46 The following quotation is from R. H. S. Grossman, Plato To-Day 1937), pp. 71 f.
A very similar doctrine is expressed by M. R. Cohen and E. Nagel in their book, An Introduction to Logic and Scientific Method (1936), p. 232: ‘Many of the disputes about the true nature of property, of religion, of law,.. would assuredly disappear if the precisely defined equivalents were substituted for these words.’ (See also notes 48 and 49 to this chapter.)
The views concerning this problem expressed by Wittgenstein in his Tractatus Logico-Philosophicus (1921/22) and by several of his followers are not as definite as those of Grossman, Cohen, and Nagel. Wittgenstein is an anti-metaphysician. ‘The book’, he writes in the preface, ‘deals with the problems of philosophy and shows, I believe, that the method of formulating these problems rests on the misunderstanding of the logic of our language.’ He tries to show that metaphysics is ‘simply nonsense’ and tries to draw a limit, in our language, between sense and nonsense: ‘The limit can .. be drawn in languages and what lies on the other side of the limit will be simply nonsense.’ According to Wittgenstein’s book, propositions have sense. They are true or false. Philosophical propositions do not exist; they only look like propositions, but are, in fact, nonsensical. The limit between sense and nonsense coincides with that between natural science and philosophy: ‘The totality of true propositions is the total natural science (or the totality of the natural sciences).—Philosophy is not one of the natural sciences.” The true task of philosophy, therefore, is not to formulate propositions; it is, rather, to clarify propositions: ‘The result of philosophy is not a number of “philosophical propositions”, but to make propositions clear.’ Those who do not see that, and propound philosophical propositions, talk metaphysical nonsense.
(It should be remembered, in this connection, that a sharp distinction between meaningful statements which have sense, and meaningless linguistic expressions which may look like statements but which are without sense, was first made by Russell in his attempt to solve the problems raised by the paradoxes which he had discovered. Russell’s division of expressions which look like statements is three-fold, since statements which may be true or false, and meaningless or nonsensical pseudo-statements, may be distinguished. It is important to note that this use of the terms ‘meaningless’ or ‘senseless’ partly agrees with ordinary use, but is much sharper, since ordinarily one often calls real statements ‘meaningless’, for example, if they are ‘absurd’, i.e. self-contradictory, or obviously false. Thus a statement asserting of a certain physical body that it is at the same time in two different places is not meaningless but a false statement, or one which contradicts the use of the term ‘body’ in classical physics; and similarly, a statement asserting of a certain electron that it has a precise place and momentum is not meaningless—as some physicists have asserted, and as some philosophers have repeated—but it simply contradicts modern physics.)
What has been said so far can be summed up as follows. Wittgenstein looks for a line of demarcation between sense and nonsense, and finds that this demarcation coincides with that between science and metaphysics, i.e. between scientific sentences and philosophical pseudo-propositions. (That he wrongly identifies the sphere of the natural sciences with that of true sentences shall not concern us here; see, however, note 51 to this chapter.) This interpretation of his aim is corroborated when we read: ‘Philosophy limits the .. sphere of natural science.’ (All sentences so far quoted are from pp. 75 and 77.)
How is the line of demarcation ultimately drawn? How can ‘science’ be distinguished from ‘metaphysics’, and thereby ‘sense’ from ‘nonsense’? It is the reply given to this question which establishes the similarity between Wittgenstein’s theory and that of Grossman and the rest. Wittgenstein implies that the terms or ‘signs’ used by scientists have meaning, while the metaphysician ‘has given no meaning to certain signs in his propositions’; this is what he writes (pp. 187 and 189): ‘The right method of philosophy would be this. To say nothing except what can be said, i.e. the propositions of natural science, i.e. something that has nothing to do with philosophy: and then always when someone else wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions.’ In practice, this implies that we should proceed by asking the metaphysician: ‘What do you mean by this word? What do you mean by that word?’ In other words, we demand a definition from him: and if it is not forthcoming, we assume that the word is meaningless.
This theory, as will be shown in the text, overlooks the facts (a) that a witty and unscrupulous metaphysician every time he is asked, ‘What do you mean by this word?’, will quickly proffer a definition, so that the whole game develops into a trial of patience; (b) that the natural scientist is in no better logical position than the metaphysician; and even, if compared with a metaphysician who is unscrupulous, in a worse position.
It may be remarked that Schlick, in Erkenntnis, I, p. 8, where he deals with Wittgenstein’s doctrine, mentions the difficulty of an infinite regress; but the solution he suggests (which seems to lie in the direction of inductive definitions or ‘constitutions’, or perhaps of operationalism; cp. note 50 to this chapter) is neither clear nor able to solve the problem of demarcation. I think that certain of the intentions of Wittgenstein and Schlick in demanding a philosophy of meaning are fulfilled by that logical theory which Tarski has called ‘Semantics’. But I also believe that the correspondence between these intentions and Semantics does not go far; for Semantics propounds propositions; it does not only ‘clarify’ them.—These comments upon Wittgenstein are continued in notes 51-52 to the present chapter. (See also notes 8 (2) and 32 to chapter 24; and 10 and 25 to chapter 25.)
47 It is important to distinguish between a logical deduction in general, and a proof or demonstration in particular. A proof or demonstration is a deductive argument by which the truth of the conclusion is finally established; this is how Aristotle uses the term, demanding (for example, in Anal. Post., I, 4, pp. 73a, ff.) that the ‘necessary’ truth of the conclusion should be established; and this is how Carnap uses the term (see especially Logical Syntax, § 10, p. 29, § 47, p. 171), showing that conclusions which are ‘demonstrable’ in this sense are ‘analytically’ true. (Into the problems concerning the terms ‘analytic’ and ‘synthetic’, I shall not enter here.
Since Aristotle, it has been clear that not all logical deductions are proofs (i.e. demonstrations); there are also logical deductions which are not proofs; for example, we can deduce conclusions from admittedly false premises, and such deductions are not called proofs. Non-demonstrative deductions are called by Carnap ‘derivations’ (loc. cit.). It is interesting that a name for these non-demonstrative deductions has not been introduced earlier; it shows the preoccupation with proofs, a preoccupation which arose from the Aristotelian prejudice that ‘science’ or ‘scientific knowledge’ must establish all its statements, i.e. accept them either as self-evident premises, or prove them. But the position is this. Outside of pure logic and pure mathematics nothing can be proved. Arguments in other sciences (and even some within mathematics, as I. Lakatos has shown) are not proofs but merely derivations.
It may be remarked that there is a far-reaching parallelism between the problems of derivation on the one side and definition on the other, and between the problems of the truth of sentences and that of the meaning of terms.
A derivation starts with premises and leads to a conclusion; a definition starts (if we read it from the right to the left) with the defining terms and leads to a defined term. A derivation informs us about the truth of the conclusion, provided we are informed about the truth of the premises; a definition informs us about the meaning of the defined term, provided we are informed about the meaning of the defining terms. Thus a derivation shifts the problem of truth back to the premises, without ever being able to solve it; and a definition shifts the problem of meaning back to the defining terms, without ever being able to solve it.