*22. CALCULUS OF CLASSES.
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Whitehead, Alfred North and Russell, Bertrand
The analogue of *4·78, i.e. is false. We have only
A similar remark applies to the analogue of *4·79. Cf. *22·64·65.
*22·6.
Dem.
*22·61.
*22·62.
Dem.
*22·621.
The proof proceeds as in *22·62. The proposition *22·621 is one of the most useful propositions in the present number.
*22·63.
The process of obtaining *22·63 from *4·44 is of the same kind as the process employed in the proofs that have been written out in this number. Hence only *4·44 is referred to. We shall similarly restrict references for later propositions in this number. The process is always roughly as follows: , , are replaced by , , ; then *10·11 is applied, and such further propositions of *10 as may be required, together with *22·33·34·35.
*22·631.
*22·632.
*22·633.
*22·64.
Dem.
The converse of this proposition does not hold, because the converse of *10·41 does not hold.
*22·65.
Here again the converse is untrue.
*22·66.
*22·68.
Dem.
*22·69.
The above propositions *22·68·69 are the two forms of the distributive law. Note that either results from the other by interchanging the signs of addition and multiplication.
*22·7.
*22·71.
*22·72.
*22·73.
*22·74.
Dem.
*22·8.
*22·81.
*22·811.
*22·82.
*22·83.
*22·831.
*22·84.
*22·85.
*22·86.
*22·87.