Principia mathematica, vol. 1 (of 3)

(7) Besides the above primitive propositions, we require a primitive proposition called "the axiom of identification of real variables." When we have separately asserted two different functions of , where is undetermined, it is often important to know whether we can identify the in one assertion with the in the other. This will be the case—so our axiom allow us to infer—if both assertions present x as the argument to some one function, that is to say, if is a constituent in both assertions (whatever propositional function may be), or, more generally, if is a constituent in one assertion, and is a constituent in the other. This axiom introduces notions which have not yet been explained; for a fuller account, see the remarks accompanying *3·03, *1·7, *1·71, and *1·72 (which is the statement of this axiom) in the body of the work, as well as the explanation of propositional functions and ambiguous assertion to be given shortly.

The law of tautology, in the two forms: i.e. " is true" is equivalent to " is true and is true," as well as to " is true or is true." From a formal point of view, it is through the law of tautology and its consequences that the algebra of logic is chiefly distinguished from ordinary algebra.

The law of absorption : i.e. " implies " is equivalent to " is equivalent to ." This is called the law of absorption because it shows that the factor in the product is absorbed by the factor , if implies . This principle enables us to replace an implication () by an equivalence ( whenever it is convenient to do so.

Logical addition and multiplication of propositions obey the associative and commutative laws, and the distributive law in two forms, namely The second of these distinguishes the relations of logical addition and multiplication from those of arithmetical addition and multiplication.

Propositional functions. Let be a statement containing a variable and such that it becomes a proposition when is given any fixed determined meaning. Then is called a "propositional function"; it is not a proposition, since owing to the ambiguity of it really makes no assertion at all. Thus " is hurt" really makes no assertion at all, till we have settled who is. Yet owing to the individuality retained by the ambiguous variable , it is an ambiguous example from the collection of propositions arrived at by giving all possible determinations to in " is hurt" which yield a proposition, true or false. Also if " is hurt" and " is hurt" occur in the same context, where is another variable, then according to the determinations given to and , they can be settled to be (possibly) the same proposition or (possibly) different propositions. But apart from some determination given to and , they retain in that context their ambiguous differentiation. Thus " is hurt" is an ambiguous "value" of a propositional function. When we wish to speak of the propositional function corresponding to " is hurt," we shall write " is hurt." Thus " is hurt" is the propositional function and " is hurt" is an ambiguous value of that function. Accordingly though " is hurt" and " is hurt" occurring in the same context can be distinguished, " is hurt" and " is hurt" convey no distinction of meaning at all. More generally, is an ambiguous value of the propositional function , and when a definite signification is substituted for , is an unambiguous value of .

Propositional functions are the fundamental kind from which the more usual kinds of function, such as "" or "" or "the father of ," are derived. These derivative functions are considered later, and are called "descriptive functions." The functions of propositions considered above are a particular case of propositional functions.

The range of values and total variation. Thus corresponding to any propositional function , there is a range, or collection, of values, consisting of all the propositions (true or false) which can be obtained by giving every possible determination to in . A value of for which is true will be said to "satisfy" . Now in respect to the truth or falsehood of propositions of this range three important cases must be noted and symbolised. These cases are given by three propositions of which one at least must be true. Either (1) all propositions of the range are true, or (2) some propositions of the range are true, or (3) no proposition of the range is true. The statement (1) is symbolised by "," and (2) is symbolised by "." No definition is given of these two symbols, which accordingly embody two new primitive ideas in our system. The symbol "" may be read " always," or " is always true," or " is true for all possible values of ." The symbol "" may be read "there exists an for which is true," or "there exists an satisfying ," and thus conforms to the natural form of the expression of thought.

Proposition (3) can be expressed in terms of the fundamental ideas now on hand. In order to do this, note that "" stands for the contradictory of . Accordingly is another propositional function such that each value of contradicts a value of and vice versa. Hence "" symbolises the proposition that every value of is untrue. This is number (3) as stated above.

It is an obvious error, though one easy to commit, to assume that cases (1) and (3) are each other's contradictories. The symbolism exposes this fallacy at once, for (1) is , and (3) is , while the contradictory of (1) is . For the sake of brevity of symbolism a definition is made, namely

Definitions of which the object is to gain some trivial advantage in brevity by a slight adjustment of symbols will be said to be of "merely symbolic import," in contradistinction to those definitions which invite consideration of an important idea.

For reasons which will be explained in Chapter II, we do not take negation as a primitive idea when propositions of the forms and are concerned, but we define the negation of , i.e. of " is always true," as being " is sometimes false," i.e. "," and similarly we define the negation of as being . Thus we put

In like manner we define a disjunction in which one of the propositions is of the form "" or "(" in terms of a disjunction of propositions not of this form, putting i.e. "either is always true, or is true" is to mean "' or ' is always true," with similar definitions in other cases. This subject is resumed in Chapter II, and in *9 in the body of the work.

Apparent variables. The symbol "(" denotes one definite proposition, and there is no distinction in meaning between "(" and "" when they occur in the same context. Thus the "" in "" is not an ambiguous constituent of any expression in which "(" occurs; and such an expression does not cease to convey a determinate meaning by reason of the ambiguity of the x in the "." The symbol "(" has some analogy to the symbol for definite integration, since in neither case is the expression a function of .

The range of in "(" or "(" extends over the complete field of the values of for which "" has meaning, and accordingly the meaning of "(" or "(" involves the supposition that such a field is determinate. The which occurs in "(" or "(" is called (following Peano) an "apparent variable." It follows from the meaning of "(" that the in this expression is also an apparent variable. A proposition in which occurs as an apparent variable is not a function of . Thus e.g. "(" will mean "everything is equal to itself." This is an absolute constant, not a function of a variable . This is why the is called an apparent variable in such cases.

Besides the "range" of in "(" or "(," which is the field of the values that may have, we shall speak of the "scope" of , meaning the function of which all values or some value are being affirmed. If we are asserting all values (or some value) of "," "" is the scope of ; if we are asserting all values (or some value) of "," "" is the scope of ; if we are asserting all values (or some value) of "," "" will be the scope of , and so on. The scope of is indicated by the number of dots after the "()" or ""; that is to say, the scope extends forwards until we reach an equal number of dots not indicating a logical product, or a greater number indicating a logical product, or the end of the asserted proposition in which the "()" or "" occurs, whichever of these happens first[5]. Thus e.g. will mean " always implies ," but will mean "if is always true, then is true for the argument ."

Note that in the proposition the two 's have no connection with each other. Since only one dot follows the in brackets, the scope of the first is limited to the "" immediately following the in brackets. It usually conduces to clearness to write since the use of different letters emphasises the absence of connection between the two variables; but there is no logical necessity to use different letters, and it is sometimes convenient to use the same letter.

Ambiguous assertion of this kind is a primitive idea, which cannot be defined in terms of the assertion of propositions. This primitive idea is the one which embodies the use of the variable. Apart from ambiguous assertion, the consideration of "," which is an ambiguous member of the values of , would be of little consequence. When we are considering or asserting "," the variable is called a "real variable." Take, for example, the law of excluded middle in the form which it has in traditional formal logic: Here and are real variables: as they vary, different propositions are expressed, though all of them are true. While and are undetermined, as in the above enunciation, no one definite proposition is asserted, but what is asserted is any value of the propositional function in question. This can only be legitimately asserted if, whatever value may be chosen, that value is true, i.e. if all the values are true. Thus the above form of the law of excluded middle is equivalent to i.e. to "it is always true that is either or not ." But these two, though equivalent, are not identical, and we shall find it necessary to keep them distinguished.

When we assert something containing a real variable, as in e.g. we are asserting any value of a propositional function. When we assert something containing an apparent variable, as in we are asserting, in the first case all values, in the second case some value (undetermined), of the propositional function in question. It is plain that we can only legitimately assert "any value" if all values are true; for otherwise, since the value of the variable remains to be determined, it might be so determined as to give a false proposition. Thus in the above instance, since we have And generally, given an assertion containing a real variable , we may transform the real variable into an apparent one by placing the in brackets at the beginning, followed by as many dots as there are after the assertion-sign.

When we assert something containing a real variable, we cannot strictly be said to be asserting a proposition, for we only obtain a definite proposition by assigning a value to the variable, and then our assertion only applies to one definite case, so that it has not at all the same force as before. When what we assert contains a real variable, we are asserting a wholly undetermined one of all the propositions that result from giving various values to the variable. It will be convenient to speak of such assertions as asserting a propositional function. The ordinary formulae of mathematics contain such assertions; for example does not assert this or that particular case of the formula, nor does it assert that the formula holds for all possible values of , though it is equivalent to this latter assertion; it simply asserts that the formula holds, leaving wholly undetermined; and it is able to do this legitimately, because, however may be determined, a true proposition results.

Definition and real variables. When the definiens contains one or more real variables, the definiendum must also contain them. For in this case we have a function of the real variables, and the definiendum must have the same meaning as the definiens for all values of these variables, which requires that the symbol which is the definiendum should contain the letters representing the real variables. This rule is not always observed by mathematicians, and its infringement has sometimes caused important confusions of thought, notably in geometry and the philosophy of space.

In the definitions given above of "" and "" and "," and are real variables, and therefore appear on both sides of the definition. In the definition of "" only the function considered, namely , is a real variable; thus so far as concerns the rule in question, need not appear on the left. But when a real variable is a function, it is necessary to indicate how the argument is to be supplied, and therefore there are objections to omitting an apparent variable where (as in the case before us) this is the argument to the function which is the real variable. This appears more plainly if, instead of a general function , we take some particular function, say "," and consider the definition of . Our definition gives But if we had adopted a notation in which the ambiguous value "," containing the apparent variable , did not occur in the definiendum, we should have had to construct a notation employing the function itself, namely "." This does not involve an apparent variable, but would be clumsy in practice. In fact we have found it convenient and possible—except in the explanatory portions—to keep the explicit use of symbols of the type "," either as constants [e.g. ] or as real variables, almost entirely out of this work.

(1) "When a propositional function can be asserted, so can the proposition that all values of the function are true." More briefly, if less exactly, "what holds of any, however chosen, holds of all." This translates itself into the rule that when a real variable occurs in an assertion, we may turn it into an apparent variable by putting the letter representing it in brackets immediately after the assertion-sign.