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solution
THE UNIFORM SOLUTION
I show in this paper that there is a uniform solution of the paradoxes of Self-Reference, but what has to be given up is not the Law of Non-Contradiction, as some believe; instead it is the Tractarian demand for determinateness of sense which must go.
The point is illustrated with the Heterologicality Paradox. Irving Copi derived the contradiction, but he made the one telling assumption - that 'Het' is univocal - without proceeding to derive its negation using the Reductio which is available. A similar need just to allow for ambiguity, to solve the Liar and related Paradoxes, was exposed by Charles Sayward. And that also solves Curry's Paradox, which some classify differently. With Berry's Paradox one must remember non-attributive referential terms, so 'the least number definable in less than 19 words' does not need to refer to a least number definable in less than 19 words, making its reference indeterminate. And the presupposition omitted from before the Abstraction Scheme, whose denial provides an escape from the set-theoretic paradoxes, is that the ensemble determined by 'P' has a determinate number of elements, as it doesn't if 'P' is a mass term. Mary Tiles realises, with Cantor, that this solves Cantor's Paradox, but she does not look at Cantor's diagonal procedure in the same light. Properly seen, however, that involves just another self-referential paradox, to be eliminated in the same way as above.
1. Introduction
Graham Priest has done a great service to the logical community in showing that the bulk of the Paradoxes of Self-Reference have a common structure (Priest [14]). He has also proposed, as a result, that these paradoxes ought to have a uniform solution, and tried to show that his own 'dialethism' fills that bill (Priest [15]). I show in this paper that there is indeed a uniform solution of the paradoxes in question, but what has to be given up is not the Law of Non-Contradiction, for reasons which will be stated. Instead it is the Tractarian demand for determinateness of sense (Wittgenstein [28], §3.23) which must go. Previously presumed certainties, integral to the generation of the above paradoxes, all had a form of this demand as a pre-supposition, which is what must be given up.
2. Indeterminate Sense
We obtain this result simply by being logical; specifically, in certain cases, just by making explicit the assumption(s) on which some conclusion is based.
The failure of the Satisfaction Scheme, for instance, is proved by the Heterologicality Paradox, since Priest assumes only that ([14], p31). And Copi's derivation of the contradiction (e.g. Copi [7], p301) makes explicit the more limited assumption that 'Het' is univocal, which is centrally what must go.
One interesting point here, which I shall dwell on later, is that Copi does not proceed to deduce this using the Reductio which is available. Why was that? Copi did not just study logic but also practised it - his undergraduate texts were in continuous use by many generations of students. Moreover, Copi was active before the current 'tortoise generation' was hatched, so he would not have wanted Reductio to be justified. It was well acknowledged in his time that all deduction is circular, so there could be no logical derivation of this rule which did not beg the question. Neither could he have been persuaded that contradictories could be both true. He well knew that if two of anything are both true they are at best subcontraries, by definition ([7], p67), which is what immediately shows the Law of Non-Contradiction is not at fault (Slater [24], Beziau [1]). You might as well say that deeper thinking has revealed that, contrary to previous belief, some composite numbers turn out to be prime.
So why did Copi not draw the appropriate conclusion from his deduction of the contradiction in the Heterologicality case? Why did he not see that the paradox immediately showed that his sole assumption was false, i.e. that 'Het' lacks a determinate sense? Well, nothing remains to be said except: Copi was being illogical...
A similar need just to be logical, to solve the Liar Paradox, was exposed in Sayward [16], though the point had been around since Goodstein [9]. For the failure of the T-Scheme is also shown by Priest's proof, now of the Liar Paradox, since he assumes only that ([14], p31). Under what conditions is Tarski's T-scheme
s is true iff p
true? It is true, of course, if the sentence named 's' is translated 'p' - but then, why isn't this presumption made explicit? The logical truth is that
if s unambiguously means that p, then s is true iff p,
which immediately resolves the many paradoxes involving Truth, in the same manner as above (see, e.g. Slater [18], [21]). In fact it also resolves Curry's Paradox, which Priest classifies differently. The central question for Tarskians is thus why they take to be necessary something which is plainly contingent, and whose very contingency removes the paradoxes that have bedevilled them.
3. Indeterminate Reference
So we have solved one class of semantic paradoxes quite easily: we have found that they presume the determinacy of sense of some predicate, or sentence. More tricky is formalising the presumption in the other semantic paradoxes which concern definition, and involve individual terms. Where is the indeterminacy in this case? Is it an indeterminacy of sense, or of reference?
Some individual term cases have been appropriately dealt with before: Burali-Forti's Paradox, for instance, speaks of 'the largest ordinal' when there is no such limit or totality, and that is covered in Slater [22], p353. But in Berry's Paradox, for instance, there may be some talk about 'the least number not definable in less than 19 words', where, since that phrase itself contains less than 19 words, we seem to have a contradiction. And a further problem arises then.
The further problem lies in the notion of definition when applied to a referring phrase. Thus 'the man drinking martini' may well single out the sole man in the context who is drinking martini - but also it may not for, as Donnellan showed, when used referringly and non-attributively such a phrase may not describe the object it is about. In connection with 'the least number not definable in less than 19 words', therefore, the first thing to check is whether what it refers to is indeed the least number not so definable, especially since there is a temptation to say that number is definable in less than 19 words, because of the length of the phrase. Now the length of the phrase is indisputable, but since that fact makes it non-attributive in its application what it refers to is not 'defined' by the phrase, any more than 'the man drinking a martini' defines a man, when used just referringly. That being so we should not say that the least number not definable in less than 19 words is 'definable' in less than 19 words, which resolves Berry's explicit Paradox.
Of course one could replace 'definable' in the phrase with a bare 'referrable to' and then it might seem that the paradox would reappear in another guise: the least number not referrable to in 19 words is clearly referrable to in less than 19 words. But now Donnellan's Distinction comes into its own (see also Slater [17]): for there is no paradox in the man with martini in his glass having no martini in his glass - once one appreciates the difference between reference and attribution.
Now one trouble here has been formalising such points, which first requires finding appropriate symbols for referential terms. But it has been shown, in many publications (e.g. Slater [20]), that Hilbert's epsilon terms do that job: unlike Russell's iota terms, they are complete symbols for individuals, which means that, although they are always referential, they are not necessarily attributive, and when non-attributive they refer arbitrarily to something in the universe at large. Specifically, the epsilon symbol (here 'e') symbolises the term 'that' in such expressions as 'that ordinal', or 'that y for which Oy'. It is then contingent whether O(that y for which Oy), and when this is false the epsilon term has an indefinite reference.
In Berry's Paradox, for instance, as derived in Priest 1994, there occurs the line
lon(x) is not in x,
which is meant to formalise the supposedly evident truth that the least ordinal not in a set is not in that set. Reading 'lon' as 'the least ordinal not in' Priest ([14], p29) wants to say that
lon(x) is not in x
holds for all sets x, and so, in particular, it holds for
lon(DN19) is not in DN19,
where DN19 is the set of finite ordinals definable in less than 19 words. But he also wants to say
lon(DN19) is in DN19,
since 'the least ordinal not in the set of finite ordinals definable in less than 19 words' defines a finite ordinal in less than 19 words - because of its length.
Now certainly any least ordinal not in a set is not in that set, i.e. it is necessary that
(x)(y)([Oy.y is not in x.(z)((Oz.z is not in x) only if y is less than or equal to z)] only if y is not in x).
But that does not employ a referential term 'the least ordinal'. So if we want to make a referential remark of the kind Priest had in mind, we must say, instead
(x)(ey[Oy.y is not in x.(z)((Oz.z is not in x) only if y is less than or equal to z)] is not in x),
but that is then no longer a universal truth, and so its instantiation to DN19 does not follow.
What would be a universal truth, following from the previous one, would be
(x)([Oa.a is not in x.(z)((Oz.z is not in x) only if a is less than or equal to z)] only if a is not in x),
where a=ey[Oy.y is not in x.(z)((Oz.z is not in x) only if y is less than or equal to z)]. But there is now no problem with
lon(DN19) is in DN19,
since there is now no requirement to also affirm the reverse. For the denial of the antecedent of this last conditional, in specific cases, is quite possible, allowing a way out from Berry's Paradox through the associated epsilon term not being attributive, and so not having a determinate referent. Every ordinal is then 'definable' using the associated epsilon term in less than 19 words, and so there is no determinate least ordinal not in the set so 'definable'. The epsilon term, in other words, when non-attributive functions as a kind of demonstrative, see Slater [19], p211, also, for instance, [22], p350.
Priest may have thought to avoid such demonstratives, since he says ([14], p28) that 'something is definable iff there is a (non-indexical) noun phrase that refers to it'. But the trouble is that that does not guarantee that the referring expression 'the least ordinal not in the set of those definable in less than 19 words' functions attributively. And, in fact, Priest himself wants to say, as above, that the least ordinal not in the set of those definable in less than 19 words is in the set of those definable in less than 19 words, which expressly makes the referring expression non-attributive. What Priest should do here is modify the above definition of 'definable'. For, as before, it would be preferable to say that something was defined by a referring phrase only if that referring phrase was attributively correct. But if the latter is the chosen usage then, again, we would have to withdraw from saying 'the least ordinal not definable in less than 19 words is definable in less than 19 words', so that the paradox would be resolved the other way. For that would allow the attributive condition to be true.
4. Indeterminate Number
So what forces the paradoxes of Set Theory? What presumption is omitted from before the Abstraction Scheme
(Ey)(x)(Px iff x belongs to y)
which, when made explicit, will allow its denial to provide an escape from paradox? We have seen the presupposition of determinate sense, or determinate reference has to be made explicit in the case of the semantic paradoxes, what is the comparable presupposition which ought to be expressed in the case of the Set-Theoretic ones?
The presupposition in Set Theory is that
(P)(En)(nx)Px.
i.e. that, for each P, there is a determinate number, such that there is that number of P's. But that implies that all predicates are count terms. There are also mass terms: if 'P' is a mass term, we can maybe only say
(Ey)(x)(Px iff x is a part of y),
where y is a mereological sum (Bunt [6], see especially pp262-3). In fact, as Lewis has shown (Lewis [12]), Set Theory can also be formulated in mereological terms, so the appropriately conditionalised Abstraction Scheme becomes:
If (En)(nx)Px, then (Ey)(x)(Px iff {x} is a part of y).
The impossibility on the right hand side in the case of Russell's Paradox, and the like, then only means that there isn't a determinate number of the associated P's.
If (En)(nx)Px, and so 'P' is count, then (mx)Px, where m=en(nx)Px, since in general we have: (Ex)Fx if and only if FexFx. In the reverse case (mx)Px is false ([19], p215), and the epsilon term has an indefinite reference as a result. In both cases m is (referentially) the number of P's, but only in the first case is it (attributively) the number of P's. The indefinition in the second case relates to the fact that, when one is dealing with an amount of stuff a number can only be assigned relative to some unit - gram, ampere, etc. For a treatment of numbers and amounts which respects this see Bostock [4], [5].
Now parallel to Set Theory it has recently been pointed out there is monadic second order logic, which Boolos has interpreted in terms of plural quantification (Boolos [3], Chs 3, 4). And for count terms, we can invariably say
(Ev)(x)(Px iff x is one of v),
where 'v' is a plural variable . But because of this it turns out that there are cases where 'P' is count, and so we can talk of 'the P's', but the P's do not form a set, as in the case of Russell's Paradox ([3], p66). As a result, what has also been commonly presupposed by Set Theory is not just that, for any given 'P', there are elements to be counted, but also that those elements are not just referrable to severally.
In fact, by definition a set consists in some discrete individuals which form a further individual (Black [2]). But that means that when there are no discrete items, or some discrete items which cannot be collectively given, there is not a 'set', but just 'a continuum', or 'those items', respectively. In the former case we ask 'how much?' not 'how many?', because a proportion is involved, not a cardinal number. In the latter case we talk about 'them', not 'it', since no collective reference, only plural reference is possible.
The fact that some discrete items might lack a determinate number, this being connected with the possibility of them being given as a complete whole, was, of course, the traditional, Aristotelian point of view, which Intuitionists, more recently, have still held to. But many others now doubt this fact. Is there any way to show that Aristotle was right? I believe there is.
For when discrete items do clearly collect into a further individual, and we have a finite set, then we determine the number in that set by counting. But what process will determine what the number is, in any other case? The newly revealed independence of the Continuum Hypothesis shows there is no way to determine the number in certain well known infinite sets. Is the number of the real numbers greater or less than the number of sets of natural numbers? It appears there is no way to determine an answer - save by postulation, maybe. But is there even a determinate number in any infinite set? Is the number of real numbers even greater than the number of natural numbers? If discrete items can lack a number, and the proper analysis of continua is in terms of amounts, then there is no basis for comparison, since that requires determinate numbers on each side. The key question therefore is: if there is a determinate number of natural numbers, then by what process is it determined? Replacing 'the number of natural numbers' with 'Aleph zero' does not make its reference any more determinate. The natural numbers can be put into one-one correspondence with the even numbers, it is well known, but does that settle that they have the same number? We have equal reason to say that they have a different number, since there are more of them. So can we settle the determinate number in a set of discrete items just by stipulation?
No! For postulation and stipulation are arbitrary processes. The only way we have to settle the number in any set is by counting. That is why the last number reached in counting is invariably the number in a set, with the consequence that items without an end, like '1, 2, 3, ...', have no determinate number (Kaufmann [11], p114). Hence emerges why it was always thought that for there to be a determinate number of P's, there must be a finite number of them: such items as the natural numbers cannot be completely given, so while they are countable, they cannot be counted. The rigorisation of the calculus in the nineteenth century followed Aristotle, and rejected 'actual infinities', in place of 'potential infinities', in line with this - it was, in fact, an earlier way of separating Boolos' 'it' from 'them'. But 'Set Theory' traditionally has made no such distinctions: it has conceived the continuum to involve just a very large, discriminable number, and indeed presumed all predicates to have a determinate number of individuals in their extension. That is the general form the demand for determinacy of sense took in this case, and as we can now see, that is again a demand which logically cannot be met. The Abstraction Scheme is contingent, like the Satisfaction Scheme, and the T-Scheme, and the assumption that some referential term is attributive. And just as in those cases, problems with the Abstraction Scheme are immediately resolved through denying the comparable determinateness presumption on which this contingent statement rests. The logical truth is the Abstraction Scheme conditionalised as before, but also there is a determinate number of P's only if there is a finite number of them which can be determined by counting - this latter fact takes care of the paradoxes associated with infinity.
Why didn't the set-theoreticians, at the turn of the twentieth century, allow for the alternative possibilities, and instead merely presumed that P's always formed into a totality - even that 'P' could always be pluralised? From where came the presumption that there was always a determinate number of P's? Even Cantor realised that Cantor's Paradox shows there is certainly not a determinate number of things in total. For if there were a determinate number of things in all (sic), then not only would the universe contain its own power set - because it contains everything - but also, contradictorily, the number of things in that power set would be larger than the number of things in the universe - by Cantor's Theorem. So here, at least, even Cantor realised that the now-inserted, determinate-number precondition must be supplied, entirely in line with the uniform presuppositions of determinateness, in the other cases before.
5. Non-denumerability
Now Mary Tiles saw many of the above facts about Set Theory. She realised it was crucial to Finitism to distinguish continuous from discrete wholes (Tiles [27], pp68, 151); and she realised, with Cantor, that Cantor's Paradox demonstrates the possibility of infinities with no number ([27], pp115-6). But she was still inclined to stay with Cantor's belief that at least some infinities are numerable. She said ([27], pp97-8, see also p63):
Indeed, if all infinite sets could be put into one-one correspondence with each other, one would be justified in treating the classification 'infinite' as an undifferentiated refusal of numerability. But given Cantor's discovery that there are infinite sets which cannot be put into one-one correspondence with each other, this conclusion is less compelling.
But, first, as Boolos suspected, Hume's Principle is false ([3], Chs 13, 19, see especially pp306-7). Boolos pointed out that no justification has been given for either the continuum, or the natural numbers themselves having a number. 'The present king of France strikes again' he said ([3], p205), and that would hold even if in some story it is said that he has founded a dynasty, under the novel name 'Alphonso Zero'.
But we can say more than this: for no justification for saying the natural numbers have a number can be given. For Dedekind defined infinite sets as those that could be put into one-one correlation with proper subsets of themselves, so the criteria for 'same number' bifurcate: if any two such infinite sets were numerable, then while, because of the correlation, their numbers would be the same, still, because there are items in the one not in the other, their numbers would be different. Hence such 'sets' are not numerable, and one-one correlation does not equate with equal numerosity, as Hume's Principle supposes. Cantor offered several proofs that there is no one-one correlation between the real numbers and the natural numbers, but only the presumption that there are infinite numbers can turn whatever impossibility there is here into a seeming demonstration that the number of the real numbers is greater than the number of natural numbers. Skolem's Paradox brings this latter result into some doubt, but the facts about proper parts entirely defeat the presumption on which it is based (Slater [23]).
And what continua have in place of a number of points is most easily seen in a correlation between two of them. Two parallel lines, for instance, one maybe double the length of the other, would be said by Cantor to contain the same number, or power of points, as seemingly can easily be seen by drawing a projection of the one line onto the other. But what is demonstrated by such a projection is not that, mysteriously, there is the same number, or power of things in each line, but simply that on each line there is the same proportion of it as on the other. Ratios and proportions aren't points in the first place (c.f. Bostock 1979). They are not even vanishingly small, or theoretical points, and in any numerable sense of 'point', say 'minuscule interval', there are clearly twice as many points on the longer line - for every size of unit no matter how small. Material continua do not consist in sets, or numbers of things, except in such a 'small interval' sense, since amounts always need to be measured in units, as we saw before. And then all such sets have a finite number of members. Indeed there are no determinate proportions of infinity, so if there were not a finite number of members there would be no way of dividing them.
The key point, therefore, is that in some material stuff there are just proportions. Indeed, it is forgetting that proportions rather than cardinal numbers are involved with continuous quantities which is the first error that makes Cantor's diagonal argument seem to show that there is no one-one correlation between the natural numbers and, say, the decimals. For what is the decimal '0.1111', for instance, but a proportion, namely '1111/10000'? So while it makes sense to talk about '0.111...1' with n 1's, for any finite n - and even say that the limit of such is 1/9 - there is no sense at all in saying there is still a decimal, or indeed any proportion, when n is (somehow, actually) infinite. For '111.../1000...' is what fraction of what? All fractions, and hence all decimal fractions must have finite denominators.
Now clearly all the decimals, if finite, can be ordered, but any anti-diagonal product of them, in the Cantorian manner, being endless would not be a ratio of anything. That is what the diagonal argument, therefore, shows in this connection: it does not construct another decimal not in the ordering of the decimals, but shows merely that there is no anti-diagonal decimal, entirely in parallel with Russell's Paradox showing there is no anti-diagonal set of non-self-membered sets. Indeed both paradoxes are now seen to be entirely parallel to the Barber's Paradox, which shows there is simply no barber who shaves all and only those who do not shave themselves. The same point can be made with respect to the sets (now all finite) of natural numbers ([23], p7). The diagonal argument does not show that there is a set of numbers, namely the set of those x's for which x does not belong to the xth set, which escapes any ordering of the sets. It shows merely that there is no such set: those x's for which x does not belong to the xth set do not themselves form into a set, there is again only 'them' and no 'it'.
So, also, the irreducibility of certain plurals must not be forgotten. Dedekind's cuts in the rationals simply may lack completion. Cantor defined real numbers in terms of Cauchy sequences of rationals, specifically in terms of equivalence classes of such sequences (Suppes [26], pp161, 181). But he merely presumed there were such equivalence classes, not realising that Russell's Paradox, amongst other things, required such a presumption to be justified in any particular case. Now it was on the basis of this assumption that Cantor was able to prove that the real number system, unlike the rational number system, was complete, i.e. that every Cauchy sequence of real numbers, unlike every Cauchy sequence of rational numbers, has a limit ([26], p185). And that leads to the representation of every real number as not just the would-be limit of a sequence of finite decimals, but also a limit which is actually reached, so that a real number is identical with a certain infinite decimal ([26], pp189, 191). The fact that there cannot be such completed decimal expansions therefore shows that the sets Cantor used to define the reals do not exist: the Cauchy sequences of rationals equivalent to a given one do not form a set (c.f. [27], p92).
Now in addition to the above, well-known 'proofs' of the non-denumerability of certain sets, in terms of unending decimals, and in terms of the subsets of the natural numbers, Cantor also gave a proof of the non-denumerability of the reals which rested solely on the completeness of the real number system (Dauben [8], p51, Grattan-Guinness [10], pp185-6, see also §6.2). But the Platonically real limit he there presumed to exist is now shown not to exist, which means we remain compelled to see the infinite as an undifferentiated lack of number.
6. Conclusion.
Does that 'solve the paradoxes'? Does the above provide a uniform solution to them? In one sense, of course, yes. But what has appeared in general, I believe, is more that there were no logical or semantical paradoxes in the first place, only rather a lot of repeatedly blind thinking. Many 'solutions to the paradoxes' have been offered over the last hundred years, but to show instead, as I have done, that straightforward rigorous analysis reveals there were no problems in the relevant areas is hardly to 'solve the paradoxes'. It just shows that it was only illogical thinking which made it appear there were paradoxes, when in fact there were none.
Why ever didn't Copi deduce that 'Het' is not univocal? On all his principles it followed that this was so; and yet he didn't see it. Maybe Copi was moved explicitly by the Tractarian demand for determinateness of sense, since he was of that generation. Maybe Tarskians (which would include Priest) do not like the intensionality of the needed antecedent in their case, since their prime hope has been to extensionalise semantics. Maybe the previously unformalised nature of Donnellan's Distinction is to blame in the individual term case, since there is a definite need to be symbolic, and Russell's alternative, symbolic analysis of definite descriptions has been dominant. But none of these facts make the seeming truth, that there are contradictions elsewhere than in people's minds, anything other than a illusion.
The situation with the Set-Theoretic Paradoxes is more intricate, and the mix-ups which might have led people to overlook mass and plural terms, and the determinable facts about denumerability, are correspondingly more complex. What is it that inclines set-theoreticians to forget material amounts, and the irreducibly plural? It starts, I think, with losing contact with the material world, for certainly even finite sets are not directly perceptual. A flock is not the birds in it, nor even the mereological sum of them, and that can lead one to think that sets are quite abstract objects, not connected with the material world. But the flock is the birds seen together rather than separately, so it is a perceptual aspect of the mereological object, something it is 'seen as' (c.f. Maddy [13]). So while sets are never material things, as such, the link with the material world is not lost entirely, and in fact the number in such a flock of birds is a second order relation between a material thing (the mereological sum) and some property (being a bird) (Slater [25], Chs 8, 9).
The temptation has been to treat numbers as objects, and certainly they can be subjects, like any other property or relation; but number words are still attributive adjectives, with 'The pack is 4 suits, but 52 cards' paralleling 'It is a small elephant, but a large animal'. One central grammatical fact associated with this is that numbers (even 0 and 1) are always numbers of things (with 'things' crucially in the plural), and so a thing, in itself, does not have a number. That is indeed why the unit set '{x}', and not 'x' itself appropriately appears in Lewis' formulation of Set Theory. In its guise as '{y|y=x}' it clearly contains a plural term 'those things which are x' - i.e. 'y|y=x' - and so has a determinate number of things in it, which means it can be a proper part of a set. It gathers the thing x into a whole, i.e. sees it as a unity.
Of course, an infinite series of things could never be gathered into such a seen unity. The totality of its members could never be available to any kind of intuition, since they cannot be completely given. But why the latter fact should mean there aren't any infinite totalities was also forgotten recently - except by the Intuitionists, of course. Indeed, against them (c.f. [27] p92), cannot consistent theories be formulated, which state that there are even non-denumerable sets? But, of course, Skolem's Paradox shows such theories must have a denumerable model. And the points above now show that they cannot have any other.
What is non-denumerable are not numbers but functions of numbers, as is
readily seen through diagonalisation. For if, for all n, fn(x) is a
function of the natural numbers then for no m can one have, for instance,
fm(n) = fn(n) + 1. If we were to define 'real numbers' not in terms of
Platonic limits, but merely convergent sequences of rationals, as the
Intuitionists have done, then we would be identifying 'real numbers' with
certain functions, since sequences are functions from the natural numbers.
But the name 'real number' is then a mis-nomer, since a function is not a
number, even if each of its values is one. There is no numerical
representation, as a result, of the length of the diagonal in a unit
square, for instance. This length is available geometrically, but all
arithmetic can do is produce a function which generates a series of
approximations. The function is then a representation of the geometric
ratio, but naturally is not equal to it. The 'irrational number' is not
available 'extensionally' only 'intensionally' it might be said. But the
difficulty for arithmetic is strictly more acute, since properly there is
no number available at all, in this case.
And the functions which should replace them are 'non-denumerable' merely
because there is no enumerating function of them, as Skolem's Paradox has
otherwise indicated. For while all computable functions of one variable
are enumerable, there is no way to specifically enumerate just those which
have completely defined values, otherwise the halting problem would be
solved. Hence the ordinal numbers of those functions which generate the
'real numbers', although denumerable, are not enumerable. There is, in
other words, a further kind of expression, which is like that for a 'real
number' except certain decimal places are undefined. These are enumerable,
but diagonalisation does not produce a further one of them, since neither
fm(n), nor fn(n) + 1 need equal anything. Amongst the functions which
generate these expressions are all the real-number functions, since, for
any argument, all values, say between 0 and 9, must be taken. But we
cannot, in general, determine which these real-number functions are. Even
if fn(x) determines a 'real number', which function it is is only
determinable from its ordinal place amongst all computable functions, not
from its ordinal place amongst the real-number functions, with the result
that, if the latter is 'n', then fn(x) is not a calculable function of n.
Of course, if one specifies a sequence just of real-number functions that
makes it the case that which function is the nth in that sequence is
determinable from n, and fn(n) + 1 (modulo the base) will then be a
further, distinct real-number function of n. But it is only the
specification of such a sequence which makes fn(x) a function both of x and
of n, and so there is no further diagonal function in the general case,
much as there was no diagonal set, or diagonal decimal, in the cases
before.
REFERENCES
[1] Beziau, J-Y, 'What is Paraconsistent Logic?' Proceedings of the First World Congress on Paraconsistent Logic, to appear.
[2] Black, M., 'The Elusiveness of Sets' Review of Mataphysics 24 (1970/1), pp614-36.
[3] Boolos, G., Logic, Logic, and Logic, Harvard U. P., Cambridge MA, 1998.
[4] Bostock, D., Logic and Arithmetic, Vol I, O.U.P. Oxford, 1974.
[5] Bostock, D., Logic and Arithmetic, Vol II, O.U.P. Oxford, 1979.
[6] Bunt, H.C., Mass Terms and Model Theoretic Semantics, C.U.P. Cambridge, 1985.
[7] Copi, I. M., Symbolic Logic, 4th ed., Macmillan, New York, 1973.
[8] Dauben, J.W., Georg Cantor Princeton U.P., Princeton, 1990.
[9] Goodstein, R.L., 'On the Formalisation of Indirect Discourse', Journal of Symbolic Logic, 23 (1958), pp417-9.
[10] Grattan-Guinness, I., From the Calculus to Set Theory 1630-1910, Duckworth, London, 1980.
[11] Kaufmann, F., The Infinite in Mathematics, Reidel, Dordrecht, 1978.
[12] Lewis, D., Parts of Classes, Blackwell, Oxford, 1991.
[13] Maddy, P., Realism in Mathematics, Clarendon Press, Oxford, 1990.
[14] Priest, G.G., 'The Structure of the Paradoxes of Self-Reference' Mind, 103 (1994), pp25-34.
[15] Priest, G.G., Beyond the Limits of Thought, C.U.P., Cambridge, 1995.
[16] Sayward, C., 'Prior's Theory of Truth', Analysis, 47 (1987), 83-7.
[17] Slater, B.H., 'Talking about Something' Analysis, 93 (1963), pp49-53.
[18] Slater, B.H., 'Prior's Analytic' Analysis, 46 (1986), pp76-81.
[19] Slater, B.H., Prolegomena to Formal Logic, Gower, Aldershot 1988.
[20] Slater, B.H., 'Hilbertian Reference', Nous, 22 (1988), pp283-97
[21] Slater, B.H., 'Liar Syllogisms and Related Paradoxes' Analysis, 51 (1991), pp146-53
[22] Slater, B.H., 'Thought Unlimited' Mind, 101 (1992), pp347-53.
[23] Slater, B.H., 'Some Wittgensteinian Theses Proved' Wittgenstein Studien, 2 (1994), pp1-11.
[24] Slater, B.H., 'Paraconsistent Logics?' Journal of Philosophical Logic, 24 (1995), pp1-4.
[25] Slater B.H., Against the Realisms of the Age, Ashgate, Aldershot, 1998.
[26] Suppes, P., Axiomatic Set Theory, Van Nostrand, Princeton N.J., 1960.
[27] Tiles, M., The Philosophy of Set Theory, Blackwell, Oxford, 1989.
[28] Wittgenstein, L. Tractatus Logico-Philosophicus trans. D.F.Pears and B.F.McGuinness, Routledge and Kegan Paul, London, 1961.
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