second contrapositive; for we have
and substituting for B its equivalent A we have
Hence from the single identity A = B we can draw the two propositions
and observing that these propositions have a common term ab we can make a new substitution, getting
This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it is not a self-evident result, independent of the steps of deduction by which we have reached it. For where two classes are coincident like A and B, whatever is true of the one is true of the other; what is excluded from the one must be excluded from the other similarly. Now as a bears to A exactly the same relation that b bears to B, the identity of either pair follows from the identity of the other pair. In every identity, equality, or similarity, we may argue from the negative of the one side to the negative of the other. Thus at ordinary temperatures
hence obviously
or since
it follows that whatever star is not the brightest is not Sirius, and vice versâ. Every correct definition is of the form A = B, and may often require to be applied in the equivalent negative form.
Let us take as an