important facts in astronomy, and the laws of light and vision, branches wonderfully calculated to arrest and occupy the attention, in certain stages of mental development, especially when faith in the universality of God's providence goes with them.

The respect and companionship, in a degree I never could have anticipated, of those who dwell by the fountains of knowledge, have cheered me; men whose virtues and accomplishments I can never think to emulate, and whose envy, if they had any, could not descend upon me.

Mr. President, I have said nothing of methods. When first questioned by the Rumford Committee as to what was original in my mode of working, I proposed that they should visit my shop, where I could show and explain to them the course pursued, and they might judge for themselves of the value or originality of any part of it. They have done so; and I accept their decision, trusting the egotism displayed in this little history of achievements may be excused.

You will not fail to perceive, that, after so many years of hopeful, cheerful, and patient toil, mingled with no ordinary share of painstaking, a high appreciation on the part of the recipient must follow this award, as naturally as light comes with the rising of the sun.

Five hundred and eightieth Meeting.

March 12, 1867.- ADJOURNED STATUTE MEETING.

The PRESIDENT in the chair.

The President called the attention of the Academy to the necessity of some action for the accommodation of the Academy, and proposed that a committee be appointed to consider this subject.

On the motion of Professor Rogers, it was voted, that a committee of three be appointed, and on the nomination of the President, Colonel Lyman, Mr. C. M. Warren, and Professor C. W. Eliot were chosen.

On the motion of Dr. Clark, this committee were instructed to inquire concerning the co-operation of other societies in providing a building for their common accommodation. The following paper was presented.

On an Improvement in Boole's Calculus of Logic. By C. S. PEIRCE.

THE principal use of Boole's Calculus of Logic lies in its application to problems concerning probability. It consists, essentially, in a system of signs to denote the logical relations of classes. The data of any problem may be expressed by means of these signs, if the letters of the alphabet are allowed to stand for the classes themselves. From such expressions, by means of certain rules for transformation, expressions can be obtained for the classes (of events or things) whose frequency is sought in terms of those whose frequency is known. Lastly, if certain relations are known between the logical relations and arithmetical operations, these expressions for events can be converted into expressions for their probability.

It is proposed, first, to exhibit Boole's system in a modified form, and second, to examine the difference between this form and that given by Boole himself.

66

Let the letters of the alphabet denote classes whether of things or of occurrences. It is obvious that an event may either be singular, as this sunrise," or general, as "all sunrises." Let the sign of equality with a comma beneath it express numerical identity. Thus a = b is to mean that a and b denote the same class, the same collection of individuals.

Let a + b denote all the individuals contained under a and b together. The operation here performed will differ from arithmetical addition in two respects: 1st, that it has reference to identity, not to equality; and 2d, that what is common to a and b is not taken into account twice over, as it would be in arithmetic. The first of these differences, however, amounts to nothing, inasmuch as the sign of identity would indicate the distinction in which it is founded; and therefore we may say that

and also, that the process denoted by +, and which I shall call the process of logical addition, is both commutative and associative. That is to say

and

(4.)

(a + b) + c = a + (b + c).

Let a, b denote the individuals contained at once under the classes a and b; those of which a and b are the common species. If a and b were independent events, a, b would denote the event whose probability is the product of the probabilities of each. On the strength of this analogy, (to speak of no other,) the operation indicated by the comma may be called logical multiplication. It is plain that

Logical multiplication is evidently a commutative and associative That is,

process.

Logical addition and logical multiplication are doubly distributive, so that

(8.) and

(9.)

Proof.

(a + b), ca, c + b,c

a, b + c = (a + c), (b + c).

Let aa' + x + y + o

b=b+x+x+o

c = c + y +2 +0.

where any of these letters may vanish. These formulæ comprehend every possible relation of a, b and c; and it follows from them, that

a+b= a'+b'+x+y+z+o

But

a,c=y+o

So

But

b,c=z+○

(a+b),c=y+z+0.

a, b = x+o a, b + c = c + x + y +z+ 0.

(a+c)=a'+c'+x+y+z+o

(b+c)=b'+c'+x+y+z+o

(a + c), (b + c) = c' + x + y + z + o

.. (9).

Let T

(10.)

be the sign of logical subtraction; so defined that
x = αT b.

If b + x = a

Here it will be observed that x is not completely determinate. It may vary from a to a with b taken away. This minimum may be denoted by ab. It is also to be observed that if the sphere of b reaches at all beyond a, the expression ab is uninterpretable. If then we denote the contradictory negative of a class by the letter which denotes the class itself, with a line above it,* if we denote by v a wholly indeterminate class, and if we allow [01] to be a wholly uninterpretable symbol, we have

If we define zero by the following identities, in which x may be any class whatever,

then, zero denotes the class which does not go beyond any class, that is nothing or nonentity.

Let a; b be read a logically divided by b, and be defined by the condition that

(13.)

If b, x = a

x = a; b

x is not fully determined by this condition. It will vary from a to a + and will be uninterpretable if a is not wholly contained under b. Hence, allowing [1; 0] to be some uninterpretable symbol,

(14.)

a; b = a, b + v, ā, b + [1; 0] a,ō

which is uninterpretable unless

a,b=0.

Unity may be defined by the following identities in which x may be any class whatever.

Then unity denotes the class of which any class is a part; that is, what is or ens.

* So that, for example, a denotes not-a.

It is plain that if for the moment we allow a:b to denote the maximum value of a; b, then

The rules for the transformation of expressions involving logical subtraction and division would be very complicated. The following method is, therefore, resorted to.

It is plain that any operations consisting solely of logical addition and multiplication, being performed upon interpretable symbols, can result in nothing uninterpretable. Hence, if + × x signifies such an operation performed upon symbols of which x is one, we have

Φ

9 + x x =α, x + b, (1 − x)

where a and b are interpretable.

It is plain, also, that all four operations being performed in any way upon any symbols, will, in general, give a result of which one term is interpretable and another not; although either of these terms may disappear. We have then

We have seen that if either of these coefficients and j is uninterpretable, the other factor of the same term is equal to nothing, or else the whole expression is uninterpretable. But

Hence (18.)

❤ (1) i and p (0) — j.

❤x = ∞ (1), x + ø (0), (1 − x)

❤ (x and y) = (1 and 1), x, y +❤ (1 and 0), x,ÿ + (0 and 1), x, y

(18'.)

❤x = (ø (1) + x), (❤ (0) + x)

(x and y) = ( (1 and 1) + x + y), (p (1 and 0) + x + y), (9 (0 and 1) + x + y), (9 (0 and 0) + x + y).

Developing by (18) xy, we have,

x − y = (1 − 1), x, y+(1, 0), x‚ÿ+(0 − 1)‚ñ‚y+(070)‚ñ‚ÿ. So that, by (11),

(19.) (1, 1) — v 10—1 01 [01] 0 0 0.