Principia mathematica, vol. 1 (of 3)

The following are some propositions concerning classes which are analogues of propositions previously given concerning propositions: i.e. the common part of and is the negation of "not or not"; i.e., " is a member of or not"; i.e. " is not a member of both and not";

This expresses the ordinary syllogism in Barbara (with the premisses interchanged); for "" means the same as "all 's are 's," so that the above proposition states: "If all 's are 's, and all 's are 's, then all 's are 's." (It should be observed that syllogisms are traditionally expressed with "therefore," as if they asserted both premisses and conclusion. This is, of course, merely a slipshod way of speaking, since what is really asserted is only the connection of premisses with conclusion.)

The syllogism in Barbara when the minor premiss has an individual subject is e.g. "if Socrates is a man, and all men are mortals, then Socrates is a mortal." This, as was pointed out by Peano, is not a particular case of "," since "" is not a particular case of "." This point is important, since traditional logic is here mistaken. The nature and magnitude of its mistake will become clearer at a later stage.

Generally, when we require analogous but different symbols for relations and for classes, we shall choose for relations the symbol obtained by adding a dot, in some convenient position, to the corresponding symbol for classes. (The dot must not be put on the line, since that would cause confusion with the use of dots as brackets.) But such symbols require and receive a special definition in each case.

A class is said to exist when it has at least one member: " exists" is denoted by "" Thus we put The class which has no members is called the "null-class," and is denoted by "." Any propositional function which is always false determines the null-class. One such function is known to us already, namely " is not identical with ," which we denote by "." Thus we may use this function for defining , and put

For relations we use similar notations. We put i.e. "" means that there is at least one couple , between which the relation holds. will be the relation which never holds, and the relation which always holds. is practically never required; will be the relation . We have

There are no classes which contain objects of more than one type. Accordingly there is a universal class and a null-class proper to each type of object. But these symbols need not be distinguished, since it will be found that there is no possibility of confusion. Similar remarks apply to relations.

Descriptions. By a "description" we mean a phrase of the form "the so-and-so" or of some equivalent form. For the present, we confine our attention to the in the singular. We shall use this word strictly, so as to imply uniqueness; e.g. we should not say " is the son of " if had other sons besides . Thus a description of the form "the so-and-so" will only have an application in the event of there being one so-and-so and no more. Hence a description requires some propositional function which is satisfied by one value of and by no other values; then "the which satisfies " is a description which definitely describes a certain object, though we may not know what object it describes. For example, if is a man, " is the father of " must be true for one, and only one, value of . Hence "the father of " is a description of a certain man, though we may not know what man it describes. A phrase containing "the" always presupposes some initial propositional function not containing "the"; thus instead of " is the father of " we ought to take as our initial function " begot "; then "the father of " means the one value of which satisfies this propositional function.

If is a propositional function, the symbol "" is used in our symbolism in such a way that it can always be read as "the which satisfies ." But we do not define "" as standing for "the which satisfies ," thus treating this last phrase as embodying a primitive idea. Every use of "," where it apparently occurs as a constituent of a proposition in the place of an object, is defined in terms of the primitive ideas already on hand. An example of this definition in use is given by the proposition "" which is considered immediately. The whole subject is treated more fully in Chapter III.

The symbol should be compared and contrasted with "" which in use can always be read as "the 's which satisfy ." Both symbols are incomplete symbols defined only in use, and as such are discussed in Chapter III. The symbol "" always has an application, namely to the class determined by ; but "" only has an application when is only satisfied by one value of , neither more nor less. It should also be observed that the meaning given to the symbol by the definition, given immediately below, of does not presuppose that we know the meaning of "one." This is also characteristic of the definition of any other use of .

We now proceed to define "" so that it can be read "the satisfying exists." (It will be observed that this is a different meaning of existence from that which we express by ""). Its definition is i.e. "the satisfying exists" is to mean "there is an object such that is true when is but not otherwise."

A proposition such as "Apollo exists" is really of the same logical form, although it does not explicitly contain the word the. For "Apollo" means really "the object having such-and-such properties," say "the object having the properties enumerated in the Classical Dictionary[9]." If these properties make up the propositional function , then "Apollo" means "," and "Apollo exists" means "." To take another illustration, "the author of Waverley" means "the man who (or rather, the object which) wrote Waverley." Thus "Scott is the author of Waverley" is Here (as we observed before) the importance of identity in connection with descriptions plainly appears.

The notation "," which is long and inconvenient, is seldom used, being chiefly required to lead up to another notation, namely "" meaning "the object having the relation to ." That is, we put The inverted comma may be read "of." Thus "" is read "the of ." Thus if is the relation of father to son, "" means "the father of "; if is the relation of son to father, "" means "the son of ," which will only "exist" if has one son and no more. is a function of , but not a propositional function; we shall call it a descriptive function. All the ordinary functions of mathematics are of this kind, as will appear more fully in the sequel. Thus in our notation, "" would be written "," and "sin" would stand for the relation which has to . Instead of a variable descriptive function , we put , where the variable relation takes the place of the variable function . A descriptive function will in general exist while belongs to a certain domain, but not outside that domain; thus if we are dealing with positive rationals, will be significant if is a perfect square, but not otherwise; if we are dealing with real numbers, and agree that "" is to mean the positive square root (or, is to mean the negative square root), will be significant provided is positive, but not otherwise; and so on. Thus every descriptive function has what we may call a "domain of definition" or a "domain of existence," which may be thus defined:

If is any relation, we will speak of as the "associated descriptive function." A great many of the constant relations which we shall have occasion to introduce are only or chiefly important on account of their associated descriptive functions. In such cases, it is easier (though less correct) to begin by assigning the meaning of the descriptive function, and to deduce the meaning of the relation from that of the descriptive function. This will be done in the following explanations of notation.

Various descriptive functions of relations. If is any relation, the converse of is the relation which holds between and whenever holds between and . Thus greater is the converse of less, before of after, cause of effect, husband of wife, etc. The converse of is written[10] or . The definition is The second of these is not a formally correct definition, since we ought to define "Cnv" and deduce the meaning of . But it is not worth while to adopt this plan in our present introductory account, which aims at simplicity rather than formal correctness.

A relation is called symmetrical if , i.e. if it holds between and whenever it holds between and (and therefore vice versa). Identity, diversity, agreement or disagreement in any respect, are symmetrical relations. A relation is called asymmetrical when it is incompatible with its converse, i.e. when , or, what is equivalent,

Before and after, greater and less, ancestor and descendant, are asymmetrical, as are all other relations of the sort that lead to series. But there are many asymmetrical relations which do not lead to series, for instance, that of wife's brother[11]. A relation maybe neither symmetrical nor asymmetrical; for example, this holds of the relation of inclusion between classes: and will both be true if , but otherwise only one of them, at most, will be true. The relation brother is neither symmetrical nor asymmetrical, for if is the brother of , may be either the brother or the sister of .

In the propositional function , we call the referent and the relatum. The class , consisting of all the 's which have the relation to , is called the class of referents of with respect to ; the class , consisting of all the 's to which has the relation , is called the class of relata of with respect to . These two classes are denoted respectively by and . Thus The arrow runs towards in the first case, to show that we are concerned with things having the relation to ; it runs away from in the second case to show that the relation goes from to the members of . It runs in fact from a referent and towards a relatum.

The notations , are very important, and are used constantly. If is the relation of parent to child, = the parents of , = the children of . We have These equivalences are often embodied in common language. For example, we say indiscriminately " is an inhabitant of London" or " inhabits London." If we put "" for "inhabits," " inhabits London" is " London," while " is an inhabitant of London" is "."

Instead of and we sometimes use , , where "" stands for "sagitta," and "" is "" backwards. Thus we put These notations are sometimes more convenient than an arrow when the relation concerned is represented by a combination of letters, instead of a single letter such as . Thus e.g. we should write , rather than put an arrow over the whole length of ().

The class of all terms that have the relation to something or other is called the domain of . Thus if is the relation of parent and child, the domain of will be the class of parents. We represent the domain of by "." Thus we put Similarly the class of all terms to which something or other has the relation is called the converse domain of ; it is the same as the domain of the converse of . The converse domain of is represented by ""; thus The sum of the domain and the converse domain is called the field, and is represented by : thus

The field is chiefly important in connection with series. If is the ordering relation of a series, will be the class of terms of the series, will be all the terms except the last (if any), and will be all the terms except the first (if any). The first term, if it exists, is the only member of , since it is the only term which is a predecessor but not a follower. Similarly the last term (if any) is the only member of . The condition that a series should have no end is , i.e. "every follower is a predecessor"; the condition for no beginning is . These conditions are equivalent respectively to and .

The relative product of two relations and is the relation which holds between and when there is an intermediate term such that has the relation to and has the relation to . The relative product of and is represented by ; thus we put Thus "paternal aunt" is the relative product of sister and father; "paternal grandmother" is the relative product of mother and father; "maternal grandfather" is the relative product of father and mother. The relative product is not commutative, but it obeys the associative law, i.e. It also obeys the distributive law with regard to the logical addition of relations, i.e. we have

A relation is called transitive when , i.e. when, if and , we always have , i.e. when Relations which generate series are always transitive; thus e.g. If is a relation which generates a series, may conveniently be read "precedes"; thus "" becomes "if precedes and precedes , then always precedes ." The class of relations which generate series are partially characterized by the fact that they are transitive and asymmetrical, and never relate a term to itself.

If is a relation which generates a series, and if we have not merely , but , then generates a series which is compact (überall dicht), i.e. such that there are terms between any two. For in this case we have i.e. if precedes , there is a term such that precedes and precedes , i.e. there is a term between and . Thus among relations which generate series, those which generate compact series are those for which .

Many relations which do not generate series are transitive, for example, identity, or the relation of inclusion between classes. Such cases arise when the relations are not asymmetrical. Relations which are transitive and symmetrical are an important class: they may be regarded as consisting in the possession of some common property.