The Principles of Mathematics (1903)

§ 61
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§ 61

In the case of a class a which has a finite number of terms--say a1, a2, a3, ... an, we can illustrate these various notions as follows:(§ 61 ¶ 1)

(1) All a's denotes a1 and a2 and ... and an.(§ 61 ¶ 2)

(2) Every a denotes a1 and denotes a2, and ... and denotes an.(§ 61 ¶ 3)

(3) Any a denotes a1 or a2 or ... or an, where or has the meaning that it is irrelevant which we take.(§ 61 ¶ 4)

(4) An a denotes a1 or a2 or ... or an, where or has the meaning that no one in particular must be taken, just as in all a's we must not take any one in particular.(§ 61 ¶ 5)

(5) Some a denotes a1 or denotes a2 or ... or denotes an, where it is not irrelevant which is taken, but on the contrary some one particular a must be taken.(§ 61 ¶ 6)

As the nature and properties of the various ways of combining terms are of vital importance to the principles of mathematics, it may be well to illustrate their properties by the following important examples.(§ 61 ¶ 7)

  1. Let a be a class, and b a class of classes. We then obtain in all six possible relations of a to b from various combinations of any, a and some. All and every do not, in this case, introduce anything new. The six cases are as follows.(§ 61 ¶ 8)

    1. Any a belongs to any class belonging to b, in other words, the class a is wholly contained in the common part or logical product of the various classes belonging to b.(§ 61 ¶ 9)

    2. Any a belongs to a b, i.e. the class a is contained in any class which contains all the b's, or, is contained in the logical sum of all the b's.(§ 61 ¶ 10)

    3. Any a belongs to some b, i.e. there is a class belonging to b, in which the class a is contained. The difference between this case and the second arises from the fact that here there is one b to which every a belongs, whereas before it was only decided that every a belonged to a b, and different a's might belong to different b's.(§ 61 ¶ 11)

    4. An a belongs to any b, i.e. whatever b we take, it has a part in common with a.(§ 61 ¶ 12)

    5. An a belongs to a b, i.e. there is a b which has a part in common with a. This is equivalent to some (or an) a belongs to some b.(§ 61 ¶ 13)

    6. Some a belongs to any b, i.e. there is an a which belongs to the common part of allthe b's, or a and all the b's have a common part. These are all the cases that arise here.(§ 61 ¶ 14)

  2. It is instructive, as showing the generality of the type of relations here considered, to compare the above case with the following. Let a, b be two series of real numbers; then six precisely analogous cases arise.(§ 61 ¶ 15)

    1. Any a is less than any b, or, the series a is contained among numbers less than every b(§ 61 ¶ 16)

    2. Any a is less than a b, or, whatever a we take, there is a b which is greater, or, the series a is contained among numbers less than a (variable) term of the series b. It does not follow that some term of the series b is greater than all the a's.(§ 61 ¶ 17)

    3. Any a is less than some b, or, there is a term of b which is greater than all the a's. This case is not to be confounded with (2).(§ 61 ¶ 18)

    4. An a is less than any b, i.e. whatever b we take, there is an a which is less than it.(§ 61 ¶ 19)

    5. An a is less than a b, i.e. it is possible to find an a and a b such that the a is less than the b. This merely denies that any a is greater than any b.(§ 61 ¶ 20)

    6. Some a is less than any b, i.e. there is an a which is less than all the b's. This was not implied in (4), where the a was variable, whereas here it is constant.(§ 61 ¶ 21)

    In this case, actual mathematics have compelled the distinction between the variable and the constant disjunction. But in other cases, where mathematics have not obtained sway, the distinction has been neglected; and the mathematicians, as was natural, have not investigated the logical nature of the disjunctive notions which they employed.(§ 61 ¶ 22)

  3. I shall give one other instance, as it brings in the difference between any and every, which has not been relevant in the previous cases. Let a and b be two classes of classes; then twenty different relations between them arise from different combinations of the terms of their terms. The following technical terms will be useful. If a be a class of classes, its logical sum consists of all the terms belonging to any a, i.e. all terms such that there is an a to which they belong, while its logical product consists of all terms belonging to every a, i.e. to the common part of all the a's. We have then the following cases.(§ 61 ¶ 23)

    1. Any term of any a belongs to every b, i.e. the logical sum of a is contained in the logical product of b.(§ 61 ¶ 24)

    2. Any term of any a belongs to a b, i.e. the logical sum of a is contained in the logical sum of b.(§ 61 ¶ 25)

    3. Any term of any a belongs to some b, i.e. there is a b which contains the logical sum of a.(§ 61 ¶ 26)

    4. Any term of some (or an) a belongs to every b, i.e. there is an a which is contained in the product of b.(§ 61 ¶ 27)

    5. Any term of some (or an) a belongs to a a, i.e. there is an a which is contained in the sum of b.(§ 61 ¶ 28)

    6. Any term of some (or an) a belongs to some b, i.e. there is a b which contains one class belonging to a.(§ 61 ¶ 29)

    7. A term of any a belongs to any b, i.e. any class of a and any class of b have a common part.(§ 61 ¶ 30)

    8. A term of any a belongs to a b, i.e. any class of a has a part in common with the logical sum of b.(§ 61 ¶ 31)

    9. A term of any a belongs to some b, i.e. there is a b with which any a has a part in common.(§ 61 ¶ 32)

    10. A term of an a belongs to every b, i.e. the logical sum of a and the logical product of b have a common part.(§ 61 ¶ 33)

    11. A term of an a belongs to every b, i.e. given any b, an a can be found with which it has a common part.(§ 61 ¶ 34)

    12. A term of an a belongs to a b, i.e. the logical sums of a and b have a common part.(§ 61 ¶ 35)

    13. Any term of every a belongs to every b, i.e. the logical product of a is contained in the logical product of b.(§ 61 ¶ 36)

    14. Any term of every a belongs to a b, i.e. the logical product of a is contained in the logical sum of b.(§ 61 ¶ 37)

    15. Any term of every a belongs to some b, i.e. there is a term of b in which the logical product of a is contained.(§ 61 ¶ 38)

    16. A (or some) term of every a belongs to every b, i.e. the logical products of a and of b have a common part.(§ 61 ¶ 39)

    17. A (or some) term of every a belongs to a b, i.e. the logical product of a and the logical sum of b have a common part.(§ 61 ¶ 40)

    18. Some term of any a belongs to every b, i.e. any a has a part in common with the logical product of b.(§ 61 ¶ 41)

    19. A term of some a belongs to any b, i.e. there is some term of a with which any b has a common part.(§ 61 ¶ 42)

    20. A term of every a belongs to any b, i.e. any b has a part in common with the logical product of a.(§ 61 ¶ 43)

The above examples show that, although it may often happen that there is a mutual implication (which has not always been stated) of corresponding propositions concerning some and a, or concerning any and every, yet in other cases there is no such mutual implication. Thus the five notions discussed in the present chapter are genuinely distinct, and to confound them may lead to perfectly definite fallacies.(§ 61 ¶ 44)

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