Volume 1 / Part 1 — *22. CALCULUS OF CLASSES.
Reading edition
*22. CALCULUS OF CLASSES.
Summary of *22.
In this number we reach what was historically the starting-point of symbolic logic. The Greek letters used (except , , , ) are always to stand for expressions of the form , or, where the Greek letters are not apparent variables, . The small Latin letters may either be such as have a meaning in isolation, or may represent classes or relations; this is possible in virtue of the notes at the ends of *20 and *21. We put:
*22·01.
This defines "the class is contained in the class ," or "all 's are 's."
*22·02.
This defines the logical product or common part of two classes and .
*22·03.
This defines the logical sum of two classes; it is the class consisting of all the members of one together with all the members of the other.
*22·04.
This defines the negation of a class. It is read "not-." It does not contain every object concerning which "" is not true, but only those objects concerning which "" is false; i.e. it excludes those objects for which "" is meaningless. Thus it consists of all objects, of the type next below , which are not members of ; but it does not contain objects of any other type but this.
*22·05.
This definition gives an abbreviation which is often convenient.
The postulates required for the algebra of logic have been enumerated by Huntington[54]. In our notation, they are as follows.
We assume a class , with two rules of combination, namely and ; and we then require the following ten postulates:
I a. is in the class whenever and are in the class.
I b. is in the class whenever and are in the class.
II a. There is an element such that for every element .
II b. There is an element such that for every element .
III a. whenever , , and are in the class.
III b. whenever , , and are in the class.
IV a. whenever , , , , , , , and are in the class.
IV b. whenever , , , , , , , and ( are in the class.
V. If the elements and in postulates II a and II b exist and are unique, then for every element there is an element such that and .
VI. There are at least two elements, and , in the class, such that .
The form of the above postulates is such that they are mutually independent, i.e. any nine of them are satisfied by interpretations of the symbols which do not satisfy the remaining one.
For our purposes, "" must be replaced by "." and will be the null-class and the universal class, which are defined in *24. Then the above ten postulates are proved below, as follows:
I a. in *22·37, namely ""
I b. in *22·36, namely ""
II a. in *24·24, namely ""
II b. in *24·26, namely ""
III a. in *22·57, namely ""
III b. in *22·51, namely ""
IV a. in *22·69, namely ""
IV b. in *22·68, namely ""
V. in *24·21·22, namely "" and ""
VI. in *24·1, namely ""
Hence, assuming Huntington's analysis of the postulates for the formal algebra of logic, the propositions proved in what follows suffice to establish that this algebra holds for classes. The corresponding propositions of *23 and *25 prove that it holds for relations, substituting , , , , for , , , , .
The principal propositions of the present number are the following:
(1) Those embodying the formal rules:
*22·51.
*22·57.
These embody the commutative law.
*22·52.
*22·7.
These embody the associative law.
*22·5.
*22·56.
These embody the law of tautology.
*22·68.
*22·69.
These embody the distributive law. It will be seen that the second results from the first by everywhere interchanging the signs of addition and multiplication.
*22·8.
This is the principle of double negation.
*22·81.
This is the principle of transposition.
(2) Other useful propositions:
*22·44.
*22·441.
These embody the two forms of the syllogism in Barbara.
*22·62.
*22·621.
These two propositions enable us to transform any inclusion ( into an equation.
*22·91.
I.e. " or " is identical with " or the part of which is excluded from ."
*22·01.
*22·02.
*22·03.
*22·04.
*22·05.
*22·1.
*22·2.
*22·3.
*22·31.
*22·32.
*22·33.
*22·34.
*22·35.
*22·351.
Dem.
This proposition is used in proving that the null-class is not identical with the class containing everything (*24·1), which is used to show that at least two classes exist. Our axioms do not suffice to prove that more than one individual exists, but they prove the existence of at least two classes and at least two relations.
*22·36.
*22·37.
*22·38.
*22·39.
Dem.
*22·391.
*22·392.
*22·4.
Dem.
*22·41.
*22·42.
*22·43.
*22·44.
This is one form of the syllogism in Barbara. Another form is the following:
*22·441.
*22·45.
Dem.
*22·46.
*22·47.
*22·48.
*22·481.
Dem.
*22·49.
*22·5.
Dem.
The above is the law of tautology for the logical multiplication of classes.
*22·51.
*22·52.
Thus logical multiplication of classes obeys the commutative and associative laws. References to *22·33·34·35 and to *20·43 will in future often be omitted.
*22·53.
This definition serves merely for the avoidance of brackets.
*22·54.
*22·55.
*22·551.
*22·56.
The above is the law of tautology for the logical addition of classes.
*22·57.
*22·58.
*22·59.
Dem.
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.
*22·84·85·86·87 are De Morgan's formulae.
*22·88.
This is a form of the law of excluded middle.
*22·89.
This is a form of the law of contradiction.
*22·9.
*22·91.
Dem.
*22·92.
*22·93.
Dem.
*22·94.
Dem.
This proposition is used in connection with mathematical induction, in *90·102, which is required for the proof of *90·132, which is one of the fundamental propositions in the theory of mathematical induction.
*22·95.
Dem.