pg078TABLE IX.Fig. I.xm0†ym′0¶xy0Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.Fig. II.xm0†ym1¶x′y1A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.Fig. III.xm0†ym0†m1¶x′y′1Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
pg078TABLE IX.Fig. I.xm0†ym′0¶xy0Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.Fig. II.xm0†ym1¶x′y1A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.Fig. III.xm0†ym0†m1¶x′y′1Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
Fig. I.
xm0†ym′0¶xy0
Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.
A Retinend, asserted in the Premisses to exist, may be so asserted in the Conclusion.
Fig. II.
xm0†ym1¶x′y1
A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign.
Fig. III.
xm0†ym0†m1¶x′y′1
Two Nullities, with Like Eliminands asserted to exist, yield an Entity, in which both Retinends change their Signs.
I will now work out, by these Formulæ, as models for the Reader to imitate, some Problems in Syllogisms which have been already worked, by Diagrams, inBook V., Chap. II.
“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.
xm′0†m1y′0¶xy′0[Fig. I.
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.
m1x′0†ym1¶xy1[Fig. II.
i.e.“Some chickens understand French.”
“All diligent students are successful;All ignorant students are unsuccessful.”
Univ. “students”;m= successful;x= diligent;y= ignorant.
x1m′0†y1m0¶x1y0†y1x0[Fig. I (β).
i.e.“All diligent students are learned; and all ignorant students are idle.”
“All soldiers are strong;All soldiers are brave.Some strong men are brave.”
Univ. “men”;m= soldiers;x= strong;y= brave.
m1x′0†m1y′0¶xy1[Fig. III.
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;y= things which I wish to examine thoroughly.
x1m′0†m1y′0¶x1y′0[Fig. I (α).
Hence proposed Conclusion,xy1, 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.
m′x0†ym′1¶x′y1[Fig. II.
Hence proposed Conclusion is right.
”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.
m′x′0†y1m′0do not come under any of the three Figures. Hence it is necessary to return to the Method of Diagrams, as shown atp. 69.
Hence there is no Conclusion.
[Work Examples §4, 12–20 (p. 100); §5, 13–24 (pp. 101,102); §6, 1–6 (p. 106); §7, 1–3 (pp. 107,108). Also readNote (A), at p. 164.]
[Work Examples §4, 12–20 (p. 100); §5, 13–24 (pp. 101,102); §6, 1–6 (p. 106); §7, 1–3 (pp. 107,108). Also readNote (A), at p. 164.]
Any argument whichdeceivesus, by seeming to prove what it does not really prove, may be called a ‘Fallacy’ (derived from the Latin verbfallo“I deceive”): but the particular kind, to be now discussed, consists of a Pair of Propositions, which are proposed as the Premisses of a Syllogism, but yield no Conclusion.
When each of the proposed Premisses is a Proposition inI, orE, orA, (the only kinds with which we are now concerned,) the Fallacy may be detected by the ‘Method of Diagrams,’ by simply setting them out on a Triliteral Diagram, and observing that they yield no information which can be transferred to the Biliteral Diagram.
But suppose we were working by the ‘Method ofSubscripts,’ and had to deal with a Pair of proposed Premisses, which happened to be a ‘Fallacy,’ how could we be certain that they would not yield any Conclusion?
Our best plan is, I think, to deal withFallaciesin the same was as we have already dealt withSyllogisms: that is, to take certain forms of Pairs of Propositions, and to workpg082them out, once for all, on the Triliteral Diagram, and ascertain that they yieldnoConclusion; and then to record them, for future use, asFormulæ for Fallacies, just as we have already recorded our threeFormulæ for Syllogisms.
Now, if we were to record the two Sets of Formulæ in thesameshape, viz. by the Method of Subscripts, there would be considerable risk of confusing the two kinds. Hence, in order to keep them distinct, I propose to record the Formulæ forFallaciesinwords, and to call them “Forms” instead of “Formulæ.”
Let us now proceed to find, by the Method of Diagrams, three “Forms of Fallacies,” which we will then put on record for future use. They are as follows:—
(1)Fallacy of Like Eliminands not asserted to exist.(2)Fallacy of Unlike Eliminands with an Entity-Premiss.(3)Fallacy of two Entity-Premisses.
These shall be discussed separately, and it will be seen that each fails to yield a Conclusion.
It is evident that neither of the given Propositions can be anEntity, since that kind asserts theexistenceof both of its Terms (seep. 20). Hence they must both beNullities.
Hence the given Pair may be represented by (xm0†ym0), with or withoutx1,y1.
These, set out on Triliteral Diagrams, are
Here the given Pair may be represented by (xm0†ym′1) with or withoutx1orm1.
These, set out on Triliteral Diagrams, are
Here the given Pair may be represented by either (xm1†ym1) or (xm1†ym′1).
These, set out on Triliteral Diagrams, are
Let us suppose that we have before us a Pair of Propositions of Relation, which contain between them a Pair of codivisional Classes, and that we wish to ascertain what Conclusion, if any, is consequent from them. We translate them, if necessary, into subscript-form, and then proceed as follows:—
(1) We examine their Subscripts, in order to see whether they are
(a) a Pair of Nullities;or (b) a Nullity and an Entity;or (c) a Pair of Entities.
(2) If they are a Pair of Nullities, we examine their Eliminands, in order to see whether they are Unlike or Like.
If their Eliminands areUnlike, it is a case of Fig. I. We then examine their Retinends, to see whether one or both of them are asserted toexist. If one Retinend is so asserted, it is a case of Fig. I (α); if both, it is a case of Fig. I (β).
If their Eliminands are Like, we examine them, in order to see whether either of them is asserted to exist. If so, it is a case of Fig. III.; if not, it is a case of “Fallacy of Like Eliminands not asserted to exist.”
(3) If they are a Nullity and an Entity, we examine their Eliminands, in order to see whether they are Like or Unlike.
If their Eliminands are Like, it is a case of Fig. II.; ifUnlike, it is a case of “Fallacy of Unlike Eliminands with an Entity-Premiss.”
(4) If they are a Pair of Entities, it is a case of “Fallacy of two Entity-Premisses.”
[Work Examples §4, 1–11 (p. 100); §5, 1–12 (p. 101); §6, 7–12 (p. 106); §7, 7–12 (p. 108).]
[Work Examples §4, 1–11 (p. 100); §5, 1–12 (p. 101); §6, 7–12 (p. 106); §7, 7–12 (p. 108).]
When a Set of three or more Biliteral Propositions are such that all their Terms are Species of the same Genus, and are also so related that two of them, taken together, yield a Conclusion, which, taken with another of them, yields another Conclusion, and so on, until all have been taken, it is evident that, if the original Set were true, the last Conclusion wouldalsobe true.
Such a Set, with the last Conclusion tacked on, is called a ‘Sorites’;the original Set of Propositions is called its ‘Premisses’;each of the intermediate Conclusions is called a ‘Partial Conclusion’ of the Sorites;the last Conclusion is called its ‘Complete Conclusion,’ or, more briefly, its ‘Conclusion’; the Genus, of which all the Terms are Species, is called its ‘Universe of Discourse’, or, more briefly, its ‘Univ.’;the Terms, used as Eliminands in the Syllogisms, are called its ‘Eliminands’;and the two Terms, which are retained, and therefore appear in the Conclusion, are called its ‘Retinends’.
[Note that eachPartialConclusion contains one or twoEliminands; but that theCompleteConclusion containsRetinendsonly.]
[Note that eachPartialConclusion contains one or twoEliminands; but that theCompleteConclusion containsRetinendsonly.]
The Conclusion is said to be ‘consequent’ from the Premisses;for which reason it is usual to prefix to it the word “Therefore” (or the symbol “∴”).
[Note that the question, whether the Conclusion is or is notconsequentfrom the Premisses, is not affected by theactualtruth or falsity of any one of the Propositions which make up the Sorites, by depends entirely on theirrelationship to one another.pg086As a specimen-Sorites, let us take the following Set of 5 Propositions:—(1)”Noaareb′;(2)Allbarec;(3)Allcared;(4)Noe′area′;(5)Allharee′”.Here the first and second, taken together, yield “Noaarec′”.This, taken along with the third, yields “Noaared′”.This, taken along with the fourth, yields “Nod′aree′”.And this, taken along with the fifth, yields “Allhared”.Hence, if the original Set were true, this wouldalsobe true.Hence the original Set, with this tacked on, is aSorites; the original Set is itsPremisses; the Proposition “Allhared” is itsConclusion; the Termsa,b,c,eare itsEliminands; and the Termsdandhare itsRetinends.Hence we may write the whole Sorites thus:—”Noaareb′;Allbarec;Allcared;Noe′area′;Allharee′.∴ Allhared”.In the above Sorites, the 3 Partial Conclusions are the Positions “Noaaree′”, “Noaared′”, “Nod′aree′”; but, if the Premisses were arranged in other ways, other Partial Conclusions might be obtained. Thus, the order 41523 yields the Partial Conclusions “Noc′areb′”, “Allhareb”, “Allharec”. There are altogetherninePartial Conclusions to this Sorites, which the Reader will find it an interesting task to make out for himself.]
[Note that the question, whether the Conclusion is or is notconsequentfrom the Premisses, is not affected by theactualtruth or falsity of any one of the Propositions which make up the Sorites, by depends entirely on theirrelationship to one another.
pg086As a specimen-Sorites, let us take the following Set of 5 Propositions:—
(1)”Noaareb′;(2)Allbarec;(3)Allcared;(4)Noe′area′;(5)Allharee′”.
Here the first and second, taken together, yield “Noaarec′”.
This, taken along with the third, yields “Noaared′”.
This, taken along with the fourth, yields “Nod′aree′”.
And this, taken along with the fifth, yields “Allhared”.
Hence, if the original Set were true, this wouldalsobe true.
Hence the original Set, with this tacked on, is aSorites; the original Set is itsPremisses; the Proposition “Allhared” is itsConclusion; the Termsa,b,c,eare itsEliminands; and the Termsdandhare itsRetinends.
Hence we may write the whole Sorites thus:—
”Noaareb′;Allbarec;Allcared;Noe′area′;Allharee′.∴ Allhared”.
In the above Sorites, the 3 Partial Conclusions are the Positions “Noaaree′”, “Noaared′”, “Nod′aree′”; but, if the Premisses were arranged in other ways, other Partial Conclusions might be obtained. Thus, the order 41523 yields the Partial Conclusions “Noc′areb′”, “Allhareb”, “Allharec”. There are altogetherninePartial Conclusions to this Sorites, which the Reader will find it an interesting task to make out for himself.]
The Problems we shall have to solve are of the following form:—
“Given three or more Propositions of Relation, which are proposed as Premisses: to ascertain what Conclusion, if any, is consequent from them.”
We will limit ourselves, at present, to Problems which can be worked by the Formulæ of Fig. I. (Seep. 75.) Those, that requireotherFormulæ, are rather too hard for beginners.
Such Problems may be solved by either of two Methods, viz.
(1)The Method of Separate Syllogisms;(2)The Method of Underscoring.
These shall be discussed separately.
The Rules, for doing this, are as follows:—
(1) Name the ‘Universe of Discourse’.(2) Construct a Dictionary, makinga,b,c, &c. represent the Terms.(3) Put the Proposed Premisses into subscript form.(4) Select two which, containing between them a pair of codivisional Classes, can be used as the Premisses of a Syllogism.(5) Find their Conclusion by Formula.(6) Find a third Premiss which, along with this Conclusion, can be used as the Premisses of a second Syllogism.(7) Find a second Conclusion by Formula.(8) Proceed thus, until all the proposed Premisses have been used.(9) Put the last Conclusion, which is the Complete Conclusion of the Sorites, into concrete form.
[As an example of this process, let us take, as the proposed Set of Premisses,(1)“All the policemen on this beat sup with our cook;(2)No man with long hair can fail to be a poet;(3)Amos Judd has never been in prison;(4)Our cook’s ‘cousins’ all love cold mutton;(5)None but policemen on this beat are poets;(6)None but her ‘cousins’ ever sup with our cook;(7)Men with short hair have all been in prison.”Univ. “men”;a= Amos Judd;b= cousins of our cook;c= having been in prison;d= long-haired;e= loving cold mutton;h= poets;k= policemen on this beat;l= supping with our cookpg089We now have to put the proposed Premisses intosubscriptform. Let us begin by putting them intoabstractform. The result is(1)”Allkarel;(2)Nodareh′;(3)Allaarec′;(4)Allbaree;(5)Nok′areh;(6)Nob′arel;(7)Alld′arec.”And it is now easy to put them intosubscriptform, as follows:—(1)k1l′0(2)dh′0(3)a1c0(4)b1e′0(5)k′h0(6)b′l0(7)d′1c′0We now have to find a pair of Premisses which will yield a Conclusion. Let us begin with No. (1), and look down the list, till we come to one which we can take along with it, so as to form Premisses belonging to Fig. I. We find that No. (5) will do, since we can takekas our Eliminand. So our first syllogism is(1)k1l′0(5)k′h0∴l′h0… (8)We must now begin again withl′h0and find a Premiss to go along with it. We find that No. (2) will do,hbeing our Eliminand. So our next Syllogism is(8)l′h0(2)dh′0∴l′d0… (9)We have now used up Nos. (1), (5), and (2), and must search among the others for a partner forl′d0. We find that No. (6) will do. So we write(9)l′d0(6)b′l0∴db′0… (10)Now what can we take along withdb′0? No. (4) will do.(10)db′0(4)b1e′0∴de′0… (11)pg090Along with this we may take No. (7).(11)de′0(7)d′1c′0∴c′e′0… (12)And along with this we may take No. (3).(12)c′e′0(3)a1c0∴a1e′0This Complete Conclusion, translated intoabstractform, is“Allaaree”;and this, translated intoconcreteform, is“Amos Judd loves cold mutton.”In actuallyworkingthis Problem, the above explanations would, of course, be omitted, and all, that would appear on paper, would be as follows:—(1)k1l′0(2)dh′0(3)a1c0(4)b1e′0(5)k′h0(6)b′l0(7)d′1c′0(1)k1l′0(5)k′h0∴l′h0… (8)(8)l′h0(2)dh′0∴l′d0… (9)(9)l′d0(6)b′l0∴db′0… (10)(10)db′0(4)b1e′0∴de′0… (11)(11)de′0(7)d′1c′0∴c′e′0… (12)(12)c′e′0(3)a1c0∴a1e′0Note that, in working a Sorites by this Process, we may begin withanyPremiss we choose.]
[As an example of this process, let us take, as the proposed Set of Premisses,
(1)“All the policemen on this beat sup with our cook;(2)No man with long hair can fail to be a poet;(3)Amos Judd has never been in prison;(4)Our cook’s ‘cousins’ all love cold mutton;(5)None but policemen on this beat are poets;(6)None but her ‘cousins’ ever sup with our cook;(7)Men with short hair have all been in prison.”
Univ. “men”;a= Amos Judd;b= cousins of our cook;c= having been in prison;d= long-haired;e= loving cold mutton;h= poets;k= policemen on this beat;l= supping with our cook
pg089We now have to put the proposed Premisses intosubscriptform. Let us begin by putting them intoabstractform. The result is
(1)”Allkarel;(2)Nodareh′;(3)Allaarec′;(4)Allbaree;(5)Nok′areh;(6)Nob′arel;(7)Alld′arec.”
And it is now easy to put them intosubscriptform, as follows:—
(1)k1l′0(2)dh′0(3)a1c0(4)b1e′0(5)k′h0(6)b′l0(7)d′1c′0
We now have to find a pair of Premisses which will yield a Conclusion. Let us begin with No. (1), and look down the list, till we come to one which we can take along with it, so as to form Premisses belonging to Fig. I. We find that No. (5) will do, since we can takekas our Eliminand. So our first syllogism is
(1)k1l′0(5)k′h0∴l′h0… (8)
We must now begin again withl′h0and find a Premiss to go along with it. We find that No. (2) will do,hbeing our Eliminand. So our next Syllogism is
(8)l′h0(2)dh′0∴l′d0… (9)
We have now used up Nos. (1), (5), and (2), and must search among the others for a partner forl′d0. We find that No. (6) will do. So we write
(9)l′d0(6)b′l0∴db′0… (10)
Now what can we take along withdb′0? No. (4) will do.
(10)db′0(4)b1e′0∴de′0… (11)
pg090Along with this we may take No. (7).
(11)de′0(7)d′1c′0∴c′e′0… (12)
And along with this we may take No. (3).
(12)c′e′0(3)a1c0∴a1e′0
This Complete Conclusion, translated intoabstractform, is
“Allaaree”;
and this, translated intoconcreteform, is
“Amos Judd loves cold mutton.”
In actuallyworkingthis Problem, the above explanations would, of course, be omitted, and all, that would appear on paper, would be as follows:—
(1)k1l′0(2)dh′0(3)a1c0(4)b1e′0(5)k′h0(6)b′l0(7)d′1c′0
(1)k1l′0(5)k′h0∴l′h0… (8)
(8)l′h0(2)dh′0∴l′d0… (9)
(9)l′d0(6)b′l0∴db′0… (10)
(10)db′0(4)b1e′0∴de′0… (11)
(11)de′0(7)d′1c′0∴c′e′0… (12)
(12)c′e′0(3)a1c0∴a1e′0
Note that, in working a Sorites by this Process, we may begin withanyPremiss we choose.]
Consider the Pair of Premisses
xm0†ym′0
which yield the Conclusionxy0
We see that, in order to get this Conclusion, we must eliminatemandm′, and writexandytogether in one expression.
Now, if we agree tomarkmandm′as eliminated, and to read the two expressions together, as if they were written in one, the two Premisses will then exactly represent theConclusion, and we need not write it out separately.
Let us agree to mark the eliminated letters byunderscoringthem, putting asinglescore under thefirst, and adoubleone under thesecond.
The two Premisses now become
xm0†ym′0
which we read as “xy0”.
In copying out the Premisses for underscoring, it will be convenient toomit all subscripts. As to the “0’s” we may alwayssupposethem written, and, as to the “1’s”, we are not concerned to knowwhichTerms are asserted toexist, except those which appear in theCompleteConclusion; and forthemit will be easy enough to refer to the original list.
pg092[I will now go through the process of solving, by this method, the example worked in§ 2.The Data are1k1l′0†2dh′0†3a1c0†4b1e′0†5k′h0†6b′l0†7d′1c′0The Reader should take a piece of paper, and write out this solution for himself. The first line will consist of the above Data; the second must be composed, bit by bit, according to the following directions.We begin by writing down the first Premiss, with its numeral over it, but omitting the subscripts.We have now to find a Premiss which can be combined with this,i.e., a Premiss containing eitherk′orl. The first we find is No. 5; and this we tack on, with a †.To get theConclusionfrom these,kandk′must be eliminated, and what remains must be taken as one expression. So weunderscorethem, putting asinglescore underk, and adoubleone underk′. The result we read asl′h.We must now find a Premiss containing eitherlorh′. Looking along the row, we fix on No. 2, and tack it on.Now these 3 Nullities are really equivalent to (l′h†dh′), in whichhandh′must be eliminated, and what remains taken as one expression. So weunderscorethem. The result reads asl′d.We now want a Premiss containinglord′. No. 6 will do.These 4 Nullities are really equivalent to (l′d†b′l). So we underscorel′andl. The result reads asdb′.We now want a Premiss containingd′orb. No. 4 will do.Here we underscoreb′andb. The result reads asde′.We now want a Premiss containingd′ore. No. 7 will do.Here we underscoredandd′. The result reads asc′e′.We now want a Premiss containingcore. No. 3 will do—in factmustdo, as it is the only one left.Here we underscorec′andc; and, as the whole thing now reads ase′a, we tack one′a0as theConclusion, with a ¶.We now look along the row of Data, to see whethere′orahas been given asexistent. We find thatahas been so given in No. 3. So we add this fact to the Conclusion, which now stands as ¶e′a0†a1,i.e.¶a1e′0; i.e. “Allaaree.”If the Reader has faithfully obeyed the above directions, his written solution will now stand as follows:—1k1l′0†2dh′0†3a1c0†4b1e′0†5k′h0†6b′l0†7d′1c′01kl′†5k′h†2dh′†6b′l†4be′†7d′c′†3ac¶e′a0†a1i.e.¶a1e′0;i.e.“Allaaree.”pg093The Reader should now take a second piece of paper, and copy the Data only, and try to work out the solution for himself, beginning with some other Premiss.If he fails to bring out the Conclusiona1e′0, I would advise him to take a third piece of paper, andbegin again!]
pg092[I will now go through the process of solving, by this method, the example worked in§ 2.
The Data are
1k1l′0†2dh′0†3a1c0†4b1e′0†5k′h0†6b′l0†7d′1c′0
The Reader should take a piece of paper, and write out this solution for himself. The first line will consist of the above Data; the second must be composed, bit by bit, according to the following directions.
We begin by writing down the first Premiss, with its numeral over it, but omitting the subscripts.
We have now to find a Premiss which can be combined with this,i.e., a Premiss containing eitherk′orl. The first we find is No. 5; and this we tack on, with a †.
To get theConclusionfrom these,kandk′must be eliminated, and what remains must be taken as one expression. So weunderscorethem, putting asinglescore underk, and adoubleone underk′. The result we read asl′h.
We must now find a Premiss containing eitherlorh′. Looking along the row, we fix on No. 2, and tack it on.
Now these 3 Nullities are really equivalent to (l′h†dh′), in whichhandh′must be eliminated, and what remains taken as one expression. So weunderscorethem. The result reads asl′d.
We now want a Premiss containinglord′. No. 6 will do.
These 4 Nullities are really equivalent to (l′d†b′l). So we underscorel′andl. The result reads asdb′.
We now want a Premiss containingd′orb. No. 4 will do.
Here we underscoreb′andb. The result reads asde′.
We now want a Premiss containingd′ore. No. 7 will do.
Here we underscoredandd′. The result reads asc′e′.
We now want a Premiss containingcore. No. 3 will do—in factmustdo, as it is the only one left.
Here we underscorec′andc; and, as the whole thing now reads ase′a, we tack one′a0as theConclusion, with a ¶.
We now look along the row of Data, to see whethere′orahas been given asexistent. We find thatahas been so given in No. 3. So we add this fact to the Conclusion, which now stands as ¶e′a0†a1,i.e.¶a1e′0; i.e. “Allaaree.”
If the Reader has faithfully obeyed the above directions, his written solution will now stand as follows:—
1k1l′0†2dh′0†3a1c0†4b1e′0†5k′h0†6b′l0†7d′1c′0
1kl′†5k′h†2dh′†6b′l†4be′†7d′c′†3ac¶e′a0†a1i.e.¶a1e′0;
i.e.“Allaaree.”
pg093The Reader should now take a second piece of paper, and copy the Data only, and try to work out the solution for himself, beginning with some other Premiss.
If he fails to bring out the Conclusiona1e′0, I would advise him to take a third piece of paper, andbegin again!]
I will now work out, in its briefest form, a Sorites of 5 Premisses, to serve as a model for the Reader to imitate in working examples.
(1)”I greatly value everything that John gives me;(2)Nothing but this bone will satisfy my dog;(3)I take particular care of everything that I greatly value;(4)This bone was a present from John;(5)The things, of which I take particular care, are things I donotgive to my dog”.
Univ. “things”;a= given by John to me;b= given by me to my dog;c= greatly valued by me;d= satisfactory to my dog;e= taken particular care of by me;h= this bone.
1a1c′0†2h′d0†3c1e′0†4h1a′0†5e1b0
1ac′ †3ce′†4ha′†2h′d†5eb¶db0
i.e. “Nothing, that I give my dog, satisfies him,” or, “My dog is not satisfied withanythingthat I give him!”
[Note that, in working a Sorites by this process, we may begin withanyPremiss we choose. For instance, we might begin with No. 5, and the result would then be5eb†3ce′†1ac′†4ha′†2h′d¶bd0]
[Note that, in working a Sorites by this process, we may begin withanyPremiss we choose. For instance, we might begin with No. 5, and the result would then be
5eb†3ce′†1ac′†4ha′†2h′d¶bd0]
[Work Examples §4, 25–30 (p. 100); §5, 25–30 (p. 102); §6, 13–15 (p. 106); §7, 13–15 (p. 108); §8, 1–4, 13, 14, 19, 24 (pp. 110,111); §9, 1–4, 26, 27, 40, 48 (pp. 112,116,119,121).]
[Work Examples §4, 25–30 (p. 100); §5, 25–30 (p. 102); §6, 13–15 (p. 106); §7, 13–15 (p. 108); §8, 1–4, 13, 14, 19, 24 (pp. 110,111); §9, 1–4, 26, 27, 40, 48 (pp. 112,116,119,121).]
pg094The Reader, who has successfully grappled with all the Examples hitherto set, and who thirsts, like Alexander the Great, for “more worlds to conquer,” may employ his spare energies on the following 17 Examination-Papers. He is recommended not to attempt more thanonePaper on any one day. The answers to the questions about words and phrases may be found by referring to the Index atp. 197.
I.§4, 31 (p. 100); §5, 31–34 (p. 102); §6, 16, 17 (p. 106); §7, 16 (p. 108); §8, 5, 6 (p. 110); §9, 5, 22, 42 (pp. 112,115,119). What is ‘Classification’? And what is a ‘Class’?
II.§4, 32 (p. 100); §5, 35–38 (pp. 102,103); §6, 18 (p. 107); §7, 17, 18 (p. 108); §8, 7, 8 (p. 110); §9, 6, 23, 43 (pp. 112,115,119). What are ‘Genus’, ‘Species’, and ‘Differentia’?
III.§4, 33 (p. 100); §5, 39–42 (p. 103); §6, 19, 20 (p. 107); §7, 19 (p. 109); §8, 9, 10 (p. 111); §9, 7, 24, 44 (pp. 113,116,120). What are ‘Real’ and ‘Imaginary’ Classes?
IV.§4, 34 (p. 100); §5, 43–46 (p. 103); §6, 21 (p. 107); §7, 20, 21 (p. 109); §8, 11, 12 (p. 111); §9, 8, 25, 45 (pp. 113,116,120). What is ‘Division’? When are Classes said to be ‘Codivisional’?
V.§4, 35 (p. 100); §5, 47–50 (p. 103); §6, 22, 23 (p. 107); §7, 22 (p. 109); §8, 15, 16 (p. 111); §9, 9, 28, 46 (pp. 113,116,120). What is ‘Dichotomy’? What arbitrary rule doesitsometimes require?
pg095VI.§4, 36 (p. 100); §5, 51–54 (p. 103); §6, 24 (p. 107); §7, 23, 24 (p. 109); §8, 17 (p. 111); §9, 10, 29, 47 (pp. 113,117,120). What is a ‘Definition’?
VII.§4, 37 (p. 100); §5, 55–58 (pp. 103,104); §6, 25, 26 (p. 107); §7, 25 (p. 109); §8, 18 (p. 111); §9, 11, 30, 49 (pp. 113,117,121). What are the ‘Subject’ and the ‘Predicate’ of a Proposition? What is its ‘Normal’ form?
VIII.§4, 38 (p. 100); §5, 59–62 (p. 104); §6, 27 (p. 107); §7, 26, 27 (p. 109); §8, 20 (p. 111); §9, 12, 31, 50 (pp. 113,117,121). What is a Proposition ‘inI’? ‘InE’? And ‘inA’?
IX.§4, 39 (p. 100); §5, 63–66 (p. 104); §6, 28, 29 (p. 107); §7, 28 (p. 109); §8, 21 (p. 111); §9, 13, 32, 51 (pp. 114,117,121). What is the ‘Normal’ form of a Proposition of Existence?
X.§4, 40 (p. 100); §5, 67–70 (p. 104); §6, 30 (p. 107); §7, 29, 30 (p. 109); §8, 22 (p. 111); §9, 14, 33, 52 (pp. 114,117,122). What is the ‘Universe of Discourse’?
XI.§4, 41 (p. 100); §5, 71–74 (p. 104); §6, 31, 32 (p. 107); §7, 31 (p. 109); §8, 23 (p. 111); §9, 15, 34, 53 (pp. 114,118,122). What is implied, in a Proposition of Relation, as to the Reality of its Terms?
XII.§4, 42 (p. 100); §5, 75–78 (p. 105); §6, 33 (p. 107); §7, 32, 33 (pp. 109,110); §8, 25 (p. 111); §9, 16, 35, 54 (pp. 114,118,122). Explain the phrase “sitting on the fence”.
XIII.§5, 79–83 (p. 105); §6, 34, 35 (p. 107); §7, 34 (p. 110); §8, 26 (p. 111); §9, 17, 36, 55 (pp. 114,118,122). What are ‘Converse’ Propositions?
XIV.§5, 84–88 (p. 105); §6, 36 (p. 107); §7, 35, 36 (p. 110); §8, 27 (p. 111); §9, 18, 37, 56 (pp. 114,118,123). What are ‘Concrete’ and ‘Abstract’ Propositions?
pg096XV.§5, 89–93 (p. 105); §6, 37, 38 (p. 107); §7, 37 (p. 110); §8, 28 (p. 111); §9, 19, 38, 57 (pp. 115,118,123). What is a ‘Syllogism’? And what are its ‘Premisses’ and its ‘Conclusion’?
XVI.§5, 94–97 (p. 106); §6, 39 (p. 107); §7, 38, 39 (p. 110); §8, 29 (p. 111); §9, 20, 39, 58 (pp. 115,119,123). What is a ‘Sorites’? And what are its ‘Premisses’, its ‘Partial Conclusions’, and its ‘Complete Conclusion’?
XVII.§5, 98–101 (p. 106); §6, 40 (p. 107); §7, 40 (p. 110); §8, 30 (p. 111); §9, 21, 41, 59, 60 (pp. 115,119,124). What are the ‘Universe of Discourse’, the ‘Eliminands’, and the ‘Retinends’, of a Syllogism? And of a Sorites?
[N.B. Reference tags for Examples, Answers & Solutions will be found in the right margin.]
[N.B. Reference tags for Examples, Answers & Solutions will be found in the right margin.]
1.I have been out for a walk.
2.I am feeling better.
3.No one has read the letter but John.
4.Neither you nor I are old.
5.No fat creatures run well.
6.None but the brave deserve the fair.
7.No one looks poetical unless he is pale.
8.Some judges lose their tempers.
9.I never neglect important business.
10.What is difficult needs attention.
11.What is unwholesome should be avoided.
12.All the laws passed last week relate to excise.
13.Logic puzzles me.
14.There are no Jews in the house.
15.Some dishes are unwholesome if not well-cooked.
16.Unexciting books make one drowsy.
17.When a man knows what he’s about, he can detect a sharper.
18.You and I know what we’re about.
19.Some bald people wear wigs.
20.Those who are fully occupied never talk about their grievances.
21.No riddles interest me if they can be solved.
1.Noxarem;Nom′arey.
2.Nox′arem′;Allm′arey.
3.Somex′arem;Nomarey.
4.Allmarex;Allm′arey′.
5.Allm′arex;Allm′arey′.
6.Allx′arem′;Noy′arem.
7.Allxarem;Ally′arem′.
8.Somem′arex′;Nomarey.
9.Allmarex′;Nomarey.
10.Nomarex′;Noyarem′.
11.Nox′arem′;Nomarey.
12.Somexarem;Ally′arem.
13.Allx′arem;Nomarey.
14.Somexarem′;Allmarey.
15.Nom′arex′;Allyarem.
16.Allxarem′;Noyarem.
17.Somem′arex;Nom′arey′.
18.Allxarem′;Somem′arey′.
19.Allmarex;Somemarey′.
20.Nox′arem;Someyarem.
21.Somex′arem′;Ally′arem.
22.Nomarex;Somemarey.
23.Nom′arex;Allyarem′.
24.Allmarex;Noy′arem′.
25.Somemarex;Noy′arem.
26.Allm′arex′;Some y arem′.
27.Somemarex′;Noy′arem′.
28.Noxarem′;Allmarey′.
29.Nox′arem;Nomarey′.
30.Noxarem;Somey′arem′.
31.Somem′arex;Ally′arem;
32.Allxarem′;Allyarem.
1.Nomarex′;Allm′arey.
2.Nom′arex;Somem′arey′.
3.Allm′arex;Allm′arey′.
4.Nox′arem′;Ally′arem.
5.Somemarex′;Noyarem.
6.Nox′arem;Nomarey.
7.Nomarex′;Somey′arem.