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Structural similarity or structuralism? Comments on Priest's analysis of the paradoxes of self-reference.

I. GRATTAN-GUINNESS

Graham Priest (1994) argued (1) that all the paradoxes of set theory and logic fall under one schema; and (2) hence they should be solved by one kind of solution. This reply addresses both claims, and counters that (1) in fact at least one paradox escapes the schema, and also some apparently "safe" theorems fall within it; and (2) even for the (considerable) range of paradoxes so captured by the schema, the assumption of a common solution is not obvious; each paradox surely depends upon the theory and context in which it arises. Details of Priest's proposed solution are also sought.

1. Priest's schema for the paradoxes

In a most attractive article in these pages, Graham Priest (1994) has shown that paradoxes of self-reference in logics and set theory carry a similarity of structure far beyond that which Bertrand Russell had exhibited for his own paradox and those of the greatest cardinal and greatest ordinal numbers. He criticises the common view that the paradoxes should be divided into the mathematical and the semantic ones, with a corresponding bifurcation of Russell's type theory into its simple and ramified branches. I shall defend the common view in certain ways (while not adhering to any particular sort of solution), and also consider the meta-philosophical status of structure vis-a-vis other factors such as objects, self-reference itself and ontology. I also address the question of whether all paradoxes are captured. While the issues are primarily philosophical, I add a few little-known details from the history of the paradoxes and their solutions (a rich and sometimes surprising story) which deserve more attention than they have received, both in practising and teaching philosophy. I abbreviate "if and only if" to "iff", and "classical bivalent logic" to "CBL".

Priest's schema for structuring paradoxes is as follows. Let a propositional function [Phi] be satisfied by at least one object, and so determine some non-empty set w; and let a denoting function [Delta] (which Russell himself wrote as "f") be such that the correlate set [Delta](x) of any non-empty subset x of w belongs to w but not to x. Then a double contradiction arises: [Delta](w) belongs to w iff it does not do so (Priest 1994, p. 27, though with an obvious misprint at line 19).

This schema depends upon a dichotomy based upon negation: true/ false, belong/does-not-belong, or whatever. Many paradoxes satisfy it (the liar, Russell's, (un)nameability, and so on); some of the main paradoxes will be analysed below. There are also variants upon them (of which a nice selection is supplied in Sainsbury 1995, App. 1)). Especially attractive is this version of the liar, due to Ushenko (1937):

All propositions written on this line of text are false; for paradox arises even from considering propositionhood here.

2. Remarks on paradoxes

The words "paradox", "antinomy" and "contradiction" are often used very carelessly, which is especially sad since some genuine logical distinctions do obtain (Quine 1962). Here are five relevant points or examples.

2.1 Not paradoxes.

Several so-called "paradoxes" are actually logically valid assertions or theorems which are surprising, or at least seemed so at the time of their debuts. Examples include certain of Kant's "antinomies", Peano's spacefilling curve, some consequences of the axioms of choice--and, in my heterodox view (Grattan-Guinness 1974), the Achilles-tortoise "paradox". Intersecting with this category are "fallacies", which arise from slips in reasoning or inconsistent premises, often not obvious in occurrence. While not the concern of the discussion to follow, these categories are not always sharply distinguished from paradoxes.

2.2 On logical connectives

One cause of the surprisingly blurred boundaries between contradiction and consistency is the dependence among the five basic logical connectives in CBL; for it permits any compound proposition to be converted into other forms. In particular, since the conditional and the biconditional are convertible, some paradoxes can be cast in the form of a double contradiction, where both the propositions M and ??M lead to trouble: for example, Epimenides's assertion "this proposition is false" is true iff it is false. This dependency also reduces the range of logical structures of the paradoxes, as is shown dramatically by Priest's schema, which itself is a double contradiction.

2.3 Contradiction [not equal to] reductio ad absurdum

As another example of the blur, there is available in mathematics based on CBL the method of indirect proof of a theorem M: assert ??M, deduce as contradictory consequents some proposition A and also ??A, and invoke the modus tollens rule of inference to infer M. Its dual is the reductio ad absurdum of M: deduce from it both some A and ??A, and (if one wishes) infer ??M. (In both cases A may be M or ??M.) Thus, for example, an indirect proof of the irrationality of [square root of]2 is a reductio argument against the rationality of [square root of] 2.(1) As my discussion largely lies within CBL, I shall not press the distinction between these two kinds of reasoning; but it will be invoked in [sections]4.3.

Some paradoxes take--or can be cast in--these forms. For example, those of the greatest cardinal or ordinal number N arise from contradictions such as "(N = N)" and "(N [is greater than] N)", which argue for the non-existence of N. Further, to exemplify [sections]2.1 above, Russell (1903, p. 323) invented "Burali-Forti's paradox" of the greatest ordinal (and most of the others also): Burali-Forti (1897) thought that he had simply shown that a particular way of ordering ordinal numbers did not satisfy trichotomy (that is, [is less than], = or [is greater than]).(2)

2.4 Two paradoxes for cardinal numbers

The paradox just stated for the greatest cardinal number is usually called "Cantor's". But there is another paradox of cardinals, arising from the unrestricted construction of power-sets P(S) from a set S (and doubtless also known to Cantor): ifs is the universal set U, then

both P(U) [subset of equal to] U and U [subset] P(U); and both [2.sup.N] [is less than or equal to] N and N < [2.sup.N] can be proved, by defining N suitably from U.

Following Russell, Priest attaches the name "Cantor's paradox" to this form, and regards Russell's own as "a stripped down version of Cantor's" (1994, p. 27); for it can be derived by applying Cantor's diagonal argument to U under the identity mapping, and finding that the set R of all sets which do not belong to themselves belongs to itself iff it does not.(3) All three paradoxes fall under Priest's schema.

2.5 Russell's overlooked paradox.

At the end of his Principles Russell (1903) stated a paradox but then forgot about it completely, as have most of his commentators. Let propositions form a type, and apply Cantor's diagonal argument to the proposition M given by "every member of a class m of propositions is true". M corresponds one-one with m, and may or may not belong to it; and if one takes the class w of non-belonging propositions M, then "every member of class w of propositions is true" belongs to w if and only if it does not (Russell 1903, pp. 527-8: in those days "class" was the technical term). It too falls under Priest's schema; and, strikingly, it is constructible in the simple type theory of Principia Mathematica (see de Rouilhan 1996, pp. 226-30).

3. Negation-free paradoxes

Attempts have been made to construct this kind of paradox. Popper (1962) claimed success by presenting the visiting card paradox (which is due to G. G. Berry) in the form of handing you a blank card and asking you to mark it in some way iff you think Popper will find it still blank when you hand it back; but the dichotomy written-on/blank seems to let negation slip in. A cognate case is "Airy's paradox" (as I call it), where the incredibly fastidious Astronomer Royal G. B. Airy (1801-1892) is said to have found an empty box at Greenwich Observatory, whereupon he wrote "empty box" on a piece of paper and put it inside--with the result that ....

More serious is the paradox due to M. H. Lob and Leon Henkin, which is based upon the proposition

If this proposition A is true, then so is B (A) for any proposition B; for then "B is true" follows, from properties of the conditional alone (L6b 1955).(4) In a recent personal communication Henkin told me that this paradox was found by noting that G6del's incompletability theorem is grounded upon the proposition "this sentence is not provable" and wondering what happened from considering "this sentence is provable"; thus it cannot fit the schema. One may introduce negation here via the usual definition of implication

(A [contain] B): = ?? A v B;

but it is not obligatory to make this move, or grant negation automatic priority (compare [sections]2.2 on the connectives), and in any case the schema is still not satisfied. Further, any solution to this paradox would require some assumption(s) to be modified and thereby negated; but such changes themselves do not fall under the schema either.

4. The place of structure

As is normal, Priest attributes to Ramsey (1926) the distinction between the simple and the ramified branches of Russell's type theory. In fact the pioneer work is Chwistek (1924, 1925), with the respective names "simplified" and "branched" used (among others). Ramsey was aware of Chwistek's work and may have known of this paper; in any case he held a clearer vision of the points at issue.

4.1 Classifications of the paradoxes

Both men split paradoxes by kind: mathematical, or at least set-theoretic ones, to be solved by the simple type theory; and linguistic, especially relating to naming and truth-values, and not necessarily mathematical, to be solved by the ramification of types into orders. As a corollary, the axiom of reducibility was banished as irrelevant to mathematics (though Ramsey did preserve it, at the heavy cost of the total extensionality of mathematical logic). The axiom has been regarded as re-instating the semantic paradoxes, although Myhill (1979) formulated the ramified theory in a way in which this disaster was avoided.

Priest finds Ramsey's distinction of the paradoxes to be "wrong" (1994, p. 25), since his own structural schema applies to both kinds. On the assumption that "if one wants to draw a fundamental distinction, this ought to be in terms of the structure of the different paradoxes" (p. 26), he puts forward a "Principle of Uniform Solution (PUS, sorry): same kind of paradox, same kind of solution" (p. 32).

A similar position was taken also by Moulder (1974), not cited by Priest. He considered Russell's paradox (in its set version) and two variants: the barber who shaves those and only those men who do not shave themselves, and the library catalogue of those and only those catalogues which do not catalogue themselves. After correctly showing that all three paradoxes display the same logical structure, he wondered why Russell's version required sophisticated solutions such as type theory whereas the variants were taken to show merely that no such barber or catalogue could exist.(5)

4.2 What is the solution?

Since paradoxes lie at the centre of logical systems, the consequences of PUS are very considerable for logic(s) in general. Why must one prefer structure over reference (or ontology for the Platonists) to solve paradoxes? Why should a common structure of paradoxes across various contexts determine the manner of avoiding them for every context?

Further, Priest does not describe in his paper The One and Onlie Solution which should be preferred under it. In a recent book on the limits of thought, where the argument of his paper is rehearsed, he states that "the only satisfactory uniform approach to all these paradoxes is the dialethic one", where certain propositions may be both true and false, so that contradictions are true (Priest 1995, p. 187). On what grounds is it to be so preferred? Further, he also shows there that several different versions are available (pp. 188-93; for more details see Priest and others (1989), pt. 2); so which version of The Solution should we choose? If two versions are metalogically equivalent, are they the "same"? May the definition of set w in the schema itself be modified in the manner of free logic to avoid assumption of existence? What happens to the dialethic solution of the liar paradox if one stipulates that "true" and "untrue" determine complementary and thereby disjoint sets (Sainsbury 1995, p. 143)?

In any case, what happens to alternative solutions, such as in some kind of constructive logic where the duality of [sections]2.3 between indirect proof and reductio is rejected? Is it good-bye to all axiomatic set theories and type theories? Or may we take (any) one of them? and stir in dialethism? Must we also reject logical pluralism and adopt logical monism, with the dialethic as the "correct" logic? How should we re-read those formal logico-scientific systems (for example in quantum mechanics) not primarily concerned with paradoxes but embodying means of avoiding them, and in that sense containing solutions?

4.3 Context first

For me, PUS is contentious, and as yet not established. In contrast, I advocate Solutions Under Contextual Kinds (SUCKs, sorry). With each paradox comes its own context, with its own content; and these govern the reaction and proposed solution(s). Dialethic logic(s) are certainly included, and due note should be taken of the pertaining logical structure. But diversity of solutions should be maintained, as in the past. For example, while Russell's paradox was inspired by one of Cantor's, his paradoxical set is not "universally" large in the style of Cantor's greatest number in [sections] 2.4, and so he never adopted a criterion of "limitation of size" (his own name for Cantor's approach) in his various solutions, including type theory. Again, his paradox (in its versions both for sets and for functions) shows that the naughty set (or function) must not exist; but this conclusion has to be taken further in the context of rebuilding theories in set theory, transfinite arithmetic and mathematical logic, especially preserving the two trails of paradox-free predecessor cardinals and ordinals and their arithmetics. To answer Moulder, there are no analogous concerns in coiffeur or in library sciences. While in a much wider meta-context, I ally myself with the general interpretations of Frege's context principle.

4.4 Further classifications of paradoxes

The range of solutions may be increased by following Ramsey in distinguishing paradoxes in still other ways. Priest himself notes the distinction between paradoxes that draw upon diagonalisation and those that do not (1994, p. 26). Again, the semantic paradoxes can be divided between those such as the liar which involve the truth-values of a proposition and those of (in)definability and heterologicality which draw only upon properties of the language in which a sentence is expressed. In another option, Behmann (1931) proposed two criteria for solving the paradoxes based on banning impredicativity from nominal symbolic definitions; in the end the proposals did not work,(6) but structure did not cause the failure.

4.5 The varieties of self-reference

In logics and philosophy, reference is a complicated matter, with many referential contexts of its own (Fitch 1946). A particularly interesting case in epistemology is "comprehensive critical rationalism", a consequence of Popperian fallibilism based upon the proposition "all propositions are conjectural" (Watkins 1969). In a fine discussion of (near-)self-reference T. S. Champlin gives many cases, such as signing a letter, using an identity card, and pitying one's own self-pity.(7) Artists such as Magritte ("ceci n'est pas une pipe") and Escher provide further instructive examples. The schema fits some examples in these contexts; but therefore its limits as well as its scope need to be made precise.

5. Relationships between logics and other subjects

The reservations over PUS deepen with Priest's remarks about the relationship between logics and other disciplines, especially mathematics.

5.1 Symbolism [is not equal to] mathematics

Priest states that after Ramsey's time "both syntactical and semantical linguistic notions became quite integral parts of mathematics" (1994, p. 26). The chronology is correct, but not necessarily the philosophy, especially with this ambiguous word "integral". Taken as a reduction, this statement asserts, as a kind of mathematicism, that syntax and semantics (hereafter, "SS") were to be placed within mathematics (like Russell's logicist thesis placing his so-called "pure" mathematics within his mathematical logic); but surely this claim cannot be sustained. The fact that SS may be symbolised does not entail that their content is within mathematics. In SS the symbols belong to SS (and their metatheories), whatever be their referents (which may include parts of mathematics on occasion). Even structure similarity between a system and its metatheory may abound, but each collection of symbols belongs to its own level (Camap (1935) is a remarkable but ignored pioneering exercise of this kind, using different fonts). This is why the paradoxes of set theory are distinct from those of naming.

Another warning against reducing a theory in SS to some part P of mathematics is the risk of admitting a vicious meta-logical circle, at least within CBL. For, in order that the enterprise be purposeful, P will have to be consistent; but that property belongs to SS and its metatheory, which is under reduction in the first place.

5.2 Overuse of symbols

When Augustus de Morgan and George Boole began to develop algebraic logics in the mid 19th century, they naturally used symbols from common algebra and arithmetic such as "=", "0", "=0", and "+" (Grattan-Guinness 1997). But the new referents of these symbols had no connection with those of the mathematical originals, and such multiple use has steadily declined in logics; Russell himself (and Frege earlier) saw it as a source of ambiguity and misleading analogy. Now, and indeed even then, we recognise formal(ish) languages in many contexts with no special link to mathematics ("[H.sub.2][SO.sub.4]" belongs to chemistry, "??." to music, and so on). These changes weaken links between logics and mathematics, and thereby the generality of solutions of, for example, paradoxes.

5.3 Structure-similarity in mathematics

A worthwhile comparison with mathematics concerns the various structuralist philosophies that have been proffered for it (Vercelloni 1989); for they stress logical bones over the flesh-like objects (using this word widely and neutrally). Structure-similarity is indeed a powerful philosophy of mathematics, capable of reaching across the range of the subject instead of just set theory and arithmetic (Grattan-Guinness 1993); but the relationship is basically one of analogies, with at most only a modest measure of reduction. Multiplying rational numbers is structurally similar to compounding ratios Euclid-style, but neither theory reduces to the other one; group theory has all sorts of interpretations both within mathematics and on its physical applications, none of which however is group theory; and so on, in various guises. Even within set theory and logics, enough plurality is present to allow for various structures, as is hinted by the few examples given in [sections] 4.4.

6. Conclusions

Priest is to be congratulated on extending structural similarity far beyond Russell's original conception by encompassing many paradoxes of self-reference in his schema. However, negation-free paradoxes show that is not sufficient to characterise all such paradoxes; and the seemingly consistent G6del's incompletability theorem exemplifies the fact that it is not necessary either.

PUS claims that the paradoxes falling under the schema should have the same solution; but Priest's paper contains none of the details, although one can recover some features from other of his and colleagues' writings. An amplification would be welcome, to answer the queries raised in [sections] 4.2.

Above all, whatever form The Solution takes, the claim of its dominance through PUS raises meta-philosophical issues which need further explanation, especially the preference for of structure over reference, ontology and the related factors, and of dialethic logic(s?) over all other logics, as noted in [sections] 4. Further issues involving relationships between logics and other disciplines were raised in [sections] 5. They occur against an advocacy of pluralities, including logical pluralism.

The difference between Priest's PUS and my SUCKS hinges upon the status of metatheory in general. Priest closes his recent book with "Whereof one cannot speak, thereof one has just contradicted oneself" (1995, p. 256), thereby allying himself with the views of Wittgenstein, who rejected the notion of metalanguage. My advocacy of SUCKS is inspired by Godel and Tarski and their many followers, for whom the distinctions between meta-language/-theory/-logics and the corresponding language/theory/logics are central to philosophy. Whereof one has just contradicted oneself, thereof speaking somewhere is particularly worthwhile.(8)

I. GRATTAN-GUINNESS Middlesex University at Enfield Middlesex EN3 4SF UK IVOR2@MDX.AC.UK

(1) In either form, the proof of irrationality usually takes the form of assuming that [square root of] 2 =p / q and then showing that the co-prime positive integers p and q actually are both even. However, contradiction arises already in the intermediate equation 2[q.sup.2] = [p.sup.2], as the Greeks may well have known; assuming the prime factorisation of an integer, the left hand side has an odd number of such factors (because of the presence of 2 there) while the right hand side has an even number.

Dedekind found a lovely but poorly received proof, of the reductio form, by assuming that q was the smallest positive integer satisfying this equation and then finding a still smaller one, a process which can be repeated indefinitely many times. (Dedekind 1862, pp. 24-5; generalised to [square root of]D, D not a square, in 1872, Art. 4.)

(2) The history thereafter is quite messy (Garciadiego 1992, pp. 21-32); luckily it is not of concern here. Around the same time, in 1898, the American mathematician E. H. Moore also found the paradox, and moreover thought that it was one; but he only told Cantor in a letter (found by me many years ago, and published in Garciadiego, pp. 205-6).

(3) This relationship between these paradoxes holds not only logically (Crossley 1973) but also historically (Grattan-Guinness 1978, Coffa 1979).

(4) A similar though more complicated paradox had been in circulation for some time, based upon propositions of the form A = (.4 [contain] B), with "=" for identity; see Black (1970) for a discussion, and also (Priest 1994, p. 33).

(5) Elsewhere Priest gives a version of the barber paradox in which his schema is not satisfied (1995, p. 193); however, if it is modified so that (in the symbols of [sections] 1) x = w, then it is accommodated.

(6) After an incomplete criticism in Dubislav (1931, p. 94), G6del showed that unrestricted type theory would generate paradoxes independently of the forms of nominal definition. Godel's manuscript, and the attendant story, will be revealed in his Collected Works, Vol. 4 (in preparation). When presenting his approach in Mind, Behmann (1937) anticipated G6del (1944, pp. 132-3) in exemplifying Russell's paradox with "Does the quality `not applying to itself' apply to itself or not?".

(7) See Champlin (1988, esp. Ch. 9); however, the claimed counter-examples to self-reference involving mathematical functions (p. 124) are wrong.

(8) I am much indebted to a referee for sharp reading of the draft, and to Leon Henkin for fruitful contact. I also thank Graham Priest for correspondence--very friendly and helpful, though ultimately incommensurate.

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Title Annotation:response to Graham Priest, Mind, vol. 103, p. 27, 1994
Author:Grattan-Guinness, I.
Publication:Mind
Date:Oct 1, 1998
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