*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*3·2.
*3·21.
*3·22.
This is one form of the commutative law for logical multiplication. A more complete form is given in *4·3.
Dem.
Note that, in the above proof, "(1)" stands for the proposition as was explained in the proof of *2·31.
*3·24.
Dem.
The above is the law of contradiction.
*3·26.
Dem.
*3·27.
Dem.
*3·26·27 will both be called the "principle of simplification," like *2·02, from which they are deduced. They will be referred to as "Simp."
*3·3.
Dem.
*3·31.
Dem.
*3·33.
*3·34.
These two propositions will hereafter be referred to as "Syll"; they are usually more convenient than either *2·05 or *2·06.
*3·35.
*3·37.
Dem.
This is another form of transposition.
*3·4.
*3·41.
*3·42.
*3·43.
Dem.
*3·44.
This principle is analogous to *3·43. The analogy between *3·43 and *3·44 is of a sort which generally subsists between formulae concerning products and formulae concerning sums.
Dem.
*3·45.
This principle shows that we may multiply both sides of an implication by a common factor; hence it is called by Peano the "principle of the factor." We shall refer to it as "Fact." It is the analogue, for multiplication, of the primitive proposition *1·6.
Dem.
*3·47.
This proposition, or rather its analogue for classes, was proved by Leibniz, and evidently pleased him, since he calls it "præclarum theorema[48]."
Dem.
*3·48.