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ר File: 11-2-94.TXT - 33 KB ר
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ר B.H. Slater, Nedlands - Western Australia ר
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ר *Some Wittgensteinian theses proved* ר
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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
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ר Abstract: ר
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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
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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