Symbolic Logic

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, in Book V., Chap. II.

(1) [see p. 64]

“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.”

pg079(2) [see p. 64]

“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.”

(3) [see p. 64]

“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.”

(4) [see p. 66]

“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.

(5) [see p. 67]

“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, is incomplete, the complete one being “I wish to examine all these pictures thoroughly.”

pg080(6) [see p. 67]

“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.

(7) [see p. 69]

”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′₀ do not come under any of the three Figures. Hence it is necessary to return to the Method of Diagrams, as shown at p. 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 read Note (A), at p. 164.]

pg081§ 3.

Fallacies.

Any argument which deceives us, by seeming to prove what it does not really prove, may be called a ‘Fallacy’ (derived from the Latin verb fallo “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.

Our best plan is, I think, to deal with Fallacies in the same was as we have already dealt with Syllogisms: that is, to take certain forms of Pairs of Propositions, and to work pg082 them out, once for all, on the Triliteral Diagram, and ascertain that they yield no Conclusion; and then to record them, for future use, as Formulæ for Fallacies, just as we have already recorded our three Formulæ for Syllogisms.

Now, if we were to record the two Sets of Formulæ in the same shape, 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 for Fallacies in words, 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.

It is evident that neither of the given Propositions can be an Entity, since that kind asserts the existence of both of its Terms (see p. 20). Hence they must both be Nullities.

Hence the given Pair may be represented by (xm₀ † ym₀), with or without x₁, y₁.

These, set out on Triliteral Diagrams, are

pg083(2) Fallacy of Unlike Eliminands with an Entity-Premiss.

(3) Fallacy of two Entity-Premisses.

pg084§ 4.

Method of proceeding with a given Pair of Propositions.

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:—

[Work Examples § 4, 1–11 (p. 100); § 5, 1–12 (p. 101); § 6, 7–12 (p. 106); § 7, 7–12 (p. 108).]

pg085BOOK VII.

SORITESES.

CHAPTER I.

INTRODUCTORY.

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’.

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 not consequent from the Premisses, is not affected by the actual truth or falsity of any one of the Propositions which make up the Sorites, by depends entirely on their relationship to one another.

pg086As a specimen-Sorites, let us take the following Set of 5 Propositions:—

(1) ”No a are b′;
(2)   All b are c;
(3)   All c are d;
(4)   No e′ are a′;
(5)   All h are e′”.

Here the first and second, taken together, yield “No a are c′”.

This, taken along with the third, yields “No a are d′”.

This, taken along with the fourth, yields “No d′ are e′”.

And this, taken along with the fifth, yields “All h are d”.

Hence, if the original Set were true, this would also be true.

Hence the original Set, with this tacked on, is a Sorites; the original Set is its Premisses; the Proposition “All h are d” is its Conclusion; the Terms a, b, c, e are its Eliminands; and the Terms d and h are its Retinends.

Hence we may write the whole Sorites thus:—

”No a are b′;
  All b are c;
  All c are d;
  No e′ are a′;
  All h are e′.
          ∴ All h are d”.

In the above Sorites, the 3 Partial Conclusions are the Positions “No a are e′”, “No a are d′”, “No d′ are e′”; but, if the Premisses were arranged in other ways, other Partial Conclusions might be obtained. Thus, the order 41523 yields the Partial Conclusions “No c′ are b′”, “All h are b”, “All h are c”. There are altogether nine Partial Conclusions to this Sorites, which the Reader will find it an interesting task to make out for himself.]

pg087CHAPTER II.

PROBLEMS IN SORITESES.

§ 1.

Introductory.

The Problems we shall have to solve are of the following form:—

We will limit ourselves, at present, to Problems which can be worked by the Formulæ of Fig. I. (See p. 75.) Those, that require other Formulæ, are rather too hard for beginners.

Such Problems may be solved by either of two Methods, viz.

pg088§ 2.

Solution by Method of Separate Syllogisms.

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

[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 into subscript form. Let us begin by putting them into abstract form. The result is

  (1) ”All k are l;
  (2)   No d are h′;
  (3)   All a are c′;
  (4)   All b are e;
  (5)   No k′ are h;
  (6)   No b′ are l;
  (7)   All d′ are c.”

And it is now easy to put them into subscript form, 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 take k as our Eliminand. So our first syllogism is

  (1) k₁l′₀
  (5) k′h₀
∴ l′h₀ … (8)

We must now begin again with l′h₀ and find a Premiss to go along with it. We find that No. (2) will do, h being 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 for l′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 with db′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 into abstract form, is

“All a are e”;

and this, translated into concrete form, is

“Amos Judd loves cold mutton.”

In actually working this 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 with any Premiss we choose.]

pg091§ 3.

Solution by Method of Underscoring.

Let us agree to mark the eliminated letters by underscoring them, putting a single score under the first, and a double one under the second.

In copying out the Premisses for underscoring, it will be convenient to omit all subscripts. As to the “0’s” we may always suppose them written, and, as to the “1’s”, we are not concerned to know which Terms are asserted to exist, except those which appear in the Complete Conclusion; and for them it 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 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 either k′ or l. The first we find is No. 5; and this we tack on, with a †.

To get the Conclusion from these, k and k′ must be eliminated, and what remains must be taken as one expression. So we underscore them, putting a single score under k, and a double one under k′. The result we read as l′h.

We must now find a Premiss containing either l or h′. 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 which h and h′ must be eliminated, and what remains taken as one expression. So we underscore them. The result reads as l′d.

We now want a Premiss containing l or d′. No. 6 will do.

These 4 Nullities are really equivalent to (l′d † b′l). So we underscore l′ and l. The result reads as db′.

We now want a Premiss containing d′ or b. No. 4 will do.

Here we underscore b′ and b. The result reads as de′.

We now want a Premiss containing d′ or e. No. 7 will do.

Here we underscore d and d′. The result reads as c′e′.

We now want a Premiss containing c or e. No. 3 will do—in fact must do, as it is the only one left.

Here we underscore c′ and c; and, as the whole thing now reads as e′a, we tack on e′a₀ as the Conclusion, with a ¶.

We now look along the row of Data, to see whether e′ or a has been given as existent. We find that a has 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. “All a are e.”

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 † a₁     i.e. ¶ a₁e′₀;

i.e. “All a are e.”

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 Conclusion a₁e′₀, I would advise him to take a third piece of paper, and begin 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.

[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 than one Paper on any one day. The answers to the questions about words and phrases may be found by referring to the Index at p. 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 does it sometimes require?

pg095   VI. § 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 ‘in I’? ‘In E’? And ‘in A’?

   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?

pg096  XV. § 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?

pg097BOOK VIII.

EXAMPLES, ANSWERS, AND SOLUTIONS.

CHAPTER I.

EXAMPLES.

EX1§ 1.

Propositions of Relation, to be reduced to normal form.

pg098
EX2§ 2.

Pairs of Abstract Propositions, one in terms of x and m, and the other in terms of y and m, to be represented on the same Triliteral Diagram.

pg099
EX3§ 3.

Marked Triliteral Diagrams, to be interpreted in terms of x and y.

pg100
EX4§ 4.

Pairs of Abstract Propositions, proposed as Premisses: Conclusions to be found.