Principia mathematica, vol. 1 (of 3)

THE notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico. His use of dots as brackets is adopted, and so are many of his symbols.

Variables. The idea of a variable, as it occurs in the present work, is more general than that which is explicitly used in ordinary mathematics. In ordinary mathematics, a variable generally stands for an undetermined number or quantity. In mathematical logic, any symbol whose meaning is not determinate is called a variable, and the various determinations of which its meaning is susceptible are called the values of the variable. The values may be any set of entities, propositions, functions, classes or relations, according to circumstances. If a statement is made about "Mr A and Mr B," "Mr A" and "Mr B" are variables whose values are confined to men. A variable may either have a conventionally-assigned range of values, or may (in the absence of any indication of the range of values) have as the range of its values all determinations which render the statement in which it occurs significant. Thus when a text-book of logic asserts that " is ," without any indication as to what may be, what is meant is that any statement of the form " is " is true. We may call a variable restricted when its values are confined to some only of those of which it is capable; otherwise, we shall call it unrestricted. Thus when an unrestricted variable occurs, it represents any object such that the statement concerned can be made significantly (i.e. either truly or falsely) concerning that object. For the purposes of logic, the unrestricted variable is more convenient than the restricted variable, and we shall always employ it.

We shall find that the unrestricted variable is still subject to limitations imposed by the manner of its occurrence, i.e. things which can be said significantly concerning a proposition cannot be said significantly concerning a class or a relation, and so on. But the limitations to which the unrestricted variable is subject do not need to be explicitly indicated, since they are the limits of significance of the statement in which the variable occurs, and are therefore intrinsically determined by this statement. This will be more fully explained later[1]. To sum up, the three salient facts connected with the use of the variable are: (1) that a variable is ambiguous in its denotation and accordingly undefined: (2) that a variable preserves a recognizable identity in various occurrences throughout the same context, so that many variables can occur together in the same context each with its separate identity: and (3) that either the range of possible determinations of two variables may be the same, so that a possible determination of one variable is also a possible determination of the other, or the ranges of two variables may be different, so that, if a possible determination of one variable is given to the other, the resulting complete phrase is meaningless instead of becoming a complete unambiguous proposition (true or false) as would be the case if all variables in it had been given any suitable determinations.

The uses of various letters. Variables will be denoted by single letters, and so will certain constants; but a letter which has once been assigned to a constant by a definition must not afterwards be used to denote a variable. The small letters of the ordinary alphabet will all be used for variables, except and after *40, in which constant meanings are assigned to these two letters. The following capital letters will receive constant meanings: , , , , , and . Among small Greek letters, we shall give constant meanings to , and (at a later stage) to , and . Certain Greek capitals will from time to time be introduced for constants, but Greek capitals will not be used for variables. Of the remaining letters, , , will be called propositional letters, and will stand for variable propositions (except that, from *40 onwards, must not be used for a variable); , , , , , and (until *33) will be called functional letters, and will be used for variable functions.

The small Greek letters not already mentioned will be used for variables whose values are classes, and will be referred to simply as Greek letters. Ordinary capital letters not already mentioned will be used for variables whose values are relations, and will be referred to simply as capital letters. Ordinary small letters other than , , , , , will be used for variables whose values are not known to be functions, classes, or relations; these letters will be referred to simply as small Latin letters.

After the early part of the work, variable propositions and variable functions will hardly ever occur. We shall then have three main kinds of variables: variable classes, denoted by small Greek letters; variable relations, denoted by capitals; and variables not given as necessarily classes or relations, which will be denoted by small Latin letters.

In addition to this usage of small Greek letters for variable classes, capital letters for variable relations, small Latin letters for variables of type wholly undetermined by the context (these arise from the possibility of "systematic ambiguity," explained later in the explanations of the theory of types), the reader need only remember that all letters represent variables, unless they have been defined as constants in some previous place in the book. In general the structure of the context determines the scope of the variables contained in it; but the special indication of the nature of the variables employed, as here proposed, saves considerable labour of thought.

The fundamental functions of propositions. An aggregation of propositions, considered as wholes not necessarily unambiguously determined, into a single proposition more complex than its constituents, is a function with propositions as arguments. The general idea of such an aggregation of propositions, or of variables representing propositions, will not be employed in this work. But there are four special cases which are of fundamental importance, since all the aggregations of subordinate propositions into one complex proposition which occur in the sequel are formed out of them step by step.

They are (1) the Contradictory Function, (2) the Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive Function, (4) the Implicative Function. These functions in the sense in which they are required in this work are not all independent; and if two of them are taken as primitive undefined ideas, the other two can be defined in terms of them. It is to some extent—though not entirely—arbitrary as to which functions are taken as primitive. Simplicity of primitive ideas and symmetry of treatment seem to be gained by taking the first two functions as primitive ideas.

The Contradictory Function with argument , where is any proposition, is the proposition which is the contradictory of , that is, the proposition asserting that is not true. This is denoted by . Thus is the contradictory function with as argument and means the negation of the proposition . It will also be referred to as the proposition not-. Thus means not-, which means the negation of .

The Logical Sum is a propositional function with two arguments and , and is the proposition asserting or disjunctively, that is, asserting that at least one of the two and is true. This is denoted by . Thus is the logical sum with and as arguments. It is also called the logical sum of and . Accordingly means that at least or is true, not excluding the case in which both are true.

The Logical Product is a propositional function with two arguments and , and is the proposition asserting and conjunctively, that is, asserting that both and are true. This is denoted by , or—in order to make the dots act as brackets in a way to be explained immediately—by , or by , or by . Thus is the logical product with and as arguments. It is also called the logical product of and . Accordingly means that both and are true. It is easily seen that this function can be defined in terms of the two preceding functions. For when and are both true it must be false that either is true. Hence in this book is merely a shortened form of symbolism for If any further idea attaches to the proposition "both and are true," it is not required here.

The Implicative Function is a propositional function with two arguments and , and is the proposition that either not- or is true, that is, it is the proposition . Thus if is true, is false, and accordingly the only alternative left by the proposition is that is true. In other words if and are both true, then is true. In this sense the proposition will be quoted as stating that implies . The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting and without the intervention of . But "implies" as used here expresses nothing else than the connection between and also expressed by the disjunction "not- or ." The symbol employed for " implies ," i.e. for "" is "." This symbol may also be read "if , then ." The association of implication with the use of an apparent variable produces an extension called "formal implication." This is explained later: it is an idea derivative from "implication" as here defined. When it is necessary explicitly to discriminate "implication" from "formal implication," it is called "material implication." Thus "material implication" is simply "implication" as here defined. The process of inference, which in common usage is often confused with implication, is explained immediately.

These four functions of propositions are the fundamental constant (i.e. definite) propositional functions with propositions as arguments, and all other constant propositional functions with propositions as arguments, so far as they are required in the present work, are formed out of them by successive steps. No variable propositional functions of this kind occur in this work.

Equivalence. The simplest example of the formation of a more complex function of propositions by the use of these four fundamental forms is furnished by "equivalence." Two propositions and are said to be "equivalent" when implies and implies . This relation between and is denoted by "" Thus "" stands for "." It is easily seen that two propositions are equivalent when, and only when, they are both true or are both false. Equivalence rises in the scale of importance when we come to "formal implication" and thus to "formal equivalence." It must not be supposed that two propositions which are equivalent are in any sense identical or even remotely concerned with the same topic. Thus "Newton was a man" and "the sun is hot" are equivalent as being both true, and "Newton was not a man" and "the sun is cold" are equivalent as being both false. But here we have anticipated deductions which follow later from our formal reasoning. Equivalence in its origin is merely mutual implication as stated above.

Truth-values. The "truth-value" of a proposition is truth if it is true, and falsehood if it is false[2]. It will be observed that the truth-values of , , , , depend only upon those of and , namely the truth-value of "" is truth if the truth-value of either or is truth, and is falsehood otherwise; that of "" is truth if that of both and is truth, and is falsehood otherwise; that of "" is truth if either that of is falsehood or that of is truth; that of "" is the opposite of that of ; and that of "" is truth if and have the same truth-value, and is falsehood otherwise. Now the only ways in which propositions will occur in the present work are ways derived from the above by combinations and repetitions. Hence it is easy to see (though it cannot be formally proved except in each particular case) that if a proposition occurs in any proposition which we shall ever have occasion to deal with, the truth-value of will depend, not upon the particular proposition , but only upon its truth-value; i.e. if , we shall have . Thus whenever two propositions are known to be equivalent, either may be substituted for the other in any formula with which we shall have occasion to deal.

We may call a function a "truth-function" when its argument is a proposition, and the truth-value of depends only upon the truth-value of . Such functions are by no means the only common functions of propositions. For example, " believes " is a function of which will vary its truth-value for different arguments having the same truth-value: may believe one true proposition without believing another, and may believe one false proposition without believing another. Such functions are not excluded from our consideration, and are included in the scope of any general propositions we may make about functions; but the particular functions of propositions which we shall have occasion to construct or to consider explicitly are all truth-functions. This fact is closely connected with a characteristic of mathematics, namely, that mathematics is always concerned with extensions rather than intensions. The connection, if not now obvious, will become more so when we have considered the theory of classes and relations.

Assertion-sign. The sign "," called the "assertion-sign," means that what follows is asserted. It is required for distinguishing a complete proposition, which we assert, from any subordinate propositions contained in it but not asserted. In ordinary written language a sentence contained between full stops denotes an asserted proposition, and if it is false the book is in error. The sign "" prefixed to a proposition serves this same purpose in our symbolism. For example, if "" occurs, it is to be taken as a complete assertion convicting the authors of error unless the proposition "" is true (as it is). Also a proposition stated in symbols without this sign "" prefixed is not asserted, and is merely put forward for consideration, or as a subordinate part of an asserted proposition.

Inference. The process of inference is as follows: a proposition "" is asserted, and a proposition " implies " is asserted, and then as a sequel the proposition "" is asserted. The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error. Accordingly whenever, in symbols, where and have of course special determinations, have occurred, then "" will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of "." It is of course convenient, even at the risk of repetition, to write "" and "" in close juxtaposition before proceeding to "" as the result of an inference. When this is to be done, for the sake of drawing attention to the inference which is being made, we shall write instead which is to be considered as a mere abbreviation of the threefold statement Thus "" may be read ", therefore ," being in fact the same abbreviation, essentially, as this is; for ", therefore " does not explicitly state, what is part of its meaning, that implies . An inference is the dropping of a true premiss; it is the dissolution of an implication.

The use of dots. Dots on the line of the symbols have two uses, one to bracket off propositions, the other to indicate the logical product of two propositions. Dots immediately preceded or followed by "" or "" or "" or "," or by "," "," "" ... or "," "," "" ... or "" or "" or analogous expressions, serve to bracket off a proposition; dots occurring otherwise serve to mark a logical product. The general principle is that a larger number of dots indicates an outside bracket, a smaller number indicates an inside bracket. The exact rule as to the scope of the bracket indicated by dots is arrived at by dividing the occurrences of dots into three groups which we will name I, II, and III. Group I consists of dots adjoining a sign of implication () or of equivalence () or of disjunction () or of equality by definition ( Df). Group II consists of dots following brackets indicative of an apparent variable, such as or or or or or analogous expressions[3]. Group III consists of dots which stand between propositions in order to indicate a logical product. Group I is of greater force than Group II, and Group II than Group III. The scope of the bracket indicated by any collection of dots extends backwards or forwards beyond any smaller number of dots, or any equal number from a group of less force, until we reach either the end of the asserted proposition or a greater number of dots or an equal number belonging to a group of equal or superior force.

Dots indicating a logical product have a scope which works both backwards and forwards; other dots only work away from the adjacent sign of disjunction, implication, or equivalence, or forward from the adjacent symbol of one of the other kinds enumerated in Group II.

"" means the proposition "' or ' implies ' or .'" When we assert this proposition, instead of merely considering it, we write where the two dots after the assertion-sign show that what is asserted is the whole of what follows the assertion-sign, since there are not as many as two dots anywhere else. If we had written "," that would mean the proposition "either is true, or implies ' or .'" If we wished to assert this, we should have to put three dots after the assertion-sign. If we had written "," that would mean the proposition "either ' or ' implies q, or p is true." The forms "" and "" have no meaning.

"" will mean "if implies , then if implies implies ." If we wish to assert this (which is true) we write Again "" will mean "if ' implies ' implies ' implies ,' then implies ." This is in general untrue. (Observe that "" is sometimes most conveniently read as " implies ," and sometimes as "if , then .") "" will mean "if implies , and implies , then implies ." In this formula, the first dot indicates a logical product; hence the scope of the second dot extends backwards to the beginning of the proposition. "" will mean " implies ; and if implies , then implies ." (This is not true in general.) Here the two dots indicate a logical product; since two dots do not occur anywhere else, the scope of these two dots extends backwards to the beginning of the proposition, and forwards to the end.

"" will mean "if either or is true, then if either or ' implies ' is true, it follows that either or is true." If this is to be asserted, we must put four dots after the assertion-sign, thus: (This proposition is proved in the body of the work; it is *2·75.) If we wish to assert (what is equivalent to the above) the proposition: "if either or is true, and either or ' implies ' is true, then either or is true," we write Here the first pair of dots indicates a logical product, while the second pair does not. Thus the scope of the second pair of dots passes over the first pair, and back until we reach the three dots after the assertion-sign.

Other uses of dots follow the same principles, and will be explained as they are introduced. In reading a proposition, the dots should be noticed first, as they show its structure. In a proposition containing several signs of implication or equivalence, the one with the greatest number of dots before or after it is the principal one: everything that goes before this one is stated by the proposition to imply or be equivalent to everything that comes after it.

Definitions. A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known. Or, if the defining combination of symbols is one which only acquires meaning when combined in a suitable manner with other symbols[4], what is meant is that any combination of symbols in which the newly-defined symbol or combination of symbols occurs is to have that meaning (if any) which results from substituting the defining combination of symbols for the newly-defined symbol or combination of symbols wherever the latter occurs. We will give the names of definiendum and definiens respectively to what is defined and to that which it is defined as meaning. We express a definition by putting the definiendum to the left and the definiens to the right, with the sign "=" between, and the letters "Df" to the right of the definiens. It is to be understood that the sign "=" and the letters "Df" are to be regarded as together forming one symbol. The sign "=" without the letters "Df" will have a different meaning, to be explained shortly.

It is to be observed that a definition is, strictly speaking, no part of the subject in which it occurs. For a definition is concerned wholly with the symbols, not with what they symbolise. Moreover it is not true or false, being the expression of a volition, not of a proposition. (For this reason, definitions are not preceded by the assertion-sign.) Theoretically, it is unnecessary ever to give a definition: we might always use the definiens instead, and thus wholly dispense with the definiendum. Thus although we employ definitions and do not define "definition," yet "definition" does not appear among our primitive ideas, because the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous.

In spite of the fact that definitions are theoretically superfluous, it is nevertheless true that they often convey more important information than is contained in the propositions in which they are used. This arises from two causes. First, a definition usually implies that the definiens is worthy of careful consideration. Hence the collection of definitions embodies our choice of subjects and our judgment as to what is most important. Secondly, when what is defined is (as often occurs) something already familiar, such as cardinal or ordinal numbers, the definition contains an analysis of a common idea, and may therefore express a notable advance. Cantor's definition of the continuum illustrates this: his definition amounts to the statement that what he is defining is the object which has the properties commonly associated with the word "continuum," though what precisely constitutes these properties had not before been known. In such cases, a definition is a "making definite": it gives definiteness to an idea which had previously been more or less vague.

Some important remarks must be made respecting the variables occurring in the definiens and the definiendum. But these will be deferred till the notion of an "apparent variable" has been introduced, when the subject can be considered as a whole.

Summary of preceding statements. There are, in the above, three primitive ideas which are not "defined" but only descriptively explained. Their primitiveness is only relative to our exposition of logical connection and is not absolute; though of course such an exposition gains in importance according to the simplicity of its primitive ideas. These ideas are symbolised by "" and "," and by "" prefixed to a proposition.

Primitive propositions. Some propositions must be assumed without proof, since all inference proceeds from propositions previously asserted. These, as far as they concern the functions of propositions mentioned above, will be found stated in *1, where the formal and continuous exposition of the subject commences. Such propositions will be called "primitive propositions." These, like the primitive ideas, are to some extent a matter of arbitrary choice; though, as in the previous case, a logical system grows in importance according as the primitive propositions are few and simple. It will be found that owing to the weakness of the imagination in dealing with simple abstract ideas no very great stress can be laid upon their obviousness. They are obvious to the instructed mind, but then so are many propositions which cannot be quite true, as being disproved by their contradictory consequences. The proof of a logical system is its adequacy and its coherence. That is: (1) the system must embrace among its deductions all those propositions which we believe to be true and capable of deduction from logical premisses alone, though possibly they may require some slight limitation in the form of an increased stringency of enunciation; and (2) the system must lead to no contradictions, namely in pursuing our inferences we must never be led to assert both and not-, i.e. both "" and "" cannot legitimately appear.