ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר ר ר ר File: 11-2-94.TXT - 33 KB ר ר ר ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר ר ר ר B.H. Slater, Nedlands - Western Australia ר ר ר ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר ר ר ר *Some Wittgensteinian theses proved* ר ר ר ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר ר ר ר Slater, B.H. (1994) Some Wittgensteinian theses proved; in: ר ר Wittgenstein Studies 2/94, File: 11-2-94; hrsg. von ר ר K.-O. Apel, F. Bצrncke, N. Garver, P. Hacker, R. Haller, ר ר G. Meggle, K. Puhl, Th. Rentsch, A. Roser, J.G.F. Rothhaupt, ר ר J. Schulte, U. Steinvorth, P. Stekeler-Weithofer, ר ר W. Vossenkuhl (3 1/2'' Diskette), ISBN 3-211-82655-6, ר ר ISSN 0943-5727 ר ר ר ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר ר ר ר Abstract: ר ר ר ר Wittgenstein's Tractarian point, at 3.332, not only can be ר ר substantiated rigorously, but can itself be used to prove ר ר several central anti-Platonist theses in Wittgenstein's ר ר later philosophy. It is also a central point which frustrates ר ר any fundamentalist demand for determinateness of sense, and ר ר so reveals important facts about what meaning is, and its ר ר relation to truth. Further forthright conclusions are ר ר derivable, in this manner, as well, notably in the philosophy ר ר of Arithmetic, and centrally in connection with Goedel's ר ר Theorems. These not only make the Wittgensteinian position ר ר more clear cut, but they also show that that position is quite ר ר thoroughgoing, and not at all accidental. ר ר ר ררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררררר This paper first shows that Goedel's two incompleteness results do not hold when 'it is provable that' is an operator expression, modifying a used sentence, as in the tradition of Prior and Goodstein, rather than a predicative expression, describing a mentioned sentence, as in the tradition from Tarski and Quine. It follows that Goedel's theorems do not hold in natural language, and so Wittgenstein's neglectful attitude to Goedel's results has a firm basis. Indeed Wittgenstein, even in the Tractatus, saw the essential point which needs to be made to escape the opposing tradition. The relevant paragraph (3.332) reads: No proposition can make a statement about itself, since a propositional sign cannot be contained in itself (that is the whole of the 'theory of types'). This paper goes on to discuss this point more directly in the Tractarian connection - showing how it can be substantiated without question, and thus proving one further, now explicit Wittgensteinian thesis. But the point about Goedel has met some interesting comment - notably at Cambridge, Oxford and Lincoln, Nebraska - and this has led me to realise that further Wittgensteinian theses may be similarly given a firm foundation, in direct connection with the one above. These theses may gain a more detailed, and maybe a more revealing defense elsewhere. But there can be no defense more irrevocable than that given here. Moreover, as we shall see, with a securer foundation for them, one becomes more confident about certain consequences of these theses, which otherwise might remain obscure. Now I have met people who believe there are no proofs in philosophy, and who therefore would doubt even one proof of the above sort could be given. This is perhaps even especially true of Wittgensteinians. But if there are any such sceptics in my present audience I trust they will read on. For they must, at least, realise the uncertainty of their position. Certainly they cannot believe it can be proved! So, by their own understanding, they should not be presumptive and aprioristic about the matter, but must wait to pass judgement on the offered demonstrations. *1* With respect to the proof of Goedel's results, the following theorem is provable in K4: L(p = -Lp) > (Lp > Lq). For if this statement were false in some world, then L(p = -Lp) would be true and (Lp > Lq) would be false there, and so Lp would be true and Lq false there. Hence there would have to be a further world in which q is false, and in that world both p and (p = -Lp) would be true, requiring -Lp to be true there, too. But if Lp is false in the second world, there must be a third world in which p is false, yet the accessibility relation is transitive in K4, making p true there, since Lp is true in the first world. Hence the statement cannot be false. In a similar manner the following statement is also provable in K4:L(p = -Lp) > (L-Lq > Lr).But Goedel's results hold for K4G, where G='L(Lp > p) > Lp', since there is no difficulty within that system of obtaining a 'p' for which L(p = - Lp) (Smullyan 1988). Hence, by detachment, we can get Goedel's first result, (Lp > Lq), i.e. that if p is provable then anything is (which means the system is inconsistent). And also we can get his second result, (L-Lq > Lr), i.e. that the system can prove it is consistent only if it is in fact inconsistent (proving it is consistent means proving there is something it cannot prove). Now Goedel himself realised that the natural notion of proof obeyed the axioms of S4 (=K4T) rather than those of K4G, which formalises the meta-mathematical notion he was mainly concerned with (Goedel 1969, Boolos 1979). Here we see that the S4 notion cannot be appropriately self-referential, and hence cannot provide a 'p' for which 'p'='it is unprovable that p'.But the operator reading of 'it is unprovable that p' is pursued, to show further that, in contrast to formal systems, natural language is both consistent and provably consistent from within itself, directly against the second of Goedel's results. For the self-referential antecedent L(p = -Lp), although it can be true in K4G, cannot be true even in KT, where T='Lp > p', since if it is true in some world then either p and -Lp are both true there, or both are false there. But if both are false then Lp is true, and so p is true, since the accessibility relation is reflexive in KT. On the other hand if both are true then Lp is false, which requires there to be another world, like that before, in which p is false. Either way we face a contradiction, so the statement cannot be true. But this means it cannot be true in KT4 (=S4), the system Goedel himself identified as the system for natural provability. Hence in that system we cannot detach L(p = -Lp) for any 'p' to get Goedel's incompleteness theorems. Q.E.D. But what is the grammatical reason why natural language lacks the appropriate self-reference? It lacks this because the notion of provability there is an operator, not a metalinguistic predicate, as in the cases Goedel originally studied. Provability in natural language obeys axiom T because it is of facts, not formulae, and it is the provability of facts not formulae simply because, as was indicated at the start, it is a modifier of a used sentence. Since the subordinate sentence is used in operator constructions that automatically gives it an interpretation, so it is not then just a string of symbols. And that is all that is needed to show that natural language is consistent, and provably so. For, in place of the law of non-contradiction being a predicative fact about a certain language - i.e. being of the form '-(p.-p)' is truewhere this might be said to hold contingently say of natural languages but not of some other languages, notably paraconsistent ones - its operator expression shows it is not a fact about language at all, but a fact about what is expressed in language. Properly put the law of non-contradiction says it is not both true that p and that -p,and that fact is necessary simply because its reverse is inconsistent. That shows that what is provable in natural language is consistent, and provably so within natural language, against Goedel's second result. But even more important, for understanding the irrelevance of Goedel's results, is the following grammatical fact about operators, which is what specifically prevents appropriate self-reference. For, as we shall shortly see, while 1='1 is unprovable'is possible,'p'='it is unprovable that p'is not, since the latter would require a sentence, and not just its name, to be a proper part of itself. This is the point which was realised by Wittgenstein in the Tractatus, and so it is undoubted that it lies behind Wittgenstein's later neglect of Goedel's results. For certainly, as we have seen, Goedel's results do not apply to natural language. *2* With respect specifically to the Tractatus remark 3.332, I should first make clear that I want to show merely that there is a foolproof ground for the claim that Wittgenstein makes there about propositions. The consequences for the theory of types are more debatable, though I shall later just mention some of them. More historically, whether one can go further, and show that my proof of the major assertions in 3.332 was exactly what Wittgenstein had in mind I also regard as somewhat debatable. Given the brevity of what Wittgenstein even elsewhere said, I do not think any final judgement on that matter can be reached, although I do not see, myself, what other basis he could have had than that I shall give. But showing, in any case, just that 3.332 is true is a notable advance, especially since it is a number rarely commented on, unlike its successor 3.333. Now the major premise of 3.332 is true simply because no thing can be longer than itself. This principle is enshrined in Mereology (Martin 1988), where it emerges as the axiom requiring that things not be proper parts of themselves. But it strictly has an even wider application. For no thing, either, can have any duplicate of itself as a proper part, since that duplicate then would be as long as itself. I shall refer to all such facts as 'the mereological principle'. It remains to see why the major conclusion of 3.332 is true, i.e. why no proposition can make a statement about itself. One casts around, to find the answer, for a ground for 3.332 amongst the semantic paradoxes, where it is plausible to say that propositions seem to be constructed which do talk about themselves. But one has to be rather careful, since what might first come to mind, namely such a self-referential paradox as 1 = '1 is false',of course constructs a sentence which is about a sentence, i.e. a 'propositional sign', and so again not a proposition. Moreover, there is no attempt in this paradox to make a 'propositional sign' somehow contain itself, since what '1 is false' relevantly contains is merely its own name, and in no way some form of its own self. The only rational ground for Wittgenstein's 3.332, therefore, must be, as above, the corresponding possibility - or rather impossibility - with operator expressions, e.g. it is false that p.For first, as Prior notably emphasised (Prior 1963, 1971), such an expression contains a propositional sign rather than the name of one as a proper part - since ''p'' there is used, not mentioned. In addition, it can obviously be said to be 'about' (the proposition that) p - though, note, in a quite different sense from that in which, say, '1 is false' is about 1, since '1' is a referring term, whereas 'p' (sic) is not. But while such an expression may say something about its contained proposition, p, there is no way such an expression can say something about itself. For that would require the Wittgensteinian impossibility 'p' = 'it is false that p',i.e., as before, that a propositional sign should contain not it own name, but its own self. What general relevance this has to the problems of self-reference we shall look at in section 6. But I have now achieved my second objective: to defend the major assertions of paragraph 3.332 as true, and as true unquestionably. They are true because it is operator expressions which are about propositions, and, since in operator expressions the contained sentence is used rather than mentioned, no such can be about itself, because the part contained then cannot be the whole sentence. *3* Now I am indebted, in a roundabout way, for my next two points, as I said, to some suggestions made at Cambridge and Oxford. So I want first to acknowledge this background, even though what I shall go on to do with the provided suggestions, as will be seen, is entirely my own. For when I made the above point at Cambridge, and at Oxford, I found, strangely, that there were objections to the mereological principle about parts and wholes. Thus Timothy Smiley, at Cambridge, pointed out that Goedel himself explicitly doubted the basic mereological principle, because of its conflict with certain seeming facts in Infinitary Logic (Schilpp 1944, p139). Notably Smiley restrained himself from endorsing Goedel, and confessed (in a subsequent letter) to finding Infinitary Logic meaningless (as, remarkably, Tarski did, originally (Tarski 1956, p244)). But if one does allow 'infinite sentences' which, for instance, as Smiley suggested, iterate the same element, e.g. p v p v p v... LLLL.....p,then each of these would seem to have many proper parts which are identical with the whole. For after the initial 'p v' in the first there would be a repetition of the whole and likewise after the initial 'L' in the second.Of course, this point presumes each infinity is 'actual' rather than 'potential' in Aristotle's terminology; and centrally Wittgenstein also ruled against such infinities being realities (Klenk 1976, p98). But arguments from such authorities carry no weight against a platonist such as Goedel. For a platonist it will seem that 'abstract objects' merely behave differently from 'concrete objects' - and certainly the extensive work which has been done on Infinitary Logic would seem to presume some such thing (Karp 1964). But a further platonic object which seems to defeat the basic mereological principle was presented to me at Oxford (and in subsequent letters), by Jonathan Cohen. Objecting to the general impossibility of things like'p' = 'Op',where 'O' is some operator expression, Cohen pointed out that, in the specific case where 'O' is 'it is true that', we must have the strict equivalence p = it is true that p,and therefore, it seems, the proposition that it is true that p does contain as a proper part something, namely the proposition that p, which is identical with its whole self. For by reifying the propositions in the above equivalence it may seem we can get the referential identity the proposition that p = the proposition that p is true,in which 'the proposition that p' occurs on both sides. Of course Wittgenstein's specific use of the mereological principle was about propositional signs, and not propositions. And so Cohen needs a defense of his reification of propositions before he can step from the above strict equivalence to the referential identity, and so apparently defeat the mereological principle. But believers in 'intensional objects' commonly make this kind of move, so it seems they can easily insist on their point, just like the infinitary logicians. But, of course, while someone may insist on such a point, and so reject the mereological principle, that does not show they can do so truly. For while people may define abstract objects, both of the intensional and the infinite variety, so that they simply behave differently from concrete objects, that in no way shows that there are any objects that really do so. Unicorns, to exist, would have to have well known features, but that in no way guarantees there actually are creatures with those features. So Smiley's and Cohen's suggestions now present us, by reversal, with some proofs of the non-reality of infinities, and the non-objecthood of meanings - further Wittgensteinian theses, as I promised at the beginning. *4* For, of course, the mereological principle is correct, because of the meaning of the word 'part'. We might allow a whole to be, by extension of the term, a part of itself, but there is no way it is a proper part, since proper parts expressly are parts which are not the whole. Hence there is an instant reductio of Platonism in both of the above cases. Thus if indeed the meaning of a sentence, i.e. a proposition, were an object, then, by Cohen's construction, there would be an object which was a proper part of itself. In fact all propositions would be in this category, since, for all p, the proposition that p would be identical with the proposition that it is true that p. But no thing is a proper part of itself. Hence it follows, conclusively, that the meaning of a sentence cannot be an object. And more follows even than that. For if, further, the meaning of an individual word were an object, then the meaning of a sentence could be assembled, as a larger object, just as compositional semantics requires, from the meanings of its separate words. But since the meaning of a sentence is not an object, the meanings of its constituent words cannot be objects, either. Q.E.D. Again, in the other case, if an infinitary series of objects were itself a further object then, by Smiley's construction, we would have objects which were proper parts of themselves. For taking an endless iteration of the same kind of element would have this result. But, patently, as we have seen, no objects can be proper parts of themselves. Hence an infinitary series of objects cannot itself be an object. Q.E.D. *5* In connection with this proof that infinities cannot be actual, it is not always appreciated how radical was Wittgenstein's rejection of, for instance, real numbers as objects. Indeed, he did not spell out all the consequences of this view, and it is possibly only with the firmer base for his ideas provided above that the full strength of his finitist position can start to become clear. For clearly the possibility of an endless construction of a decimal does not make possible the construction of an endless decimal. And so we must say that real numbers, in general, do not exist. But neither do infinite sets, for instance, exist. For, by parity of reasoning, there is no collection of an endless series of objects of a certain type, even if there is the endless possibility of collecting objects of that type. Thus all collections are finite, and all objects are finite, even though there is no a priori limit to the size of collections, or the number of objects. Russell would not have had his trouble with the Axiom of Infinity had the existence of an infinite set been more certain. But Russell's postulate was indeed just that, for the keener eye of the later Wittgenstein saw the postulate was, in fact, false.And it is no use pragmatists saying here, for instance, that such an axiom is 'needed', maybe to develop certain parts of Mathematics. If certain parts of Mathematics cannot be developed then that is as it should be: what is dependent on a mere presumption cannot be a necessary truth. But, as with the theory of real numbers, there is no reason to believe a priori that the theory of infinity will collapse totally, once it is appreciated clearly that there are no infinite sets. The theory may only need remodelling, as in classical epsilon-delta analysis, that is all. Pragmatists ignore the possibility that there might be some truth to the matter. Thus a pragmatist might be pleased to note that the above denial of infinite sets is all that is needed to escape Russell's Paradox, For, if '(En)(nx)Px' (c.f. Bostock 1974) says that 'P' has a finite number of instances, then, with the Abstraction Axiom as a consequence of Finitism reduced to(En)(nx)(Px) > (Ex)(y)(Py = y (En)(y)(y