George Boole

“ With these laws there is connected a general axiom. We have seen that algebraic operations performed with elective symbols represent mental processes. Thus the connexion of two symbols by the sign + represents the aggregation of two classes into a single class, the connexion of two symbols as in multiplication, represents the mental operation of selecting from a class those members which belong also to another class , and so on. By such operations the conception of a class is modified. But beside this the mind has the power of perceiving relations of equality among classes. ”

“ We have now arrived at this step,—that we are in possession of a class of symbols , , , &c. satisfying certain laws, and applicable to the rigorous expression of any categorical proposition whatever. It will be our next business to exhibit a few of the general theorems of the calculus which rest upon the basis of those laws, and these theorems we shall afterwards apply to the discussion of particular examples.
Of the general theorems I shall only exhibit two sets: those which relate to the development of functions, and those which relate to the solution of equations. ”

“ It therefore represents a class. We may replace it by an elective symbol , we have then
the interpretation of which is
All not-s are not-s.
This is a known transformation in logic, and is called conversion by contraposition, or negative conversion. But it is far from exhausting the solution we have obtained. Logicians have overlooked the fact, that when we convert the proposition All s are (some) s into All not-s are (some) not-s there is a relation between the two (somes) , understood in the predicates. ”

“ Functions of three symbols,
in which (111) , (110) , &c. are the moduli, and , (1 - ) , &c. the constituents.
From these examples the general law of development is obvious. And I desire it to be noted that this law is a mere consequence of the primary laws which have been expressed in (1) , (2) , (3) .
THEOREM. If we have any equation (..) = 0, and fully expand the first member, then every constituent whose modulus does not vanish may be equated to 0.
This enables us to interpret any equation by a general rule. ”

“ There may be but one individual in a class, or there may be a thousand. There may be individuals common to different classes, or the classes may be mutually exclusive. All elective symbols are distributive, and commutative, and all elective symbols satisfy the law expressed by (3) . ”

“ Page— 119, line— 6, the letter imperfect, like .
Page— 000,„ line— 10, for read .
Page— 120, line— 1, last term, for read .
Page— 128, line— 4 from bottom, for read .
Page— 000,„ line— 6 from bottom, for read .
Page— 000,„ line— 8 from bottom, the letter imperfect, like .
Page— 129, line— 2, for read .
Page— 221, line— 18, for read .
Page— 231, line— 10 from bottom, for read .
Page— 261, line— 2 from bottom, for , read .
Page— 262, line— 22, for and read and .
Page— 270, line— 6 from bottom, for read .
Page— 274, line— 11, for read .
Page— 282, line— 10, dele (1) gives. ”

“ This belongs to Case 2, and satisfies the second condition. The extreme Some s becomes All not-s,
∴ All not-s are s.
The other extreme treated in the same way would give
All not-s are s,
which is an equivalent result.
If we confine ourselves to the Aristotelian premises A, E, I, O, the second condition of inference in Case 2 is not needed. The conclusion will not necessarily be confined to the Aristotelian system. ”