexist and noxy′exist,” remembering that it was stated, atp. 33, to be equivalent to the Double Proposition “Somexareyand noxarey′” (i.e. “Somexyexist and noxy′exist”). The explanation is that the Proposition “Somexyexist” containssuperfluous information. “Somexexist” is enough for our purpose.]

This expression may be written in a shorter form, viz. “x1y′0”, sinceeachSubscript takes effect back to thebeginningof the expression.

Similarly we may represent the seven similar Propositions “Allxarey′”, “Allx′arey”, “Allx′arey′”, “Allyarex”, “Allyarex′”, “Ally′arex”, and “Ally′arex′”.

[The Reader should make out all these for himself.]

It will be convenient to remember that, in translating a Proposition, beginning with “All”, from abstract form into subscript form, orvice versâ, the Predicatechanges sign(that is, changes from positive to negative, or else from negative to positive).

[Thus, the Proposition “Allyarex′” becomes “y1x0”, where the Predicate changes fromx′tox.Again, the expression “x′1y′0” becomes “Allx′arey”, where the Predicate changes fory′toy.]

[Thus, the Proposition “Allyarex′” becomes “y1x0”, where the Predicate changes fromx′tox.

Again, the expression “x′1y′0” becomes “Allx′arey”, where the Predicate changes fory′toy.]

We already know how to represent each of the three Propositions of a Syllogism in subscript form. When that is done, all we need, besides, is to write the three expressions in a row, with “†” between the Premisses, and “¶” before the Conclusion.

[Thus the Syllogism“Noxarem′;Allmarey.∴ Noxarey′.”may be represented thus:—xm′0†m1y′0¶xy′0When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it intoabstractform, andthenceinto subscript form. But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.]

[Thus the Syllogism

“Noxarem′;Allmarey.∴ Noxarey′.”

may be represented thus:—

xm′0†m1y′0¶xy′0

When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it intoabstractform, andthenceinto subscript form. But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.]

When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have aFormula, by which we can at once find, without having to use Diagrams again, the Conclusion to anyotherPair of Premisses having thesamesubscript forms.

[Thus, the expressionxm0†ym′0¶xy0is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms arexm0†ym′0For example, suppose we had the Pair of Propositions“No gluttons are healthy;No unhealthy men are strong”.proposed as Premisses. Taking “men” as our ‘Universe’, and makingm= healthy;x= gluttons;y= strong; we might translate the Pair into abstract form, thus:—“Noxarem;Nom′arey”.These, in subscript form, would bexm0†m′y0which are identical with those in ourFormula. Hence we at once know the Conclusion to bexy0that is, in abstract form,“Noxarey”;that is, in concrete form,“No gluttons are strong”.]

[Thus, the expression

xm0†ym′0¶xy0

is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are

xm0†ym′0

For example, suppose we had the Pair of Propositions

“No gluttons are healthy;No unhealthy men are strong”.

proposed as Premisses. Taking “men” as our ‘Universe’, and makingm= healthy;x= gluttons;y= strong; we might translate the Pair into abstract form, thus:—

“Noxarem;Nom′arey”.

These, in subscript form, would be

xm0†m′y0

which are identical with those in ourFormula. Hence we at once know the Conclusion to be

xy0

that is, in abstract form,

“Noxarey”;

that is, in concrete form,

“No gluttons are strong”.]

I shall now take three different forms of Pairs of Premisses, and work out their Conclusions, once for all, by Diagrams; and thus obtain some useful Formulæ. I shall call them “Fig. I”, “Fig. II”, and “Fig. III”.

This includes any Pair of Premisses which are both of them Nullities, and which contain Unlike Eliminands.

The simplest case is

In this case we see that the Conclusion is a Nullity, and that the Retinends have kept their Signs.

And we should find this Rule to hold good withanyPair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such asm1x0†ym′0(which ¶xy0)xm′0†m1y0(which ¶xy0)x′m0†ym′0(which ¶x′y0)m′1x′0†m1y′0(which ¶x′y′0).]

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

m1x0†ym′0(which ¶xy0)xm′0†m1y0(which ¶xy0)x′m0†ym′0(which ¶x′y0)m′1x′0†m1y′0(which ¶x′y′0).]

If either Retinend is asserted in thePremissesto exist, of course it may be so asserted in theConclusion.

Hence we get twoVariantsof Fig. I, viz.

(α) whereoneRetinend is so asserted;

(β) wherebothare so asserted.

[The Reader had better work out, on Diagrams, examples of these two Variants, such asm1x0†y1m′0(which provesy1x0)x1m′0†m1y0(which provesx1y0)x′1m0†y1m′0(which provesx′1y0†y1x′0).]

[The Reader had better work out, on Diagrams, examples of these two Variants, such as

m1x0†y1m′0(which provesy1x0)x1m′0†m1y0(which provesx1y0)x′1m0†y1m′0(which provesx′1y0†y1x′0).]

The Formula, to be remembered, is

xm0†ym′0¶xy0

with the following two Rules:—

(1)Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.

pg076(2)A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.

[Note that Rule (1) is merely the Formula expressed in words.]

This includes any Pair of Premisses, of which one is a Nullity and the other an Entity, and which contain Like Eliminands.

The simplest case is

In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign.

And we should find this Rule to hold good withanyPair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such asx′m0†ym1(which ¶xy1)x1m′0†y′m′1(which ¶x′y′1)m1x0†y′m1(which ¶x′y′1).]

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x′m0†ym1(which ¶xy1)x1m′0†y′m′1(which ¶x′y′1)m1x0†y′m1(which ¶x′y′1).]

The Formula, to be remembered, is,

xm0†ym1¶x′y1

with the following Rule:—

A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.

[Note that this Rule is merely the Formula expressed in words.]

This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist.

The simplest case is

xm0†ym0†m1

[Note that “m1” is here statedseparately, because it does not matter in which of the two Premisses it occurs: so that this includes thethreeforms “m1x0†ym0”, “xm0†m1y0”, and “m1x0†m1y0”.]

In this case we see that the Conclusion is an Entity, and thatbothRetinends have changed their Signs.

And we should find this Rule to hold good withanyPair of Premisses which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such asx′m0†m1y0(which ¶xy′1)m′1x0†m′y′0(which ¶x′y1)m1x′0†m1y′0(which ¶xy1).]

[The Reader had better satisfy himself of this, by working out, on Diagrams, several varieties, such as

x′m0†m1y0(which ¶xy′1)m′1x0†m′y′0(which ¶x′y1)m1x′0†m1y′0(which ¶xy1).]

The Formula, to be remembered, is

xm0†ym0†m1¶x′y′1

with the following Rule (which is merely the Formula expressed in words):—

Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.

In order to help the Reader to remember the peculiarities and Formulæ of these three Figures, I will put them all together in one Table.

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