PHIL 2511: Paradoxes
L15 A Unified Solution?
- Graham Priest established that many paradoxes, which had been traditionally divided into different families, have a structure in common – which he calls the Inclosure Schema – and, correlatively, that these paradoxes demand a uniform solution. The uniform solution favoured by Priest is a Dialetheist one. In this lecture, I shall try to show that, with minor modification, the Inclosure Schema becomes sufficiently embracing to exhibit the underlying structure not just of the logico-semantical paradoxes discussed by Priest, but of some metaphysical paradoxes too. The uniform solution advocated here is a non-Dialetheist one. Although this is not the concern of the present lecture, I am persuaded by some recent work (Bromand 2002; Simmons 1993, pp.80-2) that Dialetheism, whatever its other virtues, does not deliver a solution to the semantical paradoxes.
- Russell, in a paper for the London Mathematical Society (1905), remarked on the structural similarity between members of a certain group of logical paradoxes. In a later paper (1908), in which the Theory of Types -- first mooted in an Appendix to The Principles of Mathematics (1903) -- was developed, he more ambitiously claimed that there is a property, which he called `reflexivity’ that is common to the logical paradoxes and to a set of semantical paradoxes including the Liar. One can easily see what Russell was getting at, for it is very easy to formulate many of these paradoxes in such a way as to give prominence to the reflexive pronoun `itself’. The Russell Class is a member of itself if and only if it is not; the Liar paradox features a sentence that says of itself that it is false; the predicate `heterological’ is true of itself if and only if it is not; and so on.
- Russell did not develop this idea with any rigour, and the grammatical similarity between ways in which various paradoxes may be formulated seemed to Ramsey not to be indicative of substantial real similarity between all these paradoxes. Ramsey (1925) divided the paradoxes into two distinct groups – the logico-mathematical and those which, as he put it, `contain some reference to thought, language or symbolism’. Ramsey reckoned this division important, for he believed that, while faults in logic or mathematics are likely to be responsible for paradoxes in the first group, `faulty ideas concerning thought and language’ were likely to be responsible for paradoxes in the second. In other words, Ramsey believed that different types of solution were required for these different types of paradox.
- Although Ramsey’s bifurcation became
the orthodoxy, the last thirty years has witnessed a revival of Russell’s
unifying tendency. For example,
J.L. Mackie (1973) wrote: `I shall
assume …. that it will be a merit of a proposed
solution if it can cope in a fairly unitary way with this full range of
paradoxes, and a weakness if it cannot’.
A paper by
5. Priest (1994) claimed that the common structure underlying these paradoxes is what he called the Qualified Russell Schema, or the Inclosure Schema. Russell’s original schema can be stated as follows: Take a diagonalising function d and a totality of objects Ω, the whole set of objects having property f. Applying d to a subset X of Ω produces an object that is not in X; Priest calls this the Transcendence Condition. By the condition he calls Closure, applying d to X produces an object that is in Ω. So the result of applying d to Ω itself is an object that is both not in and in Ω. To see that this pattern underlies the Russell Paradox, let f(x) be `x Ï x’. Then Ω is the Russell Class. d, in this instance, is the identity function.
- In order to handle definability paradoxes and the semantical paradoxes, Priest introduces a slight modification to the Russell Schema, which has the effect of ensuring that Ω has a property ψ. Thus, Priest’s Inclosure Schema is as follows:
For given properties f and ψ, and (possibly partial) function d
1) Ω = {x: f(x)} exists, and ψ(Ω)
2) If X is a subset of Ω such that ψ(X):
a) d(X) Ï X
b) d(X )Î Ω
- So, for example, we retrieve König’s Paradox by having as f(x) “x is a definable ordinal”; as Ω the set of definable ordinals, as ψ(x) “x is definable” and d(x) as the least ordinal not in x. Since Priest has a Dialetheist's appetite for contradiction, he leaves matters there: When ψ(Ω), we have both d(Ω) Î Ω and d(Ω) Ï Ω and that’s fine – a true contradiction.
- Those of us superstitious enough to retain a sense of dread and veneration in the face of a contradiction will seek, in the case of each paradox, to identify and eradicate a malignant element present in either the premises or the reasoning. For example, if Priest is right that it is the co-presence of the Inclosure conditions that give rise to paradox, then we can eliminate paradox from set theory simply by ensuring that the Inclosure conditions are never co-satisfied. This would be achieved by adding the following restriction to the comprehension axioms of Naïve Set Theory:
~($d)($y)($f)(fy & (x)(((x Í y) & fx) ® (d(x) Ï x & (d(x) Î y)))
9. Here `x’ and `y’ range over sets, `d’ over functions[1] and `f’ over properties. In words, the restriction is to the effect that there cannot be a function, a set and a property all of which simultaneously satisfy the Inclosure conditions 1) and 2). Ruling out certain functions may be compared to disallowing certain mathematical operations. Borrowing an example of Nuel Belnap’s, no mathematical operator `?’ is defined by `a/b ? c/d = (a+b)/(c+d)’ because, as is easy to verify, such an operator would deliver inconsistent values. The definition of `?’ is a recipe for contradiction; it is an illegitimate stipulation. Unless we are willing to embrace contradiction, we must disallow `?’.[2] The restricting clause for Naïve Set Theory may look inelegant but, as we shall see, in the case of Russell’s Paradox, the motivation for and implementation of the restriction is childishly simple.
10. Restrictions on the application some of the rules of inference in first-order logic are required to ensure soundness. An unrestricted rule of Existential Instantiation (EI), for example, would be a recipe for a contradiction, and textbook authors standardly demonstrate the need for a restriction by showing that, without one, the derivation of a falsehood from a set of truths would be permitted by the rules. If, from `($x)Fx’, one were to infer that a particular object named `d’ is F, that would be a mistake if the name `d’ had already occurred in the deduction -- one may already have validly established that d is not F, hence we would have the contradiction `Fd & ~Fd’. To circumvent this possibility, we play safe by adding a restriction to the application of EI, to the effect that one may instantiate only with a constant not previously used in the deduction. The Inclosure Schema is, as we have seen, a (more complicated) recipe for disaster in the shape of contradiction. It embodies three existential assumptions and any particular paradox instantiating the Inclosure Schema can be solved by identifying which of these assumptions, in that particular case, are false, and giving a persuasive account of why they are false.
11. Returning to Priest, one may want to quarrel with him over points of detail. You may think that certain paradoxes escape his schema[3], or fit it only as a fat body fits into a tight corset – by being squeezed into a different shape. The Liar and the Barber, for example, have to be bent and shaken considerably in order to fit the Inclosure Schema, and Priest could be accused of, in effect, addressing himself only to set-theoretic versions of all the familiar paradoxes with which he deals. It is not obvious that his solution copes adequately with these paradoxes in their original forms. My purpose here is not, however, to raise objections to Priest but to accept and to build upon two of his conclusions that seem right -- first that there is an underlying structural similarity between the set-theoretical and the semantical paradoxes; second, that a satisfactory solution must be a uniform one.
-
Priest’s Inclosure Schema features a function d that maps a subset X of a totality Ω onto some object in Ω but not in X. Now, d need not be a diagonal function, although, of course, diagonalization is one way of achieving the desired mapping. Let us give a graphical representation of the Inclosure Schema.Ω
Diagram 1: The Inclosure Schema
13. When we apply d to Ω itself, we get the following spooky picture:
Diagram 2: The Dialetheist’s Spooky Suggestion
14. Here we have the impossible situation, where d delivers a single object that is both in and not in Ω. In Priest’s own diagram of the Inclosure Schema (1995, p.172), the result of applying d to Ω is depicted as an object lying on the circumference of the Ω circle, but this masks the fact that, for Priest, the value of the function lies both inside and outside the Ω circle.[4] Priest, of course, welcomes such impossibilities – he thinks that something’s impossibility does not impugn its possibility -- but for those who resist this Dialetheist tenet, a problem arises, and the way to solve it is to show, for each instance of the Inclosure Schema either that Ω does not exist or that any of the other two existential assumptions embedded in the Inclosure conditions do not hold.
15. Consider, for example, a computer virus-scanning program that, after checking each file for viruses, creates a report in the form of a new file, which, together with the old ones, is subjected to a new scan, after which a new report file is created… and so on, ad inf.. What corresponds to the d function here is the program that, after each scan, creates a new file (Transcendence) which belongs in the putative totality Ω of files (Closure). In this case, we should be inclined to say that, given unlimited computer memory, there is no end to the process of file-creation, and there is no totality of files.[5] As the ancients would say, there is a potential but not an actual infinity of files.
16. Take a look at Diagram 1, and, this time, think of X as a finite set of decimal numbers beween 0 and 1, each one represented by a (non-terminating) numeral. Let d be a diagonalising function which generates a new decimal O1, not in that original set. Now let X1 be the set of decimals including all those in X together with O1, and apply d to X1 obtaining a new decimal O2 and an expanded set X2. This is depicted below.
Ω
δ x x x x x x 
O2
X1
O1
Diagram 3: Iterating the Inclosure Pattern
17. The procedure can be iterated ad infinitum. In Cantor’s famous proof, the original set X was denumerably infinite, and the proof could be taken to be a demonstration that what, in this instance, substitutes for Ω, namely, the totality of decimal numbers in the range 0 to 1, is non-denumerable or that it does not exist.
18. Another instantiation of Diagram 3. In this case, d is a convergent function and X is the set of distances run by a runner after a finite number of exponentially diminishing stages. The value obtained by applying the function d to the set of distances from origin reached after the first n stages is a new distance greater than any member of that set. For example, the stages could be set such that the distance from origin after the n+1st stage is the distance dn reached at the nth stage plus half the remaining distance to the runner’s destination. dn = åi=1n D/2i, where D is the length of the course. dn+1 = (D+dn)/2. This is Zeno's Racetrack.
19. In Zeno’s paradox of Achilles and the Tortoise, there is a race in which both competitors run at constant speeds, that of Achilles being the faster, but the tortoise is given the advantage of starting some way down the track. After the first stage, Achilles has reached the point where the tortoise started, by which time the tortoise has arrived at a new point. At the next stage, Achilles reaches that latter point, by which time the tortoise is still ahead of him. And so on. At each stage, Achilles gets to where the tortoise was, but at no stage can he reach where the tortoise is. In other words, at no stage does he draw level with, let alone overtake, the tortoise. What is thought to be paradoxical is that this result is in conflict with the fact that, even if the tortoise is allowed to start half way down the track, Achilles will soon catch up, overtake, and so win the race.
20. We can transform the Racetrack into the Tortoise Paradox (the Fast Tortoise version in which the tortoise runs exactly half as fast as Achilles) by ensuring that the tortoise's rear end which, at the start of the race, is given a D/2 start over Achilles, is held static relative to the observer and marks the finishing line. In each case we are certain that the runner will reach his destination, whether that be the end of the racetrack or the rear end of the tortoise, yet what Zeno’s argument seems to show is that after no matter how many stages, the runner cannot reach the destination.
21. But, it is obvious from the way that the d function has been defined, that it can never yield the complete distance D to the destination as its value; it always delivers as value a new distance less than D. Another way of putting this is that, if V is the constant speed of the runner, then d delivers distances, each distance covered in a time less than D/V. The paradox disappears with the recognition that no information has been given about the progress of the runner at time D/V or thereafter and, in particular, no information is supplied that is inconsistent with the runner’s reaching his destination at time D/V – a point often credited to Paul Benacerraf (1962) but which appears first to have been made by William James in a work published posthumously in 1911 (James 1996, p.171). The conclusion to be drawn about Achilles is not that he will never catch the tortoise, but that he will never catch the tortoise before the time D/V.
22. In this incarnation of the Inclosure Schema, the Transcendence condition is satisfied in that, where di is the distance <D covered after the ith stage (the distance at stage i), we have that di+1 is greater than any of the distances covered up to and including the ith stage. And, the application of the d function to any subset of the set Ω of all distances generated by infinite applications of the d function, obviously produces a value that is a member of Ω, so Closure is satisfied. But d(Ω) is not defined. Alternatively, we can think of a function d* that, in the case of Owl, adds £5 to a running total. The relevant function, in the case of Achilles adds an amount to the total run, but in such a way as to never reach the distance D. This is not paradoxical, since what we can derive from the description of the race is that Achilles cannot catch the tortoise within a certain time. The whiff of paradox arises from the belief that we have an argument that Achilles cannot catch the tortoise simpliciter. But that belief is mistaken.
23. The simplest way to see the point is to merge the Achilles with another well known paradox in the same vicinity, giving the Zig-Zag Achilles shown in Diagram 4.
Diagram 4: The Zig-Zag Achilles
24. Here we have a large square with its centre at X, the square divided East-West into two rectangles, one of which is divided into two squares, one of the latter being divided East-West into two rectangles, and so on, as illustrated. Starting from 1, Achilles has to make a zig-zag run, the first six stages of which are shown. The rule for the route is that he must pass from within one rectangle [square] to within whichever adjacent square [rectangle] is farther from X. It is easy to prove that the successive legs of this run form a convergent series of distances; it is a finite run. As can be seen, Achilles zig-zags towards X. But does he ever actually reach X by obeying the rule of the route? The answer is obviously `No’, for his instructions tell him to pass from one area to another, so he can never arrive at X, which is not an area but a point. The point X is, to borrow an apt expression that Keith Simmons (following Gödel) uses in his diagnosis of the logico-semantical paradoxes, a singularity (Simmons 1993, pp.99-112 ). It lies outside the series of areas through which Achilles has to run; it is the limit of that series. (The counterpart, in the linear run, is the destination, a point that is the limit of a series of diminishing intervals.)
25. There is, of course, no question of Achilles moving from the last area in the series to X, for there is no `last area’. Equally, if we see the run as progress through a series of stages, there is no `end of the process’ (Black 2002, p.345) -- the language, carried over from discussing finite series and finite processes is crucially misleading; it embodies just the kind of false analogy that Wittgenstein identified as leading us philosophically astray.[6] By following just his instructions (the rule of the route), Achilles runs from area to area without end, but the zig-zags and the time taken to run them become progressively smaller. The distance he covers is less than that of the perimeter of the outer square, and the time he takes is less than a certain finite limit. Running only according to his instructions, he will not reach X, but, of course, a different set of instructions would permit him to reach that destination.
26. In the case of Plato’s Theory of Forms, we are willing to say that there is no all-embracing Ultimate Form. There is no totality Ω of Owl’s sponsor’s funds; his funds, we said, are inexhaustible; however much he has paid out, he can always donate £5 more. In the case of Zeno, we can readily see that there is no set Ω of distances up to and including D that are generated by the d function. Since the Inclosure Schema that underlies the Platonic, the Learish and the Zenonian conundrums also underlies Russell’s paradox, we should, for consistency, equally wish to say that there is no Russell set embracing all the non-self-membered sets. And, of course, we do wish, independently, to say just that, because we can prove, in classical, intuitionistic and relevant logics
~($x)(y)(y Î x ↔ ~(y Î y))
this is an instance of a valid schema known as Thomson’s Theorem. Another instance of the same schema is
~($x)(y)(S-1yx ↔ ~ S-1yy)
where `S-1’ abbreviates `is shaved by’. Here, with the variables ranging over the villagers of Alcala, we have the answer to the question `If the barber of Alcala shaves all and only those who do not shave themselves, who shaves the barber?’, namely, `There is no such barber’.
27. Mark Sainsbury, in his introductory text, introduces a ten-point scale for rating the difficulty level of paradoxes, the deepest scoring a 10. The Barber scores only 1 – it is the shallowest of paradoxes in Sainsbury’s estimation -- yet the Russell paradox which, as we have seen, shares the structure of the Barber, is counted a deep paradox with a score of 6 or higher (Sainsbury, p.2). Some explanation is surely called for. An incorrect explanation would be that, while a class is an abstract entity which must exist or fail to exist as a matter of necessity, the existence of a barber in Alcala is a contingent empirical matter, and that, as a matter of fact, there is no barber fitting the given description. This explanation is incorrect because it is not contingent but a logical necessity that no individual can fit that description. And, likewise, it is a logical necessity that there is no Russell class. The two paradoxes are logically on a par, so if one is genuinely shallower than the other, the essential difference must lie elsewhere.
28. To put it crudely, the difference is all in the mind. Though we can prove logically that there is no Russell class, there is nevertheless a psychological resistance to accepting this conclusion. This stems from the thought that, since there certainly are classes that are not members of themselves – the class of horses is an example – then surely we can assemble all such classes into an embracing class R. One reason why this is an appealing thought is that we tend to conceive of classes as containers (Lakoff 1987, p.458), and so imagine that we can start to `fill up’ R with the class of horses, the class of prime numbers etc.. However, this conception of a class, which is favoured by Lakoff, should be resisted. One straightforward reason for resisting is that some classes can contain themselves as members – the class of non-horses is an example – whereas no container can contain itself. As we shall see below, there is no need to impose any prohibition on the construction of non-self-membered classes.
29. A class is a mathematical entity and any particular class is specified by a biconditional of the form: `For all x, x is a member of class K if and only if q’. If the sentence substituting for `q’ is a contradiction then the biconditional specifies the null class. But if the biconditional itself is vacuous – that is to say, if the biconditional is of the form `p if and only if not-p’ then no class is specified. Compare: `You will have Spotted Dick for pudding if and only if you eat your cabbage’ with `You will have Spotted Dick for pudding if and only if you both eat and do not eat your cabbage’ and also with `You will have Spotted Dick for pudding if and only if you won’t have Spotted Dick for pudding’. The first specifies the condition for your having Spotted Dick for pudding; the second tells you that you won’t have Spotted Dick for pudding under any circumstances; the third specifies nothing. Consider, again, the Barber, and think of Alcala as a small, indeed, as a tiny village consisting of just a few women and two males, the barber (b) and Alberto (a). The biconditional purportedly specifying the shaving arrangements is
(y)( S-1yb ↔ ~ S-1yy)
which, expanded out, becomes
(S-1ab ↔ ~ S-1aa) & (S-1bb ↔ ~ S-1bb)
and, upon the early demise of Alberto,
(S-1bb ↔ ~ S-1bb)
It is immediately clear that there can be no individual satisfying this vacuous biconditional, and that fact is not contingent on the existence of Alberto, nor of any other members of the village were its population somehow to swell. Thus there can be no barber b satisfying the shaving biconditional
(y)( S-1yb ↔ ~ S-1yy)
for that biconditional is vacuous, and correlatively, there can be no totality of individuals shaved by a non-existent barber.
30. Because the Barber and the Russell paradoxes share a structure, we can run a parallel argument to show that there can be no class R satisfying the biconditional
(y)(y Î R ↔ ~(y Î y)).
and thus no totality of members of the non-existent class R. Expanding out the biconditional gives us
a Î R ↔ ~(a Î a) & ……..& R Î R ↔ ~(R Î R).
The final conjunct is vacuous and hence so is the original biconditional; no class is specified by it, just as no number is specified by the description `the number that is greater than seven, less than sixteen and three greater than itself’.
31. There is a pleasingly direct way of making this point. We said that classes are specified by biconditionals of the form: `For all x, x is a member of class K if and only if q’. Now, there is an obvious exception to this prescription. Consider: `For all x, x is a member of class J if and only if x is a member of class J’. This biconditional is appropriately termed `vacuous’, for it sets down no condition for membership of J. No class J has been defined by this bi-condtional so, in the absence of any independent specification of J, it is correct to say that there is no class J. And we see that, likewise, there is no class G, when the following is all we have as an attempted specification of G:
For all x, x is a member of class G if and only if x is NOT a member of the class G
So, a fortiori, there is no class R with the following specification:
For all x, x is a member of class R if and only if x is NOT a member of class R, and NOISE
32. But, as we saw earlier, this is exactly (an abbreviated version of) the biconditional that purports to specify the Russell Class R. Clearly, then, in order to avoid postulating the existence of R, we do not have to settle for weakly motivated restrictions, such as those incorporated in type theory, in ZF or in Quine’s NF; all we need is to properly observe the prohibition on definiens per definiendum.
33. If we glance back at Priest’s Inclosure Schema, we see that, for all the paradoxes he considers, a contradiction is derived by applying a function d to some totality. What we have observed however, are cases where the totalities are illusory, and where the illusion has been created by a biconditional that seems to specify some totality (e.g., the villagers shaved by a barber, the non-self-membered classes) but which turns out, on close inspection, to be vacuous and hence to specify nothing.
34. It is the presence of vacuous biconditionals that is the key to a unified solution of a broad range of paradoxes.[7] Let us consider two, briefly, the Grelling and (in the next section) the Liar. First, Grelling’s Paradox, which is structurally similar to Russell’s and to the Barber, and which turns on a supposed definition of the property of heterologicality. This property is so specified as to be possessed by any predicate that does not truly apply to the name of that predicate. A given predicate `g’ is heterological if and only if `g’ is not g. The specifying biconditional is
(y)(Sat (y, heterological) ↔ ~Sat (y, z)).
Where `Sat’ is short for `Satisfies’, variable `y’ holds place for names of predicates and variable `z’ for predicates, so that the predicate the name of which occupies the `y’ position has the same meaning as the predicate occupying the `z’ position. So, as part of the specification of the meaning of `heterological’, we have
“heterological” is heterological if and only if “heterological” is not heterological.
This is a vacuous biconditional. Since the `specification’ is vacuous, there
is no property of heterologicality specified, just
as, to use the Belnap example again, there is a sign
`?’ with a supposed definition of its meaning, but that `definition’ fails, and
the sign stands for no mathematical operation.
- Finally, consider the Eubulidean Liar. Whereas, in the case of the Grelling, there was a predicate with no property corresponding to it, so, in the case of the Liar, there is a sentence with no statement corresponding to it. A sentence consists of words belonging to a certain language strung together in conformity to the grammatical rules of that language; it is, in J.L. Austin’s terminology, a pheme. It is a syntactical entity. A statement is a sentence used to say something, and a sentence-in-use is categorically different from a pheme. When a statement is made, some of its elements may pick out objects. A statement is a semantical entity; statements have truth-value.
- We argued that there can be nothing satisfying a vacuous bi-conditional of the form
Fx ↔ ~ Fx.
That is to say, there can be no x such that x is F if and only if it is not-F. As one instance of this observation, substitute `Tr’ for `F’, `S’ for `x’ and we obtain the result that there is no S such that
(α) TrS ↔ ~ TrS
where `Tr’ is the truth predicate. Now, when `p’ is a putative statement and `P’ is its name, the truth conditions for P are given by
TrP ↔ p.
So we have shown that the biconditional (α) purportedly specifying the (strengthened) Liar, S, where `S’ names `S is not true’ (`~ TrS’) is vacuous and specifies nothing. Hence, there is no statement answering to the name `S’ – just as there is no class answering to the Russell specification, no property satisfying the defining conditions of `heterological’, no barber fitting the census report on Alcala.
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