Symbolic Logic

pg050CHAPTER III.

REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.

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 a Red Counter (this may be interpreted to mean “There is at least one Thing here”), and a “O” to represent a Grey Counter (this may be interpreted to mean “There is nothing here”).

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

[Let us work a few examples.

(1)

“No x are m′;
  No y′ are m”.

Let us first represent “No x are m′”. This gives us Diagram a.

Then, representing “No y′ are m” on the same Diagram, we get Diagram b.

pg051a     b

(2)

“Some m are x;
  No m are y”.

If, neglecting the Rule, we were begin with “Some m are x”, we should get Diagram a.

And if we were then to take “No m are y”, which tells us that the Inner N.W. Cell is empty, we should be obliged to take the “I” off the fence (as it no longer has the choice of two Cells), and to put it into the Inner N.E. Cell, as in Diagram c.

This trouble may be saved by beginning with “No m are y”, as in Diagram b.

And now, when we take “Some m are x”, there is no fence to sit on! The “I” has to go, at once, into the N.E. Cell, as in Diagram c.

a     b     c

(3)

“No x′ are m′;
  All m are y”.

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

(1) “No x′ are m′;
(2)   Some m are y;
(3)   No m are y′”.

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

First we take No. (1), viz. “No x′ are m′”. This gives us Diagram a.

pg052Adding to this, No. (3), viz. “No m are y′”, we get Diagram b.

This time the “I”, representing No. (2), viz. “Some m are y,” has to sit on the fence, as there is no “O” to order it off! This gives us Diagram c.

a     b     c

(4)

“All m are x;
  All y are m”.

Here we break up both Propositions, and thus get four to represent, viz.—

(1) “Some m are x;
(2)   No m are x′;
(3)   Some y are m;
(4)   No y are m′”.

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

First we take No. (2), viz. “No m are x′”. This gives us Diagram a.

To this we add No. (4), viz. “No y are m′”, and thus get Diagram b.

If we were to add to this No. (1), viz. “Some m are x”, we should have to put the “I” on a fence: so let us try No. (3) instead, viz. “Some y are m”. This gives us Diagram c.

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 Diagram already tells us that “Some m are x”.]

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

pg053CHAPTER IV.

INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS OR DIGITS.

To transfer the information, observe the following Rules:—

(1) Examine the N.W. Quarter of the Triliteral Diagram.
(2) If it contains a “I”, in either Cell, it is certainly occupied, and you may mark the N.W. Quarter of the Biliteral Diagram with a “I”.
(3) If it contains two “O”s, one in each Cell, it is certainly empty, 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)

In the N.W. Quarter, only one of the two Cells is marked as empty: so we do not know whether the N.W. Quarter of the Biliteral Diagram is occupied or empty: so we cannot mark it.

In the N.E. Quarter, we find two “O”s: so this Quarter is certainly empty; and we mark it so on the Biliteral Diagram.

In the S.W. Quarter, we have no information at all.

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

We may read off the result as “No x are y′”, or “No y′ are x,” whichever we prefer.

(2)

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

In the N.E. Quarter, we find a “I”. This shows us that it is occupied: 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 “Some x are y′”, or “Some y′ are x”, whichever we prefer.

pg055(3)

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

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 certainly empty: so we mark it as such on the Biliteral Diagram.

We may read off the results as “No x′ are y′”, or “No y′ are x′”, whichever we prefer.

(4)

The N.W. Quarter is occupied, 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.

The S.W. Quarter is certainly empty. 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 “All y are x.”]

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

pg056BOOK V.

SYLLOGISMS.

CHAPTER I.

INTRODUCTORY

When a Trio of Biliteral Propositions of Relation is such that

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 the Symbol “∴”).

pg057[Note that the ‘Eliminands’ are so called because they are eliminated, and do not appear in the Conclusion; and that the ‘Retinends’ are so called because they are retained, and do appear in the Conclusion.

Note also 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 of the Trio, but depends entirely on their relationship to each other.

As a specimen-Syllogism, let us take the Trio

“No x-Things are m-Things;
  No y-Things are m′-Things.
          No x-Things are y-Things.”

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

“No x are m;
  No y are m′.
          No x are y”.

Here the first and second contain the Pair of codivisional Classes m and m′; the first and third contain the Pair x and x; and the second and third contain the Pair y and y.

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 a Syllogism; the two Propositions, “No x are m” and “No y are m′”, are its Premisses; the Proposition “No x are y” is its Conclusion; the Terms m and m′ are its Eliminands; and the Terms x and y are its Retinends.

Hence we may write it thus:—

“No x are m;
  No y are m′.
  ∴ No x are y”.

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 at p. 64) so related that, if the first two were true, the third would be true. (The first two are, as it happens, not strictly true in our planet. But there is nothing to hinder them from being true in some other planet, say Mars or Jupiter—in which case the third would also be true in that planet, and its inhabitants would probably engage chickens as nursery-governesses. They would thus secure a singular contingent privilege, 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 a Syllogism; the Genus “creatures” is its ‘Univ.’; the two Propositions, “All cats understand French“ and ”Some chickens are cats”, are its Premisses, the Proposition “Some chickens understand French” is its Conclusion; the Terms “cats” and “cats” are its Eliminands; and the Terms, “creatures understanding French” and “chickens”, are its Retinends.

Hence we may write it thus:—

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

pg059CHAPTER II.

PROBLEMS IN SYLLOGISMS.

§ 1.

Introductory.

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

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

[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” as Species of the Genus “men”; and we may choose x to represent the peculiar Attribute (say “military”) of “soldiers,” and y to represent “brave.” Then the Proposition may be written “Some military men are brave men”; i.e. “Some x-men are y-men”; i.e. (omitting “men,” as explained at p. 26) “Some x are y.”

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

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

pg060§ 2.

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.

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

[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 now construct our 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:—

“No x are m′;
  All m are y”.

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

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

The result we read as “No x are y′” or as “No y′ are x,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose

“No x are y′”,

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

“All m are x;
  Some y are m”.

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 the three Propositions

(1) “Some m are x;
(2)   No m are x′;
(3)   Some y are m”.

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

This, however, would produce the result

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 “Some m are x” is already represented on the Diagram.

Transferring our information to a Biliteral Diagram, we get

This result we can read either as “Some x are y” or “Some y are x”.

After consulting our Dictionary, we choose

“Some y are x”,

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

“All x are m;
  All y are m′”.

These, broken up, give us the four Propositions

(1) “Some x are m;
(2)   No x are m′;
(3)   Some y are m′;
(4)   No y are m”.

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

Representing these on a Triliteral Diagram, we get

And this information, transferred to a Biliteral Diagram, is

Here we get two Conclusions, viz.

“All x are y′;
  All y are x′.”

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”. (See p. 4.)

(4)

“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”.

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

“All x are m;
  Some m are y”.

Breaking up the first, we get the three

(1) “Some x are m;
(2)   No x are m′;
(3)   Some m are y”.

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 seen only the Premisses, that the Conclusion would be

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

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

“Not true!” you exclaim. “Then who were they, who were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict ‘guilty’ returned against them, or how could they be sentenced?”

Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded ‘guilty’. So no verdict was 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.

pg064(1) [see p. 60]

“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.”

(2) [see p. 61]

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

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

i.e. “Some chickens understand French.”

(3) [see p. 62]

“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”.

pg065(4) [see p. 63]

“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).]

pg066§ 3.

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 is complete.

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

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

(2) If there be no Conclusion, say so.

(3) If there be a 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.

pg068(4)

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

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

(5)

“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 is incomplete, the complete one containing, in addition, “All selfish men are disobliging”.

(6)

”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 is another opportunity, 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 it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premisses to be true, I should be quite clear that I needn’t run—and I should walk!”

And you will 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 convenient test of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the Premisses, would make the Conclusion false, the Syllogism must be 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).]

pg070BOOK VI.

THE METHOD OF SUBSCRIPTS.

CHAPTER I.

INTRODUCTORY.

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

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

Also that “†” shall mean “and”.

When two Letters are both of them accented, or both not accented, 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’.

pg071CHAPTER II.

REPRESENTATION OF PROPOSITIONS OF RELATION.

Let us take, first, the Proposition “Some x are y”.

This, we know, is equivalent to the Proposition of Existence “Some xy exist”. (See p. 31.) Hence it may be represented by the expression “xy₁”.

The Converse Proposition “Some y are x” may of course be represented by the same expression, viz. “xy₁”.

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

Let us take, next, the Proposition “No x are y”.

This, we know, is equivalent to the Proposition of Existence “No xy exist”. (See p. 33.) Hence it may be represented by the expression “xy₀”.

The Converse Proposition “No y are x” may of course be represented by the same expression, viz. “xy₀”.

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

pg072Let us take, next, the Proposition “All x are y”.

Now it is evident that the Double Proposition of Existence “Some x exist and no xy′ exist” tells us that some x-Things exist, but that none of them have the Attribute y′: that is, it tells us that all of them have the Attribute y: that is, it tells us that “All x are y”.

[The Reader will perhaps be puzzled by the statement that the Proposition “All x are y” is equivalent to the Double Proposition “Some x exist and no xy′ exist,” remembering that it was stated, at p. 33, to be equivalent to the Double Proposition “Some x are y and no x are y′” (i.e. “Some xy exist and no xy′ exist”). The explanation is that the Proposition “Some xy exist” contains superfluous information. “Some x exist” is enough for our purpose.]

This expression may be written in a shorter form, viz. “x₁y′₀”, since each Subscript takes effect back to the beginning of the expression.

Similarly we may represent the seven similar Propositions “All x are y′”, “All x′ are y”, “All x′ are y′”, “All y are x”, “All y are x′”, “All y′ are x”, and “All y′ are x′”.

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

pg073CHAPTER III.

SYLLOGISMS.

§ 1.

Representation of Syllogisms.

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.

pg074§ 2.

Formulæ for solving Problems in Syllogisms.

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”.

pg075Fig. I.

Hence we get two Variants of Fig. I, viz.

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.]

Fig. II.

pg077Fig. III.