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Notes

MS 152: November–December 1868

Note 1

The proof offered that

is fallacious. For the identities

cannot be obtained by simply substituting 1 and 0 for x in the identity

in as much as i and j have not been proved to be independent of x. Boole’s proof is the same.

It can be proved, however, that i and j are independent of x. For let x be logically multiplied by any factor m₁ and then increased by any term a₁ and we have .

Now when the same pair of processes is performed upon an expression of this form the result is another expression of the same form for

Now to multiply by 1 or increase by 0 is the same as omitting to multiply or add at all; and therefore any series of multiplications and additions may by the intercalation of a multiplication by 1 between every pair of successive additions and of an increase by 0 between every pair of successive multiplications, be reduced to a series of alternate additions and multiplications and, therefore, the result of any operation composed only of additions and multiplications is of the form . Now . Hence when φ involves only a series of additions and multiplications

But such an expression as ax does not come under this formula.

It could easily be shown by (11) and (14) that the difference or quotient of two quantities of the form is also of this form if it is interpretable and hence the result of any series of additions, subtractions, multiplications, and divisions of x and of terms independent of x, is, if interpretable such that

Note 2.

Identity (18′) is reducible to (18) by developement of the second member by (18).

Note 3.

It should be added that

For

where cannot be less than nothing. These limits have been treated both by Boole and by De Morgan.

Note 4.

The mode proposed for the expression of particular propositions is weak. What is really wanted is something much more fundamental. Another idea has since occurred to me which I have never worked out but which I can here briefly explain.

If w denotes wise, and s denotes Solomon, then the expression ws cannot be interpreted by (18) or any principle of Boole’s calculus. It might then be used to denote wiser than Solomon. Thus, relative terms would be brought into the domain of the calculus. But if we are to have symbols for relative terms, we must have a symbol for not or other than. Let this be n. Then, ns will denote not Solomon.

But Solomon is a singular. Let m denote man. Then shall wm denote wiser than every man or wiser than some man? In either case, nwm will be ambiguous. For it may be taken as n(wm) or as (nw)m.

Now, if we simply adopt the formula

which is imitated from algebra, then wm must be wiser than every man, and (nw)m must be other than wiser (no more wise) than any man, or other than wiser-than-some-man. So that wiser than some man will be n^((nw)m). So that both conceptions will be susceptible of expression.

Then, we have the familiar-looking formulæ

But we do not have

That which we have expressed as nm can be expressed as 1 – m or 0/m without any special letter m. Can we not determine the value of n, then? Now n is fully defined by the condition that the logical product of the two expressions x and nx (whatever x may be) is zero; or

That is to say, if either x or nx is zero the other may be of any value. That is, if need not be zero; or n⁰ is not necessarily zero or unity. And, if nx is zero, x may have any value; or, in general, when x is not zero nx may be zero. Algebraical ideas at once, therefore, suggest that ; so that non-man is expressed by 0^m.

0⁰ must be taken as unity.

The difference in meaning between h(km) and (hk)^m is this: if k means king and h hater and m as before man, the former is hater of every king over all men and the latter is the hater of every king of any man.

Hence we have

Some examples of the method in which these exponents may be made use of may now be shown. Let m be man, a animal. Then, every man is an animal; or

or

or

Then,

or

And in the same way, if h denote head,

and then

That is, any man’s head is an animal’s head. This result cannot be reached by any ordinary forms of Logic or by Boole’s Calculus.

The next step requires us to notice an operation to which no specific name nor symbol has been given in algebra. We have the series of quantities

Let the nth of this series be denoted by the symbol k–n. Then, k₀ = 1 = k⁰ k₁ = 0.

Then, recurring to our calculus let k denote one foot longer. Then

km will be one foot longer than m

kkm or will be two feet longer than m

will be n feet longer than m

will be no longer than m

will be one foot shorter than m

Now, let k denote killer. Then,

km is killer of every m

kkm is killer of every killer of every m

is Every m

is killed by every m.

Then, we have, in general,

If, then,

The most curious symbol of this system must now be noticed. k_–1 and k₁ correspond to the active and passive voices respectively. But now it is also necessary to have a species of non-relative terms derived from relatives, which correspond to the middle voice. This may be indicated by a comma as exponent. Thus, if k denotes killer, k’ will be killer of himself. Then we shall have

Let p denote parent, l lover, s Sophroniscus, z Socrates. Then, take the premises

The former gives by being multiplied by s

The second gives

Hence,

This symbol is of the utmost importance.

Let ψ^a denote the case in which a does not exist or in which .

Then hypothetical propositions are expressed thus:—

Particulars thus:—

Let us now take the premises

From the latter

Note 6

An advantageous method of working Boole’s calculus is given by Jevons in his Pure Logic, or the Logic of Quality.

Note 6

I have discussed this question in the last paper contained in this volume.

Note 7.

Aristotle uses this form to prove the validity of the simple conversion of the universal negative. He says: “If No B is A, no A is B. For if not let Γ be the A which is B. Then it is false that no B is A, for Γ is some B.”

Note 8

The Aristotelean method of writing these would be

Note 9

Both these reductions are given by Aristotle. Compare 28b14 and 28b20 with 28a24.

Note 10

I neglected to refer afterwards to the form of the substitution of some-S and not-P for their definitions. But in Part III, §4 I have reduced such arguments to the first figure.

Note 11

Prescision should be spelt with another s as its etymology suggests. But in correcting the proof-sheets Hamilton’s Metaphysics and Chauvin’s Lexicon led me astray.

Note 12

It may be doubted whether it was philosophical to rest this matter on empirical psychology. The question is extremely difficult.

Note 13

Theorem VII is the associative principle, the difficulty of the demonstration of which is recognized in quaternions. I give here the demonstration in the text in syllogistic form.

To avoid a very fatiguing prolixity, I employ some abbreviated expressions. If anyone opines that these render the proof nugatory, I shall be happy to go through with him the fullest demonstration. (1). Let (Am,Bn) be any product of the series described in def. 6. By the principle of Contradiction this includes non-identical classes under it or it does not. If it does, let x denote one of these classes not identical with some other (Am,Bn).

(2) Then, by cond. 3 of def. 6, x is of the form

By Barbara.

(3) The term Am occurs in this sum or it does not.

(4) If it does the B by which it is multiplied is either Bn or it is not.

(5) If it is, by def. 2, and Barbara,

which is contradictory of the supposition in (1) that any x is (Am,Bn)

(6) If this B is not Bn, denote it by Bq. Then by def. 2, and Barbara

(7) But No Bq is Bn, by definition,

(8) And, by def. 3 and Barbara,

(9) Hence, from (7) and (8), by Celarent

(10) Hence, from (6) and (10), and Felapton

(11) But, by def. 3, and Barbara,

(12) Hence, from (10) and (11), by Baroko

which is contrary to the definition of x.

(13) This reduces the supposition that Am occurs in the Σ(A,B) defined in (2) to an absurdity. So that Am does not occur in this sum.

(14) Hence by (2) and Defs. 2 and 3

(15) Hence, by Def. 7, cond. 1,

(16) But by (1)

(17) Whence, by def. 3,

(18) And from (15) and (17)

Which is absurd. Hence

(19) The supposition that Am,Bn includes mutually coexclusive classes is absurd. Or, every thing of the form (Am,Bn) is an individual, as stated on p. 64.

(20) Every individual X, that is, whatever does not include under it mutually exclusive classes, by condition 3 of def. 7 can be expressed in the form

(21) Hence, if (Am,Bn) be a term of Σ(A,B) by def. 2

(22) Hence, by definition of an individual

Or any individual can be expressed in the form

(23) Let then any individual a be

Then by this definition

(24) Hence by def. 3

(25) Hence, by cond. 1 of def. 7,

(26) Hence, by cond. 4 of def. 7

Any Aa is a.

(27) In the same way if (Am,Bb) is any individual b

Any Bb is b.

(28) Then (Aa,Bb) belongs to the A which the individual a belongs to. For, by def. 3

Any (Aa,Bb) is Aa

And by (23) the individual a is Aa,Bn. Whence by def. 3, the individual a is Aa. Whence by cond. 1, of def. 7, the individual a belongs to no other A than Aa.

(29) Similarly (Aa,Bb) belongs to the B to which the individual b belongs.

(30) Moreover by def. 3 and (26) (Aa,Bb) is a.

(31) Similarly it is, also, b.

(32) Moreover by (19) it is an individual.

(33) And by cond. 6 of def. 7 it exists.

(34) In the same way, it could be shown that (Am,Bn) is an existent individual which belongs to the same A which the individual b belongs to, and the same B which the individual a belongs to.

Thus, for any individuals one a and the other b, there exists an individual which belongs to the same A as a and the same B as b, and an individual which belongs to the same A as b and the same B as a, provided a and b are independent.

This is the first part of our lemma.