Introduction to Mathematical Philosophy

WE have now explored, somewhat hastily it is true, that part of the philosophy of mathematics which does not demand a critical examination of the idea of class. In the preceding chapter, however, we found ourselves confronted by problems which make such an examination imperative. Before we can undertake it, we must consider certain other parts of the philosophy of mathematics, which we have hitherto ignored. In a synthetic treatment, the parts which we shall now be concerned with come first: they are more fundamental than anything that we have discussed hitherto. Three topics will concern us before we reach the theory of classes, namely: (1) the theory of deduction, (2) propositional functions, (3) descriptions. Of these, the third is not logically presupposed in the theory of classes, but it is a simpler example of the kind of theory that is needed in dealing with classes. It is the first topic, the theory of deduction, that will concern us in the present chapter. Mathematics is a deductive science: starting from certain premisses, it arrives, by a strict process of deduction, at the various theorems which constitute it. It is true that, in the past, mathematical deductions were often greatly lacking in rigour; it is true also that perfect rigour is a scarcely attainable ideal. Nevertheless, in so far as rigour is lacking in a mathematical proof, the proof is defective; it is no defence to urge that common sense shows the result to be correct, for if we were to rely upon that, it would be better to dispense with argument altogether, [Pg 144] rather than bring fallacy to the rescue of common sense. No appeal to common sense, or "intuition," or anything except strict deductive logic, ought to be needed in mathematics after the premisses have been laid down. Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called "intuition." The whole trend of modern mathematics, with its increased pursuit of rigour, has been against this Kantian theory. The things in the mathematics of Kant's day which cannot be proved, cannot be known—for example, the axiom of parallels. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic. What else is to belong to human knowledge must be ascertained otherwise—empirically, through the senses or through experience in some form, but not a priori. The positive grounds for this thesis are to be found in Principia Mathematica, passim; a controversial defence of it is given in the Principles of Mathematics. We cannot here do more than refer the reader to those works, since the subject is too vast for hasty treatment. Meanwhile, we shall assume that all mathematics is deductive, and proceed to inquire as to what is involved in deduction. In deduction, we have one or more propositions called premisses, from which we infer a proposition called the conclusion. For our purposes, it will be convenient, when there are originally several premisses, to amalgamate them into a single proposition, so as to be able to speak of the premiss as well as of the conclusion. Thus we may regard deduction as a process by which we pass from knowledge of a certain proposition, the premiss, to knowledge of a certain other proposition, the conclusion. But we shall not regard such a process as logical deduction unless it is correct, i.e. unless there is such a relation between premiss and conclusion that we have a right to believe the conclusion [Pg 145] if we know the premiss to be true. It is this relation that is chiefly of interest in the logical theory of deduction. In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is between the two a relation of the sort called "implication," i.e. that (as we say) the premiss "implies" the conclusion. (We shall define this relation shortly.) Or we may know that a certain other proposition is false, and that there is a relation between the two of the sort called "disjunction," expressed by " or ,"[32] so that the knowledge that the one is false allows us to infer that the other is true. Again, what we wish to infer may be the falsehood of some proposition, not its truth. This may be inferred from the truth of another proposition, provided we know that the two are "incompatible," i.e. that if one is true, the other is false. It may also be inferred from the falsehood of another proposition, in just the same circumstances in which the truth of the other might have been inferred from the truth of the one; i.e. from the falsehood of we may infer the falsehood of , when implies . All these four are cases of inference. When our minds are fixed upon inference, it seems natural to take "implication" as the primitive fundamental relation, since this is the relation which must hold between and if we are to be able to infer the truth of from the truth of . But for technical reasons this is not the best primitive idea to choose. Before proceeding to primitive ideas and definitions, let us consider further the various functions of propositions suggested by the above-mentioned relations of propositions. [32]We shall use the letters , , , , to denote variable propositions. The simplest of such functions is the negative, "not-." This is that function of which is true when is false, and false when is true. It is convenient to speak of the truth of a proposition, or its falsehood, as its "truth-value"[33]; i.e. truth is the "truth-value" of a true proposition, and falsehood of a false one. Thus not has the opposite truth-value to . [Pg 146] [33]This term is due to Frege. We may take next disjunction, " or ." This is a function whose truth-value is truth when is true and also when is true, but is falsehood when both and are false. Next we may take conjunction, " and " This has truth for its truth-value when and are both true; otherwise it has falsehood for its truth-value. Take next incompatibility, i.e. " and are not both true." This is the negation of conjunction; it is also the disjunction of the negations of and , i.e. it is "not- or not-." Its truth-value is truth when is false and likewise when is false; its truth-value is falsehood when and are both true. Last take implication, i.e. " implies ," or "if , then ." This is to be understood in the widest sense that will allow us to infer the truth of if we know the truth of . Thus we interpret it as meaning: "Unless is false, is true," or "either is false or is true." (The fact that "implies" is capable of other meanings does not concern us; this is the meaning which is convenient for us.) That is to say, " implies " is to mean "not- or ": its truth-value is to be truth if is false, likewise if is true, and is to be falsehood if is true and is false. We have thus five functions: negation, disjunction, conjunction, incompatibility, and implication. We might have added others, for example, joint falsehood, "not- and not-," but the above five will suffice. Negation differs from the other four in being a function of one proposition, whereas the others are functions of two. But all five agree in this, that their truth-value depends only upon that of the propositions which are their arguments. Given the truth or falsehood of , or of and (as the case may be), we are given the truth or falsehood of the negation, disjunction, conjunction, incompatibility, or implication. A function of propositions which has this property is called a "truth-function." The whole meaning of a truth-function is exhausted by the statement of the circumstances under which it is true or false. "Not-," for example, is simply that function of which is true when is false, and false when is true: there is no further [Pg 147] meaning to be assigned to it. The same applies to " or " and the rest. It follows that two truth-functions which have the same truth-value for all values of the argument are indistinguishable. For example, " and " is the negation of "not- or not-" and vice versa; thus either of these may be defined as the negation of the other. There is no further meaning in a truth-function over and above the conditions under which it is true or false. It is clear that the above five truth-functions are not all independent. We can define some of them in terms of others. There is no great difficulty in reducing the number to two; the two chosen in Principia Mathematica are negation and disjunction. Implication is then defined as "not- or "; incompatibility as "not- or not-"; conjunction as the negation of incompatibility. But it has been shown by Sheffer[34] that we can be content with one primitive idea for all five, and by Nicod[35] that this enables us to reduce the primitive propositions required in the theory of deduction to two non-formal principles and one formal one. For this purpose, we may take as our one indefinable either incompatibility or joint falsehood. We will choose the former. [34]Trans. Am. Math. Soc., vol. XIV. pp. 481-488. [35]Proc. Camb. Phil. Soc., vol. XIX., i., January 1917. Our primitive idea, now, is a certain truth-function called "incompatibility," which we will denote by . Negation can be at once defined as the incompatibility of a proposition with itself, i.e. "not-" is defined as "." Disjunction is the incompatibility of not- and not-, i.e. it is . Implication is the incompatibility of and not-, i.e. . Conjunction is the negation of incompatibility, i.e. it is . Thus all our four other functions are defined in terms of incompatibility. It is obvious that there is no limit to the manufacture of truth-functions, either by introducing more arguments or by repeating arguments. What we are concerned with is the connection of this subject with inference. [Pg 148] If we know that is true and that implies , we can proceed to assert . There is always unavoidably something psychological about inference: inference is a method by which we arrive at new knowledge, and what is not psychological about it is the relation which allows us to infer correctly; but the actual passage from the assertion of to the assertion of is a psychological process, and we must not seek to represent it in purely logical terms. In mathematical practice, when we infer, we have always some expression containing variable propositions, say and , which is known, in virtue of its form, to be true for all values of and ; we have also some other expression, part of the former, which is also known to be true for all values of and ; and in virtue of the principles of inference, we are able to drop this part of our original expression, and assert what is left. This somewhat abstract account may be made clearer by a few examples. Let us assume that we know the five formal principles of deduction enumerated in Principia Mathematica. (M. Nicod has reduced these to one, but as it is a complicated proposition, we will begin with the five.) These five propositions are as follows:— (1) " or " implies —i.e. if either is true or is true, then is true. (2) implies " or "—i.e. the disjunction " or " is true when one of its alternatives is true. (3) " or " implies " or ." This would not be required if we had a theoretically more perfect notation, since in the conception of disjunction there is no order involved, so that " or " and " or " should be identical. But since our symbols, in any convenient form, inevitably introduce an order, we need suitable assumptions for showing that the order is irrelevant. (4) If either is