Principles of the Algebra of Logic With Examples

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X=y+w, x>y, and xy=y, express the same truth. To express the truth considered by means of an equation, either three symbols must be used, or else a term of the second order introduced.
250. Inference from two disjunctive equations having a common character.
First; when the character occurs oppositely in the two equations.
x=y+z. (i) x'=i-y+z'. (2) Add; x+x'=i+z+z', that is, x i x' +z+z'.
Second; when the character occurs similarly in the two equations.
x=y+z. (i) x'=y+z f . (2) Subtract ; x x'
...= z z', that is, x=x'+z z r .
1 10 On certain forms of the Observation. When z and z' are each indefinite, the former conclusion amounts to Only 7's which are x are not x'.
The latter conclusion under the same conditions amounts to Some 7's which are x are x. For x>z, by equation (i).
251. If x=x(y+z), then x=x{y(i z)+z(i y)}- } and conversely.
that is, x{i+y-\-z 2y2z -\-2yz} =*>, that is, x{i-y(i-z)-z(i-y)}=o, that is, x=x{y(i z)+z(i y)}.
As this proof can be reversed, the converse proposition must be true.


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