The Critical Rationalist Vol. 01 No. 02 ISSN: 1393-3809 26-Nov-1996
(39) To recapitulate, World 3 is the realm of abstract, objective products of the human mind: theories, logical relationships, numbers, symphonies. Its content includes a diverse mixture from the humble Acheulian axe to the magnificent constructions of modern mathematics such as Godel's proof of the incompleteness of arithmetic. The point I wish to stress here is that World 3 is a factor of economic production that cannot be reduced to Worlds 1 or 2, and whose contribution is quite distinct.
(40) Suppose I make the following inference on paper: If it is cloudy, it will rain; it is cloudy, therefore it will rain. On Popper's model there are three aspects to this inference. There are the marks on the paper, which constitute the physical embodiment of the logical relationship between the premises and the conclusion; there is the psychological process of inference that I make (or go through) in grasping the logical relationship together with the ability to make the inference again; and there is the actual logical relationship as such.
(41) It is important to stress that World 3 makes a difference to our thought and therefore economic production. For example, the existence of logical standards makes a difference to our thought. Even a fallibilist can allow that we sometimes get things right, and not only that, but also that we get things right because they are right. You believe that you are reading an article in English (at least partly) because you are. Many people will accept the validity of the above argument (at least partly) because it is valid; they will also reject other arguments as invalid (at least partly) because they are invalid. You may include training in logic etc. as other contributing factors in our discrimination between valid and invalid inferences, but this does not make validity and invalidity causally impotent, because validity cannot be defined in terms of training or dispositions to discriminate. The logical contradiction between say, Newton's theory and Einstein's theory of gravitation, made a difference to the way scientists think.
(42) World 3 has the following properties:
(43) One of Popper's arguments for the reality of World 3 is that our grasp of a World 3 object (such as the natural numbers) can affect our interaction with the physical world and that it is common sense that only real things can affect a physical object. (However, it is wrong to see the affect on physical objects as "constituting" the reality of World 3 objects.) An architect's understanding of mathematics makes a difference to the buildings that he can construct or would even consider constructing, but such an understanding would hardly exist if the mathematics had not first been invented. Hence World 3 makes a difference to economic production. But how much of a difference can it make? To answer this you need to look closely at its infinite content.
(44) Part of the content of World 3 will always remain in principle unfathomable. World 3 has been likened to a library of human knowledge, but although this is a good metaphor as far as it goes it is misleading. When a theory is created it is written down in a book and some of its implications may even be worked out and also written down. This is the part of the theory that becomes represented in a physical form. Think of all the future worked-out implications of the theory. This is still only a part of the theory's content, the rest is the part that never gets represented in physical form. There is always a residual because the content is infinite. There are two ways of bringing out the infinite content of a theory: by talking of the information content and the logical content. The logical content of a theory is the class of all the (nontautological) consequences that can be logically derived from the theory (it may be identified with Tarski's "consequence class").
(45) The argument for the infinite logical content of a theory t can be put thus. Suppose an infinite list of statements that are pair-wise contradictory and which individually do not entail t: Then the statement "t or a or both" follows from t. The same holds for each and every one of the statements in the infinite list. Since the statements in the list are pair-wise contradictory one can infer that none of the statements "t or a or both", "t or b or both" etc., is interderivable. Thus the logical content of t must be infinite.
(46) This in itself is not so important, but when combined with the idea of information content, the two notions produce some very interesting ramifications. In the Logic of Scientific Discovery (Popper 1980), Popper put forward the idea that a statement says more the more it forbids. Carnap, accepting Popper's suggestion, defined the assertive power of a sentence as the class of possible cases it excludes (Carnap 1942, p. 151). Later Popper reformulated the intuitive idea in terms of theories, of both high and low universality (Popper 1974, see esp. note 15). The information content is the class of all those statements that are logically incompatible with the given theory. Thus since Einstein's theory contradicts Newton's theory, Einstein' theory is part of the information content of Newton's theory. Newton could hardly have known this, and so it could not have been part of his psychology.
(47) Most philosophers resist this analysis because of their adherence to psychologism and conceptual analysis. But Popper's analysis reveals that much of interest is to be discovered in the analysis of theories as such, considered as objective entities, and their logical relationships with one another.
(48) As Popper shows, when we combine this result with the idea of logical content we obtain a parallel result, for if E is part of the information content of N then Non E is part of N's logical content. Thus both the logical and information content of theories consist of an infinite number of non-trivial consequences. As Popper says, it follows that the task of understanding a theory is infinite. Furthermore, there are an infinite number of unknown theories that form part of the information content and logical content of Newton's theory, and indeed of any empirical theory. Since the concept of knowledge is often quite restrictively defined, it might be better to speak in more general terms of "representations", and say more generally that Newton's mind obviously did not have a representation of Einstein's theory or its negation, let alone most of the other theories that are part of its information or logical content.
(49) Of course, one can also mention the work of Gödel in this connection as pointing to the unfathomable content of our theories, and therefore of our inventions and resources. It can be shown that the complete system of all true propositions in the arithmetic of numbers is not axiomatizable and is undecidable. This means that there will always be an infinite number of unsolved problems in arithmetic. I wanted to focus on the ideas of logical and information content because these are rarely discussed or applied to new interesting areas. The exception has been the late Professor Bartley, who has applied these ideas in a fascinating way to art criticism and the Marxian idea of alienation (Bartley III 1990).
The Critical Rationalist Vol. 01 No. 02 ISSN: 1393-3809 26-Nov-1996
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TCR Issue Timestamp: Tue Nov 26 17:14:18 GMT 1996