In the January number of MIND, Mr. MacColl accuses me of holding premisses from which it would follow that either every red-haired Englishman is a doctor, or every English doctor is red-haired. The reason why Mr. MacColl supposes me to hold these premisses (which I do not hold) is that he has failed to grasp what seems to me a vital distinction, namely that which I call the distinction between propositions and propositional functions. The nomenclature is unimportant; but what is important is to remember that what Mr. MacColl calls a proposition is usually something which I should not call a proposition, so that when my statements about propositions are interpreted as applying to what Mr. MacColl calls propositions, the result, naturally, is apt to be curious.

Let us illustrate the distinction between propositions and propositional functions by the case which Mr. MacColl chooses. He says (p. 152):

“Take the two statements ‘He is a doctor’ and ‘He is red-haired,’ each of which, observe, is a variable, because it may be true or false. Is it really the fact that one of these statements implies the other?”

It will be observed that what I affirm (as quoted by Mr. MacColl on the previous page) is, that “of any two propositions there must be one which implies the other.” It by no means follows that of any two statements one must imply the other. Mr. MacColl assumes it undecided who ‘he’ is to be, for as soon as it is decided, the statement ceases to be ‘variable.’ Now I should not call ‘he is a doctor’ the expression of a proposition unless, from the context, it is known who ‘he’ refers to, and then it is a mere abbreviation for (say) “Mr, Smith is a doctor.” This is a proposition, but it is not variable: Mr. Smith either is or is not a doctor. Of the propositions ‘Mr. Smith is a doctor’ and ‘Mr., Smith is red-haired,’ it is easy to see that one must imply the other, using the word ‘imply’ in the sense in which I use it. (That this is not the usual sense, may be admitted; all that I affirm is that it is the sense which I most often have to speak of, and therefore for me the most convenient sense.) I say that p implies q if either p is false or q is true. This is not to be regarded as a proposition, but as a definition. Now consider ‘Mr. Smith is a doctor’ and ‘Mr. Smith is red-haired.’ Four cases are possible, namely:

(1) both are true (2) both are false (3) the first is true and the second false (4) the first is false and the second true

In cases (1) and (2) each implies the other according to the above definition. In case (3) the second implies the first; and in case (4) the first implies the second. These facts are all immediate consequences of the above definition of ‘implies,’ together with the fact that a disjunction is true when either or both of its alternatives is true. Hence in all four cases, at least one of our two propositions implies the other.

It will be seen, however, that we do not always have “‘He is a doctor’ implies ‘He is red-haired,’” nor do we always have “‘He is red-haired’ implies ‘He is a doctor.’” We have always one or other (or both), but not always one, nor yet always the other. Hence, when ‘he’ is not specified, we cannot affirm “‘He is a doctor’ implies ‘he is red-haired,’” nor yet “‘He is red-haired’ implies ‘he is a doctor.’” But when ‘he’ is unspecified, ‘he is a doctor’ and ‘he is red-haired’ have no definite meaning: they are forms awaiting determination. They are what I call ‘propositional functions,’ not what I call ‘propositions.’ It is customary to suppose that ‘propositions’ are whatever is either true or false, but ‘he is a doctor’ is neither true nor false until ‘he’ is specified; for this reason, is seems undesirable to call it a proposition. I should not, however, object to Mr. MacColl’s calling it so, if he would realise that it is something different from the things that are true or false, and from the things that I call propositions.

The formal statement of the above is as follows. Let φx, ψx, be two propositional functions, i.e., expressions which, so soon as we give a definite value to x (such as ‘Mr. Smith’), become propositions. Then for every possible value of x, either φx implies ψx or ψx implies φx. But it is not in general true that either φx implies ψx for every possible value of x, or ψx implies φx for every possible value of x. Hence, if x is not, assigned, we cannot assert ‘φx implies ψx,’ and we cannot assert ‘ψx implies φx,’ because sometimes the one is true and sometimes the other, according to the value assigned to x, and we can only assert an ambiguous expression containing an undetermined x when, however, we may determine x, the resulting proposition is true as, for example, when we say ‘x is x.’¹ We thus arrive at Mr. MacColl’s division of “statements” into the three classes of certain, doubtful and impossible, but “statements” must be understood as meaning “propositional functions,” not “propositions.” A propositional function φx is certain when all its values are true, impossible when all its values are false, and doubtful when some of its values are true and some false. But the values themselves, which are “propositions,” are subject, not to this threefold division, but merely to the twofold division into true and false.

B. RUSSELL

*  Bertrand Russell, “‘If’ and ‘Imply’: A Reply to Mr MacColl,” Mind, n.s. 17, no. 66 (Apr 1908), 300-1

¹  For further explanations on the above question, see my Principles of Mathematics, pp. 12, 13, 22