The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 17-Apr-1998
(7) Throughout this section I shall make use of a number of items of standard logical notation, together with some ideas that are less familiar. The signs , , , , are respectively the signs of negation [not], conjunction [and], disjunction [or], conditional [if ...then], universal quantification [for all]. The symbol ` ' [yields or, in the present context, implies or entails] stands for derivability or deducibility in the sense of classical elementary logic; the symbol ` ' [is logically equivalent to] stands for mutual derivability. The connectives , , , are understood to be operators, in the first instance, on syntactical items called statements. (It might be better to call them sentences, in order to emphasize their formal or morphological character, but it is less confusing if Popper's usage is followed.) Two statements that are not identical are distinct. Two statements that are not logically equivalent are logically distinct. Logical truths--statements that are derivable from all statements, or from none--will, laxly but conveniently, be called tautologies. There are infinitely many distinct tautologies, but of course no two are logically distinct.
(8) If X is a set of statements and , we shall say that x is a [logical] consequence of X. The set of consequences of X is written . A [deductive] theory is a set of statements closed under the operation ; that is to say, . A theory is inconsistent if it contains every statement; otherwise consistent. The connectives and and the relation may be extended in a natural way from statements to theories: if & only if . Whatever the terminology may suggest, it is of no importance to this paper whether the consequence operation is characterized syntactically (by rules of derivation) or semantically (in terms of models).
(9) If for some statement x, we call [finitely] axiomatizable. If is consistent, and has no consistent proper extension , then is maximal (or complete). If is both axiomatizable and maximal, has no proper consequences but tautologies (for if then ). In this case the theory , and by extension the statement , are called irreducible (Tarski 1936, section 4). It follows at once that has only finitely many distinct consequences if & only if where each of the is irreducible. It is plain that cannot have a finite content unless it is axiomatizable, and thus identical to for some statement x.
(10) The proof that Percival gives (§45) of the infinitude of the content of any interesting axiomatizable theory t comes straight from note 18 of (Popper 1974, Popper 1976). Percival writes:
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.
(11) More explicitly: if then . But by hypothesis . Hence , contrary to supposition. We may conclude that no element of the list implies any other, and hence that no two are logically equivalent.
(12) This result is rather general, since the requirement that there exist a denumerable sequence of statements so related to t is not a severe one. Popper himself notes that `[f]or most t's, something like a: ``the number of planets is 0'', b: ``the number of planets is 1'', and so on, would be adequate' (1974, p. 19; 1976, pp. 26f). The proof does not even require that each pair of elements of the sequence consist of contraries; only that each pair entail t. I do not know whether either of these conditions is necessary for t to have infinitely many logically distinct consequences, but there are closely related conditions that are separately both necessary and sufficient for this: (i) there exists a denumerable sequence of pairwise incompatible theories that individually do not imply t; (ii) there exists a denumerable sequence of theories that pairwise, but not individually, imply t. Since (i) is stronger than (ii), we need prove only the sufficiency of (ii) and the necessity of (i). These results are of only supplementary significance in the present context, and those who are not itching to see proofs of them are invited to resume the main line of argument in 2.4.
(13) The sufficiency of (ii) is established in much the same way as in 2.2. Instead of constructing a sequence of statements we construct a sequence of theories . (The disjunction of two theories consists of their common consequences.) Now if any one of these new theories is not axiomatizable, then t must have infinite content; for if t had only finite content, then its subtheories would have finite content too. In other words, we may assume that each of can be axiomatized by a single statement, and proceed as before.
(14) The proof of the necessity of (i) is a little trickier, and can only be sketched. First recall from 2.1 that t has only finitely many logically distinct consequences if & only if it is logically equivalent to the conjunction of a finite set of irreducible statements. It follows, not quite obviously, that t has finite content if & only if has only finitely many maximal extensions, all axiomatizable. In other words, it is necessary & sufficient for t to have infinite content that has either infinitely many axiomatizable maximal extensions, or at least one unaxiomatizable maximal extension. Now it is immediate from a theorem of Mostowski (1937, Theorem 8, p. 13; reported on p. 370 of Tarski 1936) that if a theory has even one unaxiomatizable maximal extension, then it has infinitely many maximal extensions. We may conclude that if t has infinite content, then has infinitely many maximal extensions. From this set of maximal extensions we may extract a denumerable sequence of theories individually implying . Since each such is consistent, none of them implies t, and since each is maximal, they are pairwise incompatible. In this way (i) is proved.
(15) Popper admits that the result of 2.2 is both `well known' and `trivial' (though he means trite), but suggests that it may appear rather more significant if it is phrased not in terms of logical content, but in terms of the closely related idea of informative content. He calls the informative content of a theory the class of statements that are incompatible with it; the class, that is, of statements that it excludes or forbids (Popper 1974, p. 18; 1976, p. 26). Now on the one hand it is quite plain that, since implies x if and only if is incompatible with , the informative content of a theory is infinite whenever its logical content is infinite. But on the other hand we can see that amongst the elements of the informative content of a theory will be many statements that the inventor of the theory may not have had in mind when he formulated it, and may indeed never come to appreciate. Kepler's theory, to vary slightly the example given in turn by Popper, by Bartley, and by Percival, excludes Newton's theory, at least in the presence of the assumption that the individual planets have non-zero mass. Kepler's first law, which says that the sun occupies one focus of the elliptical orbit of each planet, is corrected by Newton's theory in at least two ways: first, by introducing perturbations due to gravitational attraction among the planets; and second by insisting that, even in a one-planet system, it is not the sun itself but the centre of mass of the sun-planet complex that is at the focus of the elliptical orbit. The informative content of a theory, that is, is in some sense not present in its entirety to the inventor of the theory, and exactly the same therefore applies in the case of its logical content.
(16) Later in the section (in 2.11) and throughout section 3, I shall have cause to applaud this idea that the content of a theory is determined by what it rules out--an idea that goes back, as Percival notes, to the discussion of empirical content in §§31-35 of Logik der Forschung (Popper 1934). For the time being, if I may, I shall continue to investigate the significance of the theorem stated and proved in 2.2.
(17) Popper describes the situation in the words: `we never know what we are talking about' (1974, p. 19; 1976, p. 27). Bartley (1990) says: `we do not know what we are saying or ...what we are doing'. The point in each case is that if understanding a theory to the full requires understanding all its logical consequences, we cannot be said to understand to the full even our own creations. A similar evaluation was given by Ryle in his inaugural lecture at Oxford (1945, p. 7; 1971, p. 198): `Thus people can correctly be said to have only a partial grasp of most of the propositions they consider. They could usually be taken by surprise by certain of the remoter logical connexions of their most ordinary propositions.' As we shall see in 2.6, Ryle went on to qualify this judgement in an important way.
(18) Now I will admit that I am not averse to the main line of thought here, especially not to the idea that we often discover in our theories consequences that we never suspected. The history of remarkable theorems in Euclidean geometry, one of the most extensively studied of all mathematical theories, is evidence enough that our knowledge has an uncanny ability to surprise us; I need only cite my favourite theorem in Euclidean geometry, Morley's theorem (which says that the points of intersection of adjacent trisectors of adjacent angles of a triangle always form an equilateral triangle), or some of the theorems collected in (Evelyn et al. 1974). But though sympathetic to the general idea that we do not know half of what we know, I think that a few unsympathetic comments deserve a hearing.
(19) In the first place, note that the interpretation given to the result is dangerously strong. For since most of, or even all, the non-tautological consequences of a theory themselves have infinitely many logically distinct consequences--all those, in fact, that are not equivalent to the conjunction of some finite number of irreducible statements--we are forced to acknowledge that we do not understand properly most, or even any, of the non-tautological consequences of any theory that we hold. If understanding its consequences is what is important to understanding a theory, then we do not really understand theories at all; it is not just that our understanding is limited--it is unbegotten. I for one therefore want to hold on to the alternative idea, also endorsed by Popper (and by many others, such as Collingwood), that the real path to understanding a
theory is by way of understanding its response to the problem situation that provoked it. Bartley (1990, p. 34) lumps together these different varieties of understanding, but they deserve to be kept cleanly apart.
(20) The second point to be made is that Popper and Bartley can hardly be drawing attention only to the frailty of our subjective apprehension of the items of objective knowledge that we (and others) have constructed. There must be more to what they are insisting on than that we are not logically omniscient, that we are unable to recognise all the consequences of what we say. If that were really all that was meant, then it would be hard to see why the proof should bother to establish the (admittedly simple) point that under suitable conditions a theory t has infinitely many consequences that are logically distinct from each other. For it is even more straightforward to establish that every theory, even one that states only a tautology, has infinitely many syntactically distinct consequences; that is, that there are infinitely many distinct statements that are derivable from any logical truth. Everyone who has studied logic knows (though may not be able to prove) that there exist infinitely many syntactically distinct tautologies, indeed infinitely many distinct forms of tautologies. It would be plainly an exaggerated idealization to suggest that any logician, however skilled, could recognise all these distinct tautologies. But we ought to jib at the idea that the understanding of requires any ability to recognise all its equivalents in propositional logic (let alone all its equivalents in elementary predicate logic) as logical equivalents. Indeed, no one actually has the psychological prowess (or time) to recognise all tautologies of the simplest form . Once we reach instances of this scheme involving several million variables and many billions of pairs of parentheses we are beyond what is accessible even to the keenest brain. And it cannot be thought that Popper and Bartley imagine that there is a sharp and significant difference here between the transparent relation of logical equivalence and the opaque relation of logical implication. The mere possession of infinitely many equivalents, or of infinitely many consequences, though having psychological implications of the most banal kind, does not in itself make a theory unfathomable or ununderstandable.
(21) This, it seems to me, continues to hold even when we move away from the degenerate case of logical truth and consider theories that do indeed satisfy Popper's theorem. The statement , for example, says that there exists exactly one object. In elementary logic it has infinitely many distinct consequences, including for each natural number i greater than 1 a statement, which we call (the point of this notation will become clear in 2.10), to the effect that there do not exist exactly i objects. But it is surely an absurd conceit to maintain that this theory (plainly false as it is) is beyond our full comprehension. Ryle is worth quoting again: `though people's understanding of the propositions that they use is in this sense [the sense of grasping all logical connexions] imperfect, there is another sense in which their understanding of some of them may be nearly or quite complete' (1945, p. 7; 1971, p. 198). Indeed, if we restrict ourselves to the calculus of elementary logic with identity and no other predicates or relations, we can without too much difficulty give a characterization of all the consequences of (which does not mean that we can recognise presumptive consequences of great length). It might be thought that considerations of complexity or effectiveness could enable us to draw a useful distinction here. For within elementary logic with identity the theory is a decidable theory (Tarski et al. 1953, p. 19); there is a mechanical procedure for determining of any statement whether or not it is a consequence of the theory. Decidable theories, it might be conjectured, are understandable (even if we cannot recognise all their consequences), but undecidable theories are not. Percival briefly alludes to Gödel's theorem concerning the incompleteness of consistent recursively axiomatizable theories of arithmetic (§49), from which follows the undecidability of arithmetic, but he does not contrast undecidable theories with decidable ones. But the trouble with this suggestion is that even elementary logic is undecidable (Church), indeed undecidable in the highest degree; a solution to the decision problem is also a solution to the halting problem for Turing machines (this is the gist of the proof of Church's theorem, due to Büchi, presented in Chapter 10 of Boolos & Jeffrey 1974). On the other hand, elementary Euclidean geometry is decidable (Tarski 1948). Should we identify full intelligibility with decidability, we would therefore find ourselves back in the untenable position of saying that the tautology is beyond our understanding, and having to admit at the same time that Euclidean geometry is not. But nothing in this paragraph should be taken to deny that some more delicate deployment of ideas from recursion theory might provide real illumination of the problem.
(22) Whether our theories are fully understandable, and in what sense they have infinite content, are separate and, I have suggested, independent issues. For the remainder of this section I shall content myself with probing further into the latter problem, the technical one. I suspect that what led Popper (and, in his footsteps, Bartley) to think that the issue of the ununderstandability of our theories amounts to more than the incontrovertible psychological fact that we can be surprised by some of their consequences was something like this. The fact that most statements have infinitely many distinct consequences can easily be conflated with the claim that--as we might put it informally--they have infinitely many different things to say; that their contents consist of infinitely many separate nuggets of information, each distinct from and independent of the others. There is, that is to say, an objective sense in which a theory t, even though finitely axiomatizable, may be beyond us.
(23) I would certainly be prepared to consider this as a relevant difference. But unfortunately no such thing has been demonstrated by the proof in 2.2 above. If we look again at that proof, we shall see that although no one element of the sequence implies any other, any two elements of it are together logically equivalent to t, and therefore together imply all the others. For since a and b are incompatible by hypothesis, we have (using the distributive law):
The only sense, that is, in which the different elements of the sequence say different things is that no one says the same as another. But taken together, any two say exactly what all the others say. Moreover there is no possibility that a theory t such as the one with which we started should have an equivalent formulation in terms of an infinite sequence of statements each of which is genuinely independent of all the others. For suppose that t were equivalent to the infinite set of independent statements. By the principle of finitude (often called compactness), if t is derivable from this set, as we are assuming, it is derivable from some finite subset, say . But then:
and the set is not independent after all.
(24) Any independent axiomatization of an unaxiomatizable theory is infinite, and any independent axiomatization of an axiomatizable theory is finite. The difference is that the word `independent' can be dropped from the first assertion, but not from the second. Unaxiomatizable theories never look finite, but axiomatizable theories sometimes look infinite. My thesis is that this is something of an optical illusion, a logical hologram, an infinite- dimensional Necker hypercube; what is really a single thing is made to assume simultaneously an infinity of different guises. But once we sort out the dependences among them, the finiteness is restored.
(25) Yet the idea that infinite content implies the existence of infinitely many independent nuggets of information can be pressed a little further. Using a corollary stated (but not explicitly proved) by Popper (1966, p. 349), it is possible to establish the somewhat unexpected proposition that, although t is not equivalent to any infinite independent subset, under the conditions already propounded, it does include, within its content, such a set. This shows that there is one sense (though one that I shall claim to be unimportant) in which we can correctly assert that t does have infinitely many different things to say. Those who want to concentrate on the main problem of the paper, and those who can't be bothered with proofs, are once again permitted to move on, either to 2.10 or, if they are desperate, directly to section 3.
(26) The result of Popper's that we need is this: if the set of all true statements is not axiomatizable, and t is axiomatizable and false, then its truth content is not axiomatizable. Phrased more generally, this asserts that if is an unaxiomatizable maximal theory that does not imply t, then is an unaxiomatizable subtheory of t. It may be shown that if t has infinite content then there must exist such an unaxiomatizable that does not imply t. It follows that is not axiomatizable. Now by a theorem of Tarski (proved informally on p. 362 of 1935), every theory is logically equivalent to an independent set, and hence must be equivalent to an infinite independent set (for otherwise, as already noted, it would be finitely axiomatizable).
(27) To complete the proof we need to establish the result attributed to Popper, and to establish also that if t has infinite content then at least one unaxiomatizable maximal theory does not imply it.
(28) First suppose that does not imply t. Since it is maximal, it implies . Now if were axiomatizable, so would be its conjunction with . But
meaning that too would be axiomatizable. This proves Popper's corollary.
(29) If t has infinite content then, as noted in 2.3, has an infinite number of maximal extensions. That these cannot all be axiomatizable is part of Theorem 8 of (Mostowski 1937; Tarski 1936, p. 370). Here is the simple proof. If a theory (whether axiomatizable or not) has infinitely many axiomatizable maximal extensions , then it is consistent with each finite subset of , and hence (by the principle of finitude) is consistent. By Lindenbaum's theorem, has a maximal extension that is not identical with any element of (for maximal theories are pairwise incompatible). Thus has an unaxiomatizable extension , which cannot also be an extension of t.
(30) It may be concluded that when Popper's original assumption holds, so that there exists an infinite set of statements that pairwise are contradictory and individually do not entail t, then t includes amongst its consequences an infinite independent set. A simple and revealing example is provided by Kepler's three laws, augmented by a finite set of initial conditions sufficiently copious for the prediction of the positions of the planets at all future times. It is plain that within the content of this theory we can find an infinite set of statements that, taken in isolation, are logically independent of each other; statements of the positions of Venus at different times suffice for this, since it is only in the presence of universal laws that there is any logical connection between the position of Venus at one time and its position at any other time. But we should not be eager to conclude that there is any important sense in which Kepler's laws say infinitely many distinct things. On the contrary, one of the virtues of logical systematization and axiomatization is that it enables us to replace scattered sets of results by unifying principles. There is something suspiciously retrograde about the claim that Kepler's laws have an infinity of things to say (just as there is something retrograde about the claim that a theory may be replaced by its Ramsey sentence). We must note in any case that to obtain the full force of Kepler's laws (plus initial conditions) we should need to add to the infinite set of predictions concerning Venus (or any other infinite independent subset of the content of Kepler's laws) some statement (or finite set of statements) that renders all but finitely many of those predictions redundant.
(31) Lying behind the view that scientific theories are infinitely varied in their consequences there is, I suspect, an atomistic view of content: the idea that there exist minimal independent morsels of information from which the contents of all more informative statements and theories are compounded by finite or infinite conjunction. The reader is warned not to be misled by the use of the word `bit' in information theory to express what sounds like exactly this idea. It is not the same idea. Bits are not minimal, except in the logically insignificant sense of being representable by expressions of minimal length. Indeed, the atomistic thesis as it stands is untenable. For the only possible candidates for the role of atomic contents would be irreducible statements, and no axiomatizable theory with infinite content can be built entirely from irreducibles. This is easily shown. For a set of k irreducibles generates a theory with distinct consequences; and an axiomatizable theory that is logically equivalent to the conjunction of denumerably many irreducible statements is, by finitude, equivalent to the conjunction of some finite subset of them, so that we return to the previous case. A simple example is supplied by the calculus described in 2.6, elementary logic with identity as the only relation: for each positive i the statement is irreducible, but the conjunction of all these statements yields a theory (`the number of objects is not finite', or `the universe is infinite') that is not finitely axiomatizable.
(32) The objection may be strengthened by noting that in many calculi there exist no irreducible theories at all. This follows from the result of Mostowski already cited. An example is provided by ordinary classical sentential calculus with denumerably many sentence letters. In such calculi, of course, all non-tautological theories have infinite content.
(33) If an atomistic approach of content is possible at all, it will only be, I think, through a move away from logical content to a construe related to what in 2.4 was called informative content. If we identify the content of a theory not with `the class ...of statements that it excludes or forbids' but with the class of maximal theories (or, if you like, models or possible worlds) that it excludes or forbids, then most of the difficulties paraded above disappear. Some theories may exclude only finitely many maximal theories; if such theories exist, they will be just the same as those with finite contents--in other words, they will be equivalent to the conjunction of a finite number of irreducibles. But usually, and in some calculi always, non-tautological theories will succeed in excluding infinitely many maximal theories. The main difficulty lies in explaining in what sense maximal theories can be thought of as independent of each other; that is, in showing that if an axiomatizable theory has infinite content this is not simply a result of duplication. Plainly the sense required is not simple logical independence, since any maximal theory is implied by the conjunction of two others. And the hunch that maximal theories can never duplicate each other, or get in each other's way, and that any set of maximal theories can constitute a content, is unfortunately false. For example, in the calculus just mentioned, no theory can exclude the maximal theory unless it also excludes one (in fact, almost all) of the (this was in effect proved at the end of 2.8 above). Despite these worries, it is quite easy to defend the view that an axiomatizable theory can make an infinite number of independent exclusions, and this suffices for the claim that it has genuinely infinite content. Whether it entitles us to claim that the theory is infinitely applicable as well is quite a different matter. In section 3 it will be suggested that it does not so entitle us.
(34) I have been trying without success to find something defensible in the view that the infinitude of a theory's content has more than psychological significance; that there is an objective sense in which it is true that an axiomatizable theory must say more than we can ever appreciate. In his discussion of the syllogism Mill (1843, Book II, Chapter III, section II) rightly dismisses any attempt `to attach any serious scientific value to such a mere salvo as the distinction drawn between being involved by implication in the premises, and being directly asserted in them'. In other words, an axiomatizable theory t does directly (if not transparently) assert in finitely many words everything that its infinitely many consequences take infinitely many words to assert. Mill expresses puzzlement that `a science, like geometry, can be all ``wrapt up'' in a few definitions and axioms' (loc. cit.). But we should not allow ourselves to be taken in here. Although virtually all theories `wrap up' infinitely many thoughts in the sense that we can find infinitely many thoughts within them, it is a capital mistake to suppose that a theory's content is synthesized from logically more primitive (weaker) components. As we have seen, in many cases there are no weakest components. (Aristotle's treatment of Zeno's paradox of Achilles and the tortoise invites comparison.) I am not of course defending the view that the understanding of a rich scientific theory is a straightforward business, and that some acquaintance with the theory's consequences is not essential to its understanding. As I have already noted, I incline to the view that understanding a theory fundamentally means understanding the problem situation it addresses, and how well it addresses it. It is possible to go further and to recognise that understanding may be enhanced when it is realized that the theory solves, or is unable to solve, some unexpected, some newly emerged problem. I am quite happy to admit that newly identified consequences may lead to a sharp improvement in the understanding of a theory. My purpose here is only to question the doctrine that much can be explained by the infinitude of a theory's content alone. It is such a flimsy matter that we could hardly expect it to yield substantial returns.
The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 17-Apr-1998
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TCR Issue Timestamp: Fri Apr 17 07:52:54 GMT 1998