*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *3.
The logical product of two propositions and is practically the proposition " and are both true." But this as it stands would have to be a new primitive idea. We therefore take as the logical product the proposition , i.e. "it is false that either is false or is false," which is obviously true when and only when and are both true. Thus we put
*3·01.
where "" is the logical product of and .
*3·02.
This definition serves merely to abbreviate proofs.
When we are given two asserted propositional functions "" and "," we shall have "" whenever and take arguments of the same type. This will be proved for any functions in *9; for the present, we are confined to elementary propositional functions of elementary propositions. In this case, the result is proved as follows:
By *1·7, and are elementary propositional functions, and therefore, by *1·72, is an elementary propositional function. Hence by *2·11,
Hence by *2·32 and *1·01, i.e. by *3·01,
Hence by *1·11, when we have "" and "" we have "." This proposition is *3·03. It is to be understood, like *1·72, as applying also to functions of two or more variables.
The above is the practically most useful form of the axiom of identification of real variables (cf. *1·72). In practice, when the restriction to elementary propositions and propositional functions has been removed, a convenient means by which two functions can often be recognized as taking arguments of the same type is the following:
If contains, in any way, a constituent and contains, in any way, a constituent , then both and take arguments of the type of the argument in , and therefore both and take arguments of the same type. Hence, in such a case, if both and can be asserted, so can .
As an example of the use of this proposition, take the proof of *3·47. We there prove and what we wish to prove is which is *3·47. Now in (1) and (2), , , , are elementary propositions (as everywhere in Section A); hence by *1·7·71, applied repeatedly, "" and "" are elementary propositional functions. Hence by *3·03, we have whence the result follows by *3·43 and *3·33.
The principal propositions of the present number are the following:
*3·2
I.e. " implies that implies ," i.e. if each of two propositions is true, so is their logical product.
*3·26
*3·27
I.e. if the logical product of two propositions is true, then each of the two propositions severally is true.
*3·3
I.e. if and jointly imply , then implies that implies . This principle (following Peano) will be called "exportation," because is "exported" from the hypothesis. It will be referred to as "Exp."
*3·31
This is the correlative of the above, and will be called (following Peano) "importation" (referred to as "Imp").
*3·35
I.e. "if is true, and follows from it, then is true." This will be called the "principle of assertion" (referred to as "Ass"). It differs from *1·1 by the fact that it does not apply only when really is true, but requires merely the hypothesis that p is true.
*3·43
I.e. if a proposition implies each of two propositions, then it implies their logical product. This is called by Peano the "principle of composition." It will be referred to as "Comp."
*3·45.
I.e. both sides of an implication may be multiplied by a common factor. This is called by Peano the "principle of the factor." It will be referred to as "Fact."
*3·47.
I.e. if implies and implies , then and jointly imply and jointly. The law of contradiction, ")," is proved in this number (*3·24); but in spite of its fame we have found few occasions for its use.
*3·01.
*3·02.
*3·03. Given two asserted elementary propositional functions "" and "" whose arguments are elementary propositions, we have .
Dem.
*3·1.
*3·11.
*3·12.
*3·13.
*3·14.