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Naive set theory is innocent!

Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of naive set theory inevitably leads one into a severe scepticism with regard to the feasibility of giving a systematic semantics for set theory. It is further argued that this is not just a problem for philosophers of mathematics. Semantic scepticism in set theory will almost inevitably spill over into total pessimism regarding the prospects for an explanatory theory of language and meaning in general. The conclusion is that those who wish to avoid such intellectual defeatism need to look seriously at the possibility that it is the logic used in the derivation of the paradoxes, and not the naive set theory itself, which is at fault.

1. Introduction

The philosophical community is notorious for rarely reaching a consensus, even after thousands of years of discussion of a topic. This ought to be seen as evidence of the depth of the subject. Even so, it is natural to want to point to some philosophical problems which have been conclusively resolved in order to counter any suggestion that the subject is entirely empty or lacking in content; and so natural, in turn, to dismiss as mischief-makers those iconoclasts who seek to overturn even such limited consensus as exists. Though in general sympathetic to this reaction, I feel the need to demur from the consensus at, perhaps, one of its most firmly held points. This is the almost universal belief that naive set theory, a theory which had seemed well-nigh self-evident, has been shown to be false by the set-theoretic antinomies.

In this paper, therefore, I first of all discuss what naive set theory is, and where it came from ([sections]2). In [sections]3 I outline the conventional, hierarchical response and the main problem facing it, namely that it appears to be self-refuting. I then look critically in mm at a number of attempts to avoid or surmount this problem: Kreisel's argument and the appeal to reflection principles ([sections]4); appeal to "proper classes" ([sections]5); appeal to plural quantification ([sections]6).

In [sections]7, I consider the objection that all we have here is the well-known result that there can be no semantically closed theories able to express concepts of truth, validity and so on and applicable to those very languages themselves. I respond in [sections]8 that this result has been shown to hold only for very special languages, including classical bivalent languages, and that if one holds that it applies without restriction one is forced into a very nihilistic position regarding semantics quite generally, and not just the semantics of set theory.

In [sections]9 I turn to the suggestion that the problems all arise from adopting too absolutist a notion of interpretation but reject the non-absolutist schematic interpretations and appeals to "systematic ambiguity" as barely disguised forms of semantic nihilism. The penultimate section examines Dummett's view that the concept set is an indefinitely extensible one, but urges that it is unsatisfactory for the same general reasons as all hierarchical theories are unsatisfactory. I then look at his distinction between intuitive and definite totalities and his claim that it is intuitionistic not classical logic which applies, at least to the former. I reject the idea that a move to intuitionist logic can resolve the problems of set theory. In the concluding section, however, it is argued that if one wishes to avoid a destructive semantic nihilism one ought indeed to identify as the villain of the piece not the set theory but the logic used in the derivation of the antinomies, in particular the structural rather than the operational rules of classical logic. This will require a more radical revision, at least in some senses, of logic than Dummett and others envisage.

2. What is Naive Set Theory?

Naive set theory can be formulated in various ways, for example, as a first-order theory in a language containing identity and a membership predicate [is an element of], a theory whose axioms include all instances of the (generalised) na<ve comprehension schema:


(I note here that many aficionados of naive set theory restrict this axiom schema, in line with the standard Subsets schema, by debarring y from occurring free in [Phi], though this is arguably a faint-hearted concession to orthodoxy.(1)) A variant of the above is


where the free variables of [Phi] are [z.sub.1], ..., [z.sub.n], x; every instance of the generalised schema is derivable from an instance of this schema in a few relatively uncontentious steps. (Assume for existential elimination [inverted A] [z.sub.1], ..., [inverted A][z.sub.n], [inverted A]x(x [is an element of] a [equivalence] [Phi][z.sub.1], ..., [z.sub.n], x); [inverted A]E yields [inverted A][z.sub.2], ..., [inverted A][z.sub.n], [inverted A]x(x [is an element of] a [equivalence] [Phi] a, [z.sub.n], . [z.sub.n] x) from which [exists]E then a final step of [exists]E yields [exists]y[inverted A] [z.sub.2],...,[inverted A][z.sub.n], [inverted A]x(x [is an element of] y [equivalence][Phi]y, [z.sub.2], ..., [z.sub.n], x).)

The more general form is needed if we move to a second-order axiomatisation in which we quantify over the variable place occupied by schematic [Phi]. That is, if we want generalised comprehension (in a non-trivial framework) we need, as well as a monadic second-order version, the dyadic:


If we consider an expanded language with a term-forming abstraction operator we can express the first-order schema as


or generalise on [Phi]'s place to yield a second-order axiom. We can also have non-axiomatic forms, for instance an inference schema permitting us to infer [Phi]x/t from t [is an element of] {x: [Phi]x} and vice versa. In all these forms, with [is an element of] as a non-logical primitive, we need also to add an axiom of extensionality:


For a somewhat different approach, we can dispense with [is an element of] and use the rule schema permitting us to infer {x: [Phi]x} = {x: [Psi]x} from [inverted A]x([Phi]x [equivalence] [Psi]x) and vice versa. In second-order axiomatic form this is Frege's Axiom V for the value ranges of concepts:


We can then define t [is an element of] u by [exists]F(u = {x: Fx} & Ft) and derive comprehension and extensionality.

The differences between these versions can be significant, depending on the background logic presupposed, but will not matter for the purposes of the present paper in which I will maintain that all these axioms are true, and rules sound, taken at face value, that is without any restrictions on admissible instances of schematic letters nor on the scope of the quantifiers, nor via appeal to free logic. For simplicity, I will mainly consider the first form, in which we have a first-order theory with extensionality and the generalised comprehension axiom schema.

Where does this idea of sets or classes satisfying the above axioms, or schemes or rules, come from? Whilst it may well be seriously mistaken to think of Cantor's Mengonlehreas naive, and current set theory as motivated purely by desire to avoid paradox--"One Step Back from Disaster"--never the less it is also wrong to think of naive set theory as an invention of Frege or of Russell, Peano or indeed of anyone else.(3) Rather, like most of our fundamental concepts--cause, substance, person etc.--the naive notion of set and the naive notion of class (incorporating the assumption that an extension satisfying comprehension corresponds to each meaningful predicate) evolved undesigned. Consider seventeenth century cases such as Locke's "which Operations ... do furnish the Understanding with another set of Ideas" (Locke 1689, II i [sections]4 p. 105); or Evelyn's "Anenomies & Flowers and that Class should be discretely pruned" (quoted in the Oxford English Dictionary 1989, p. 279). These examples make it plausible that "set" and "class" have long had a usage in which they mean something like "natural kind". What are mathematical kinds? As Lavine points out (1994 pp. 30, 34-5), there was a process of progressive abstraction, in for example Fourier and Dirichlet, of the notion of function in mathematics, till eventually any functional expression, however convoluted, came to be thought of as standing for a function, construed as something purely extensional, the graph of an arbitrary functional expression. Even without Frege's assimilation of predicate expressions to functional ones, this process renders it almost inevitable that every mathematical predicate expression be viewed as standing for a mathematical "natural" kind, and that this be identified with the extension (whether it be called extension, domain, totality, collection, class, set or whatever is unimportant). However, even if I am wrong on the history and the naive notion of class or set did not evolve but was invented, I want to maintain that that invention was borne out of necessity and we cannot now disinvent the naive notion any more than we can disinvent nuclear weapons; the reason being, I will try to establish below, that discussion of the interpretation of set theory, even by those who reject the naive theory, inevitably invokes the naive notion.

But if that is the case, will not disaster ensue? From the first-order schematic version of naive set theory we have, for example, the following trivialising proof:
--    (1) [exists]y[inverted A]x (x [is an element
        of] y [equivalence] x [is not an equal to] x)    Axion
2     (2) [inverted A] (x [is an element of r
        [equivalence] x [is not an element of] x)        Hyp.
2     (3) r [is an element of] r [equivalence] r         2 [inverted
        [is not an element of] r                          A] E
2     (4) r [is an element of] r                         Hyp.
2,4   (5) r [is not an element of] r                     3,4 [equi-
                                                          valence] E
2,4   (6) [perpendicular]                                4,5 ~
2     (7) r [is not an element of] r                     6 ~I
2     (8) r [is an element of] r                         3,7 [equi-
                                                          valence] E
2     (9) P                                              7,8 ~E
--    (10) P                                             1,2,9,
                                                          [exists] E

Hence surely, it might be thought, the naive principles are in as bad shape as Prior's tonk and must be rejected as firmly as we refuse to embrace tonk? If such rejection is not possible then a rational belief system would seem to be impossible for us.

3. Down with hierarchies

The set-theoretic antinomies certainly brought to light, as the above proof shows, the classical inconsistency of naive set theory, indeed its inconsistency in logics considerably weaker than classical logic--intuitionistic logic, relevant logics at least as strong as (the rather weak) T or RWX etc.(5) The almost universal response has been to abandon it as false, a salutary reminder of the fallibility of mathematical intuition.(6) But there have been other responses, many motivated by dissatisfaction with the hierarchical view of set theory which underlies the standard reply. The most thoroughgoing and considered anti-hierarchical response I know of, one which has influenced strongly the position to be developed here, is to be found in Priest (1987, Chs. 2 and 10), and Priest (1995, Chs. 10 and 11).(7) However Priest revives naive set theory in the context of dialetheism, the belief that there are true contradictions, of which "the Russell set belongs and does not belong to itself" is said to be one. However since I am with orthodoxy in the matter of rejecting true contradictions, I wish to emphasise in what follows that it is possible to take seriously the possibility that naive set theory is correct without embracing dialetheism (though attempting to rebut the latter view, something I believe to be neither trivially easy nor impossible, nor unnecessary, would require an extended essay on its own). Nor do I think a single strategy for cutting off "hierarchialism" at its roots exists: each new head of this hydra must be confronted as it emerges, and below I attempt to deal with the most common forms to appear thus far.

There are, indeed, some general structural features one can discern in the hierarchical approach as currently developed. It tends to take one of two forms: (i) hierarchical restrictions on meaningfulness, for example of ascriptions of set-membership in type theory; or, more commonly now, (ii) hierarchical restrictions on principles such as Axiom V or naive comprehension. On either approach we get a picture of the "universe" of sets as partitioned into levels, with atomic non-sets at the bottom level, then sets of such atoms, then sets of sets and then the magic dots: "...". Each approach sub-divides in turn into those in which the hierarchies are supposed to be built into the deep structure of natural languages and those, surely more plausible, in which they are determined by non-linguistic factors.

More plausible, but still massively implausible, the main reason being that, as Fitch pointed out early on in connection with Russell's theory of types (1952, p. v; see also G6del, 1946, p. 149), it is very difficult to state hierarchical theories without self-refutation. Putnam makes the same point in connection with the (arguably) related problem of the classical inconsistency of the naive rules for truth:

The paradoxical aspect of Tarski's theory, indeed of any hierarchical

theory, is that one has to stand outside the whole hierarchy

even to formulate the statement that the hierarchy exists. (Putnam

1990, p. 14)

As he goes on to say:

The paradoxes themselves, however, are hardly morea paradoxical

than the solutions to which the logical community has been

driven. (ibid., p. 17)

Putnam's point applies not only to type theory but also to the equally hierarchical standard conception of set found in ZFC and NBG theories, whether or not interpreted via the "iterative" conception of set? For such theories, the problem of providing a rigorous account of the intended interpretation of the set theory (and a rigorous definition of logical consequence for the language of set theory) has been increasingly recognised as an acute one. I will call this the semantic problem.

More exactly, the semantic problem is that we are precluded from giving a model-theoretic--and so set-theoretic--characterisation of the intended interpretation (and thus of logical consequence defined over all models including the "actual" one). For the intended domain is [Union] [V.sub.][Alpha] the union of all the levels [V.sub.1], [V.sub.1] ...[V.sub.a] ... of atoms, sets of atoms, sets of sets and so forth; and this cumulative hierarchy is too big to be a set itself, since ON, the class of ordinals, is too big to be a set and [V.sub.a] [not equal to] [V.sub.b] where a [not equal to] b. So, for example, no classical ZF-style theory can permit the existence of a set U of all sets since then the Subsets axiom would give us the existence of the Russell set. Accordingly no such theory can talk of its own domain. As with type theory, it would seem that in order to give a satisfactory interpretation for standard set theory, whether motivated by the iterative conception or not, one has to stand outside that theory and utilise concepts inexpressible from within the theory. But when one is interested in the possibility of a semantic theory for our full current set theory--that is the one we use in reasoning in philosophy, formal semantics and so forth--there is no outside point of reference.

4. The Kreisel argument and reflection principles

It is sometimes argued that this expressive limitation is not a serious problem for our account of logical consequence, at least, since we can prove completeness for ?? s, i.e. logical consequence set-theoretically defined. Hence if X ?? S A, X ?? A and so, granted the intuitive soundness of the system, we know that X k A: A really does follow from X. In the converse direction we can argue that any set-theoretic interpretation is a genuine interpretation, so that if there is no counterexample to X entails A then, in particular, there is no set-theoretic counterexample, that is if X ?? A then X ?? s A. Hence putting the two together, X ?? A iff X ?? s A, and we can explain the murky k in terms of the well-understood ?? s.(10)

Or we can appeal to Reflection Principles. Thus in ZF we have theorem schemata which can tell us such things as that for any closed sentence [Phi] and ordinal [Alpha], we can prove ([Phi] [equivalence] "[Phi]" is true in [V.sub.[Beta]), for some [Beta] [is greater than] [Alpha] (for truth and satisfaction in levels [V.sub.[Gamma]] of the cumulative hierarchy is ZF definable). Boolos interprets this result as saying that if [Phi] is false, it is false in a set-theoretic model, hence if ?? s A then ?? A (Boolos 1985, p. 340).

Even if these responses enable us to give a general theoretical account of logical consequence, they still leave us unable to state what the intended interpretation of the set theory is. Moreover the Kreisel response is not open to us for second-order systems (at least given conventional, finitistic notions of provability) and the extension of Reflection Principles to second-order set theory cannot, it seems, be carried out without strengthening the standard axioms nor can we generalise this result from validity to full logical consequence, as Boolos, in his sympathetic discussion of this proposal, admits (1985 p. 343).(11) This is a problem because second-order set theory is arguably the most intuitive formulation of the theory. The argument here is that we believe every instance of axiom schemata such as the Subsets or Replacement axiom schemata of standard set theory because we believe the second-order axioms which entail all those instances (Kreisel 1967, p. 86, 1968, p. 326; Shapiro 1991, p. 127). The expressive limitations of first-order languages also tell against the thesis that ordinary set-theoretic reasoning in natural language is captured at first-order level.(12)

There is a more fundamental problem with the Kreisel/Boolos arguments, at least when seen as a response to the semantic problem. The key idea in both arguments is to link precise set-theoretic notions, for example of truth in a model, with intuitive notions of mathematical truth or validity. In the case at issue--the interpretation of the set-theory we use in model-theory and in philosophy of mathematics in general--it is not clear that the conventional theorist is entitled to use the intuitive notions with which she links precise notions such as set-theoretic [??.sub.s]. For the intuitive notions are, on the conventional picture, incoherent as shown by the Liar paradox or the medieval paradoxes of validity of Albert of Saxony or the Pseudry-Scotus (a variant of these is "this argument is valid therefore 0 = 1").

Discussions of Reflection Principles never rest at the purely schematic level. The set-theoretic community does not wait with bated breath for each fresh result from a hapless crew of graduate students charged with producing new theorems of the form ([Phi] [equivalence] "[Phi]" is true in [V.sub.[Beta]]) for endless new substitutions for [Phi]. As virtually always with purportedly schematic statements (but see [sections] 9 below for more on schematic approaches), what set-theorists accept is a universal generalisation: for all sentences s of the object language, the result of substituting s for [Phi] in the above schema is provable in ZF.

Nor is the result taken to be one which concerns the provability of uninterpreted strings of symbols. The usual explanations of reflection principles take the form: if [Phi] is true (or "holds") then it is true in set-theoretic interpretations for arbitrarily high levels of the cumulative hierarchy--this is how Boolos puts it. What notion of unrelativised truth (italicised above) is at play here? If our ordinary intuitive one, then what of the paradoxes? Should not interpreters of set theory tell us what non-standard logic they are using in their set-theoretic account of the meaning of set theory, if the naive, unrestricted notion of truth features in the account? Typically, though, when the interpreter of set theory says [Phi] is true iff [Theta] is true-in-[V.sub.[Beta], truth on the left-hand side is a meta-theoretic notion defined h la Tarski for the weaker object language. But now we are back lost in "Hierarcadia" and have been well and truly "Fitched" once more. We are interested in the notions of truth and logical consequence for our current theory of sets, the theory we use in real mathematical reasoning (e.g. in formal semantics). Telling us about truth-in-O and consequence-in-O, where O is some formal object language in which a different, weaker set theory is expressed, tells us nothing about truth and logical consequence, however interesting the links are between truth-in-O and truth-in-[V.sub.[Beta]]-in-O.(13)

A defender of the Kreisel argument might, however, eschew the semantic ascent of the Reflection Principle argument and insist on sticking with an intuitive notion of truth and validity, denying that the intuitive notions are in fact the incoherent notions featuring in the paradoxes. Can the Kreisel argument, thus interpreted, help with the semantic problem? Consider Kreisel's informal "proof" that X ?? A iff x [??.sub.S] A. In the right-to-left direction the argument appeals to completeness for [??.sub.S] and then the intuitive soundness of the classical proof system. This last, however, is something a proponent of naive set theory is bound to contest. Such a proponent will look at things in the following way: we had a package consisting of a set of classical operational inference rules, a set of classical structural rules and a naive rule or axiom scheme for the notion of class. It was discovered by Russell, Zermelo, Burali-Forti and others that the package is a trivially inconsistent one, yielding all sentences as consequences. The standard response is to abandon the last element, but the "ingenue" insists we must abandon one of the first two (most plausibly, the second; see ahead [sections] 11), that is we must reject the soundness of the standard notion of proof. The Kreisel argument completely begs the question, if directed against someone seriously considering as a response to the paradoxes and the semantic problem that one alter the logic not the set theory. Such a person had better reject the conditional: if A [??.sub.S] B then A ?? B. For, where T formalises naIve set theory(14) and P is any sentence you like, T [??.sub.S] P, since T has no set-theoretic model; whilst of course for the naive theorist ?? T. So naive theorists have to reject the above conditional otherwise they will be committed (unless they adhere only to a very rebarbative logic) to ?? P for arbitrary P.

The naive theorist may also want to deny the converse conditional--if X ?? A then X [??.sub.S] A. This will be so if she treats the class bracket operation {x: [Phi]x} as a semantic constant not up for reinterpretation from one interpretation to the next.(15) For no set-theoretic model can yield an admissible interpretation of the naive theory of sets since it is a (naive) theorem that the powerset of the universal set is a subset of the universal set(16) whereas on all set models the power set of the domain of individuals is of larger cardinality. For the naive theorist who reads "??" as "mathematical consequence", such set-theoretic interpretations are no more admissible than are models in which [inverted]Ax ranges over only a proper subset of the domain of individuals.

In sum, if one is genuinely interested in finding a general and systematic semantic theory for our overall set theory but worried about the problems this faces, specifically the paradoxes which semantic closure seems to engender, neither the Kreisel argument nor appeal to Reflection Principles are of any use, however important they may be in other spheres. The Kreisel argument makes essential use of principles rejected by proponents of alternative solutions to the problem; and both approaches either make use of semantic ascent to a metalanguage, thus refusing to face the problem, or else make uncritical use of the very naive notions it is claimed are incoherent.

5. Proper classes are proper Charlies

Another common response to the semantic problem for set theory is to appeal to proper class theory, conceived of either as a multi-sorted first-order theory or as second-order logic in disguise. In such theories, for instance NBG, proper classes are those classes c for which c [element of] s is always false and there corresponds to any extension containing only sets--defined as improper classes, classes which do belong to classes-- a proper class. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exists as a proper class. However if proper class theory is interpreted as a multi-sorted, first-order theory with two types of individual---sets and proper classes--then this is entirely unsatisfactory.(17) For one thing, a theory in which no class corresponds even to the tiny extensions of predicates such as [Lambda]x(x = c), c naming a proper class (since proper classes belong to nothing, the unit set {c} does not exist), is surely very incomplete as a theory of classes. It is not much help being able to refer to, for example the class of all ZF ordinals, if one cannot perform even such simple set-theoretic operations as forming unit sets or, in combination with other classes, ordered pairs, on such a class. Moreover, although we can refer in NBG to the entire cumulative hierarchy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of pure sets, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] since is a proper class it cannot belong to an interpretation or to an interpretation function where these are construed class-theoretically; so we are still unable to develop a class-theoretic semantics for set theory. Finally, the theory of proper classes, as a first-order theory, is supposed to contain some of our lore on class-like entities. So we have as much reason to wish to provide a semantics for its intended interpretation as for the intended interpretation of set theory. In NBG theories, though, we are unable so much as to refer to a class of all classes, as opposed to set: the domain of of pure sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not represent the entire domain of class-like entities.

However since c [element of] s is always false, for proper classes c, it is possible to exclude such contexts as illegitimate and re-interpret legitimate contexts s [element of] c as terminological variants of Cs, C a predicate. Proper class theory then becomes simply a second-order set theory whose strength is determined by the strength of the comprehension principles in the second-order logic. It is then possible to develop a semantics for first-order set theory (at least if there is an upper bound on the adicity of predicate constants in the language) using logical rather than set-theoretic notions--interpretations are relations, not first-order functions, the domain is replaced by a property true of exactly the individuals and so on. But this will require full second-order logic--in generalising over interpretations when defining logical consequence, for instance--and this raises two further problems.

Firstly, it seems psychologically impossible for us to work in higher-order logic without reifying the particular properties or relations we are working with: we can conceptually manipulate the corresponding set much more easily than the property. We would therefore be able to sleep better in our beds at night if we knew that, in doing so, we are not engaged in something highly disreputable, according to our official, publicly professed, dogma. Secondly, if we develop a semantics for first-order set theory in second-order logic then the question inevitably arises as to the semantics for that second-order theory. If we follow the same path as before, this will be given in third-order logic, or in some reduction of third-order logic into a stronger second-order theory (cf. Shapiro, 1991, [sections] 6.2) and we are off once again on the familiar ascent up a hierarchy. This response is an objectionable case of pushing the problem one step back because at each stage of the hierarchy we are still faced with the same problem we started from--the absence of a theoretical account of the language of (higher-order) set theory or a theory containing a part essentially equivalent to it.

6. Plural quantification

Supporters of plural quantification--the view that there is a separate and antecedently intelligible non-singular style of existential quantification found, for example, in "there are footballers who play well only with each other"--seek to avoid such problems by doing without functions assigning proper classes to second order variables and doing without domains of quantification in characterising validity. For example, Boolos's idea is that one replaces assignment functions with relations, in general polyvalent, between second-order variables and sets (not proper classes);(18) then, interpreting second-order quantifiers via plural quantification, one shows meta-theoretically how to define schematically "true in L" in L and how to define a schematic notion of "super-validity" (but not logical consequence--Boolos 1985, pp. 336-44).(19) However such "schematic definitions" are, on the one hand, clearly not explicit definitions, for example of truth and validity; and on the other hand their interest surely lies in their approaching as closely as we dare to explicit definitions. For a schematic definition of a concept [Phi] applicable to sentences s is essentially a generalisation of the form:

[inverted]As s is [Phi] [equivalence] [Delta] (s) is true.

Here the intended interpretation of underlining is that it represents a function mapping every expression string e of the object language (all such strings are assumed to belong to the intended domain) to a name e (also in the object language) such that e names, in the intended interpretation, e. Furthermore A represents some syntactic operation, an operation which is empty in the case of defining truth for which the schematic definition is thus the (rather unilluminating)

[inverted]As [s is [true.sub.O] [equivalence] s] is true

with s ranging only over sentences of the object language O. The distinction between object language [true.sub.O] and metalinguistic truth gives the game away. Such schematic definitions, do not, I submit, amount together to an explanatory semantic theory of how our words link with the world, any more than does the naive disquotational truth schema, which they strive towards but do not quite reach. By a semantic theory I mean a general theory of word/world relations which exhibits some systematic structural features of these relations which are likely to be of explanatory value in accounting for how speakers of the language can understand it. So a schema of the form "s is true iff p" is not useful if one is after a general semantic theory of the above type, even given substitution rules which generate only true instances; but a recursive theory of truth and satisfaction might be helpful in this task.

7. Old news?

An obvious objection to be made to all of the foregoing is that underlying the above points lies one fundamental objection, namely that one cannot give a semantics for a set theory from within that set theory; no matter how we strengthen our meta-theory so that it can handle a previously considered theory, that meta-theory itself will not be able to express its own semantics. But, the objection continues, this is not news, this is history. We have here simply the impossibility of semantic closure, as demonstrated by Tarski; and it is no objection to a theory such as set theory that it cannot achieve the impossible, however attractive that mission impossible looked prior to the demonstration of its unattainability.(20)

An initial counter-response is that the semantic problem is not the only problem with the abandonment of naive set theory. There is an apparently irresistible temptation to speak of the set of sets (often using some genteel euphemism such as the "universe" of sets or "totality" of sets) and though one can do so without returning to full naive set theory,(21) the naive theory is still the most natural home for a universal set. Furthermore there seems to be an equally irresistible temptation to use what looks suspiciously like the naive notion of class in informal explication and justification. Take, for example, justifications of the "smaller" large cardinal axioms via closure arguments: we start from some point of in the cumulative hierarchy which seems safe, and some intuitive operations for generating larger ordinals from each [Beta], [Beta] [is less than or equal to] [Alpha]. (For example, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [??.sub.[Gamma] is a ?? function with [??.sub.0] = [Beta], [??.sub.[Gamma]+1] being the cardinality of the power set of [??.sub.[Gamma]] and taking unions at limit ordinals. The closure of {[Omega]} under this operation is just the set of cardinals less than the first (uncountable) strong inaccessible.) Apply the operations repeatedly; the class which is the closure under those operations is a strictly increasing class of ordinals so it is an ordinal itself (if we use von Neumann ordinals) bigger than any member of that class. Hence there exist ordinals inaccessible by those operations.(22) Often these arguments are very persuasive but they seem on the face of it to use naive class theory in forming the closure class and assuming its members have the property specified in the defining condition.

The more important counter-response is that Tarski demonstrated the impossibility of semantic closure only for a very special type of interpreted language--roughly, classical bivalent languages. He drew, however, strongly pessimistic conclusions about the prospects for a semantic theory for natural languages. But natural language is, after all, the language in which, together with some technical adjuncts, mathematics is carried out; thus Tarskian defeatism on semantics leads to a scepticism as to the possibility of our acquiring a theoretical understanding of what mathematics is about, a scepticism uncomfortably close to Russell's definition, in his 1901 "if-thenist" period, of mathematics "as the subject in which we never know what we are talking about" (1901, p. 75). This scepticism has been challenged in the last decade or so, for example by those seeking to give semantically closed truth theories such as inductive theories of truth (Martin and Woodruff 1975, Kripke 1975, McGee 1991). The motivation here, I take it, is to lay the groundwork for a genuinely explanatory semantic theory of a natural language and confute Tarskian scepticism about semantics.

8. Contra the Pseudo-Tarski

Why did Tarskian scepticism remain the dominant position for so long? One reason might be that its extreme consequences were not clearly perceived, perhaps as a result of the seductiveness of the following "pseudo--Tarskian" line of thought. The pseudo--Tarskian starts by conceding that we are not able at any given time fully to conceptualise the semantics, the word/world relations, of our current language. However, though we cannot explicitly and fully express, for example the notions of truth and satisfaction for our current language, we can, according to the pseudo--Tarski, whistle those notions in a highly articulate manner. The idea is this: we can fully conceptualise the semantics of certain proper fragments of our current language. We might think of such fragments as [L.sub.[Beta]] [Beta] [is less than or equal to] [Alpha], where our current language is represented by [L.sub.[Alpha]+1] in some hierarchy--Tarskian, inductive, revision-theoretic or whatever--of languages (or models of language). Because of this, we have a fair idea what a truth theory or model theory for our current language [L.sub.[Alpha]+1] would look like.(23) Such a theory will look to the speakers of [L.sub.[Alpha]+2] ourselves five minutes from now, perhaps, or as we would have been in a counterfactual situation in which we spoke [L.sub.[Alpha]+2](24)--pretty much as La looks to us now from the vantage point of [L.sub.[Alpha]+1].

If this was the conventional pseudo-Tarskian view then it was a hopelessly confused one. The problem lies not with the admission that there are some concepts inexpressible in our current language--this is perfectly true. Nor does it lie with the construction of hierarchies of truth-like concepts. Thus we can conceive of a language L with a two place relation True(x, y) intuitively read as:

y is true in language x

and a partitioning of L into a cumulative hierarchy of sub-languages [L.sub.[Beta], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; here whether a iff [Phi] belongs to a given [L.sub.[Beta] is determined in some way by the constants substituted for x, or the nature of the binding of any variable substituted for x, in occurrences of True(x, y) in [Phi].(25) It will also be the case that for at least some sub-languages [L.sub.[Gamma], the theory of [L.sub.[Gamma] will contain all T-sentences of the form True([Gamma], s) [equivalence] p, where p is any member of a language [L.sub.[Alpha], [Delta] [is less than] [Gamma] and s canonically names p. In particular, this will have to be true of the sub-language which represents our current language, or any extensions of it which we will generate by, for example, reflection on the semantic concepts of our current language.

The pseudo-Tarskian supposition is then that our current language is one of the sub-languages in this hierarchy, in particular, [L.sub.@], where @ abbreviates the definite description

the ordinal [Alpha] such that [L.sub.[Alpha]] is our current language

(@ belonging, of course, to [L.sub.@], since "the ordinal [Alpha] such that [L.sub.[Alpha] is our current language" belongs to our current language). However, granted the usual diagonalisation resources, there will be, in [L.sub.@], a super-paradoxical wff [Sigma] of the form ~ True(@+ 1, [Sigma]). This generates a paradox since, if [Gamma] is in fact the level of the current language, then we must have, by our assumption about the theories of the current language and its successor, the T-sentence

True([Gamma] + 1, [Sigma]) [equivalence] ~True@ + 1, [Sigma]) as a member of the theory of [L.sub.[Gamma]+1]. For the right-hand side is a member of a language further down the hierarchy since @ is a term of [L.sub.[Gamma]] as are the canonical representations of ordinal addition, the resources for diagonalisation and so forth. But it will also be true, in the intended model for [L.sub.[Gamma]+1], that [Gamma] = @, whence

True([Gamma] + 1, [Sigma]) [equivalence] ~True([Gamma] + 1, [Sigma])

also holds in that intended model. The pseudo-Tarskian is presumably not prepared to accept that sentences of the form p [equivalence] ~p can be true (else why abandon the naive theory?). So the pseudo-Tarskian is forced by reductio to the conclusion that there is no such intended model and so there is no such hierarchy containing (a regimentation of) our current actual language and such that there is a coherent and acceptable semantics for each member of the hierarchy.(26)

We must conclude that the hierarchy is constructed in a language, the current language, with resources greater than those in any language of the hierarchy; but then we have no reason at all to think that the relation of languages from [L.sub.[Alpha]+2] and upwards to [L[Alpha]+1] tells us anything at all about the general character which a semantic theory for our current language--the language in which the hierarchy is constructed--will actually have, since this language does not occur in the hierarchy. We have no grounds at all, just from the construction of consistent hierarchies [L.sub.[Beta]] none of which contain the semantic concepts of our current language, to suppose that a being who could conceptualise the links between our language and the world (if there could be such a being) will utilise concepts which bear structural similarities to the artificially restricted truth concepts from our hierarchy. No reason, for example, to suppose that the "God's eye" analogue of ersatz Tarskian truth will be such that if a conjunction possesses the God's eye truth property, so do both conjuncts. This means that though we can show, for example, that & E--conjunction elimination--is sound with respect to preservation of ersatz truth, formal semantics utilising this notion gives us no reason at all to suppose that & E is really sound, that is sound with respect to real truth; more generally, formal semantics gives us no grounds for supposing any of our logical rules or principles are sound. For, as the super-paradoxes generable by supposing our current language is a member of the hierarchy of meta-languages show, the predicate "is true" as applicable to our current language is not one conceptualisable in any of the languages of the hierarchy, however far into the distance we imagine it extending.

This conclusion that, on the conventional hierarchical approach, we can have no explanation of the soundness of any of our logical rules in any of their applications may seem grossly overstated. Even if we cannot carry out a semantics for the entire home language, one might counter, we can do so for proper fragments. So there is no question but that we can explain how, for example, applications of & E in say, classical analysis, or ZF set theory are sound, even if we cannot say the same for applications of the rule in the semantic theory, stronger than analysis or ZF, in which we carry out the semantics of the sub-theories.

To see this is a mistake, compare naturalised epistemology. Whether or not this displaces traditional epistemology, there can be little doubt that a naturalistic account of our perceptual knowledge, for instance, is a worthwhile and non-trivial task. It is very far from accomplished, and not only because the physical story from ear drum or retina backwards into the brain is very far from complete; we also need an explanation of the relation between the physical states and changes occurring in the developing physical account and our perceptual states, and this explanation, even if wholly naturalistic, cannot be wholly non-philosophical. Never the less a large (though not unchallenged) body of opinion holds that the project of searching for a broadly physicalistic, or at any rate non-dualist, account of perceptual knowledge, one which perforce will utilise resources from perhaps our entire current body of scientific knowledge, is not a hopeless one.

Compare, now, a crude dualist. This dualist maintains that the brain activity subsequent to nerve triggerings in the ear or eye gives rise to perceptual knowledge by causing changes in the state of an entirely non-physical, non-located mental substance, the mind. Suppose that, faced with the standard challenge to explain how the physical brain and non-physical mind interact, our dualist says that they do so via transformations in an intermediate substance, [I.sub.1], a substance not well classified as physical but not properly classified as mental either. Very well, we ask, how does the brain cause changes in [I.sub.1], and [I.sub.1] changes in the mind? As to the first, our dualist says it does so via the actions of a second intermediary substance [I.sub.2], which interacts with the brain via a third intermediary [I.sub.3] and so on ad infinitum. The crude dualist then protests at our refusal to take all this seriously by maintaining that the regress is a virtuous not a vicious one. Is this protest just? Is her theory merely incomplete, as the standard physicalist story is?

Of course not. The whole project is hopeless and it is hopeless because the epistemology is supposed to explain our knowledge. An infinite regress at which there is no explanatory power at all at level of unless the explanation at level [Alpha] + 1 works is indeed vicious, for the usual reason: explanation must come to an end. What I want to suggest is that standard semantics based on the conventional hierarchical approach to set theory, or something similar to it, is in exactly the same situation as the crude dualist. If the task is to explain the soundness of the inference rules and principles in language L, an explanation in L itself, one which appeals to those very rules and principles and no more, would be bootstrapping, as soundness principles always are; but it would not be a trivial exercise, just as naturalised epistemology in general is not trivial though it cannot justify our knowledge on some basis of Cartesian certainty. Such "semantically closed" explanations of soundness are not trivial even because there is no guarantee that we will be able to achieve them the intense difficulty of producing a coherent, non-dialetheic but semantically closed semantics reveals this starkly.(27)

By contrast, an "explanation" of the soundness of L, in a metalanguage ML in which we use not only the logic of L but a stronger logic (the L logical schemata applied to all instances drawn from ML and not merely L, for example), is of absolutely no value whatsoever as an explanation of soundness. It is far too easy: take any language L you like which contains a conditional, biconditional, and a necessity operator and with, as its background logic, any old set of rules and principles you like so long as it includes the following:

[right arrow]I, [right arrow]E, [equivalence] E, []I subject to standard classical structural rules.

Form ML by adding a new predicate "is true" for which you add a disquotational truth schema with only L instances and take over all the rules of L expanded to include ML sentences as instances. Then we will be able to transform any proof of B from A into one of [](True A [right arrow] True B), that is a proof that necessarily the conclusion is true if the premiss is; this just is the intuitive definition of soundness, for one premiss/one conclusion arguments. We can do this regardless of how ridiculous the other rules of L are as follows:
   A    (1) B                                  Given
   2    (2) True A                             Hyp.
   -    (3) True A [equivalence]  A            Disquot.
   2    (4) A                                  2,3 [equivalence]E
   -    (5) A [right arrow] B                  1 [right arrow]I
   2    (6) B                                  4,5 [right arrow]E
   -    (7) True B [equivalence]  B            Disquot.
   2    (8) True B                             6,7 [equivalence]E
   -    (9) True A [right arrow] True B        8 [right arrow]I
   -    (10) [](True A [right arrow] True B)   9 []I

All this stands in sharp contrast with the semantically closed case: it is by no means easy to get a semantically closed account of "is true" plus a background logic which will yield a soundness proof and not also a proof that the system is unsound.

The conventional view that soundness proofs are of limited but never the less non-negligible worth confuses, then, soundness from within with soundness from without. The former is of limited value, just as it is of limited value to hear someone aver "I tell the truth". Limited, because an inveterate liar could say this; but still better than hearing "I tell lies" or discovering that the person is incapable of saying anything on that matter. In like fashion, naturalised epistemologies for perceptual knowledge or internal soundness proofs for a semantic closed system do not bestow absolute certainty but are not to be sneezed at; they demonstrate that the system can survive a certain internal stability test and failing to pass it would be an ominous sign. By contrast, demonstrating the soundness of a logical system from within a more powerful one is indeed a trivial result. It is as if one was to believe everything the early edition of the morning paper said solely on the basis of what one read in the later edition, where the later edition is exactly the same except for the added headline "Everything Early Edition Says True. Shock!".(28) Standard soundness proofs are indeed not entirely without value since they link up the formal rules with model-theoretic notions which are of interest for the light they throw on the power and expressibility of the system. But they do not help in the least to show that the rules of the system are truth-preserving.

Not that I think we should entertain serious doubts about, for example, & E. Never the less with respect to this--and just about every other logical rule or principle---serious philosophers can be found who do entertain doubts and provide reasons for this doubt which are not obviously idiotic: for example, relevantists who maintain that there is a genuine conjunction operator--"fusion"--for which &E fails (cf. Read 1988, Ch. 3 and pp. 133-4). One might take the Anselmian line here: "the heretics of logic must be hissed away" (quoted by Burgess 1983, p. 41) but to do so is already to move virtually the whole way down the line towards semantic nihilism. One might prefer instead another Anselmian idea: that one's initial faith in our logic be deepened by philosophical understanding. However I think we ought to allow non-dogmatically that some revision of our beliefs as to logical correctness as a result of such a process is possible. At any rate, it is simply not true that persuasive and fruitful debate is not possible between two philosophers who disagree with respect to a fundamental logical principle. The alternative semantic nihilist view results in a dogmatic blind faith in the logic one has imbued from the orthodox authorities of the day. Philosophers have complacently assumed that they could abandon naive set theory without having to embrace nihilism and blind dogmatism. My claim is that one cannot have one's cake and eat it in this fashion: either re-think opposition to naive set theory or openly embrace the impossibility of semantics and the necessity of total dogmatism in logic (and if there, of all places, why not everywhere else?).

In sum, the conventional position, including the pseudo-Tarskian standpoint, is unstable; the only remaining option for a proponent of the hierarchical position is to embrace Tarskian nihilism towards semantic theory. Some, indeed, have been attracted to such nihilism for independent reasons; but while perhaps not incoherent,(29) this view is surely highly unattractive in its intellectual defeatism and tendency towards dogmatism. Indeed the interest in developing definitions of truth-in-L, in L, for example via inductive definitions of truth, shows that the huge cost of Tarskian defeatism has become more and more evident. Not that inductive theories can succeed where others have failed, as the notorious (and thoroughly substantial) ghost of the metalanguage shows. The inductivist gives a semantics for an object language L containing a truth predicate T in which it is always true in certain fixed point models that T A [equivalence] A. However the notion of truth-in-the-fixed-point-model cannot itself be part of the object language itself, on pain of paradox, so that Tx is not a truth predicate for our entire language, but only for a proper sub-fragment of it.

This is the "holiday luggage" problem: just when you think you have semantics all squeezed into a neat packing case, for example by classing liar sentences as neither definitely true nor false--the theoretical notions you have introduced to deal with the problem burst out elsewhere (cf. Priest 1994). Thus in the case of set-theoretic semantics for standard set theory, definitely true will be characterised in terms of taking some value, 1, say, in the actual model and then we come to grief over sentences such as

[Delta]: Val [Delta] [is not equal to] 1

where Val is a set-theoretically defined function characterising truth in the model.

I should emphasise, though, that I intend no set-theoretic imperialism here. I do not assume that set theory is "the" foundation of all mathematics nor that the set-theoretic discipline of model theory is the only route to a general and systematic semantics. Other concepts, such as those of category theory, may well achieve the generality sought by set theory as well or better but there seems no reason to suppose that they will be better able to achieve semantic closure by enabling the construction in the object language of, for example, a complete category-theoretic semantics for all of category theory.(30)

The above argument strongly indicates, I suggest, that the programme of looking to hierarchies to resolve the semantic problem for set theory (and, I have argued, for language in general) is a seriously degenerating one and that we should forget our obsession with rank, and return to our naive notions of sets and class.

9. Schematism and the many pure concepts of Set Theory

Perhaps, though, the difficulties in providing an adequate semantics for set theory arise from our adopting too absolutist a perspective: from our thinking model-theoretically so that in the actual interpretation the quantifiers range over a single, fixed, sharp domain. Some, such as Charles Parsons and Shaughan Lavine, have diagnosed the source of the difficulties with the interpretation of non-naive set theory in a failure to appreciate an underlying vagueness, indeterminacy or systematic ambiguity in the notion of set.(31) It is a mistake, on this view, to think that there is a single absolute interpretation of set theory; rather there is a spread of relative interpretations where, in some interpretations, extensions which were proper classes in "less advanced" interpretations "become" sets.(32)

Now appeal to the notion of ambiguity immediately raises the expectation that we can disambiguate by relativising to some parameter; similarly indeterminacy suggests the availability of a fixed class of precisifications. However, as Parsons and Lavine recognise, it would be fatal to succumb to the temptation to think in this way, for then, once again, super-paradoxes would arise with respect to the set (class, totality or whatever) which is the union of all the domains generalising over all disambiguations or precisifications (Parsons 1977).(33) So one must see the ambiguity or indeterminacy as inescapable, as holding "all the way" up through every level of semantic ascent. Two problems, at least, arise as a consequence. Firstly one is committed to an ineluctable relativity of interpretation for mathematical sentences, including those with the minimal amount of explicit or implicit context-dependence; and this means we cannot rule out, at least for some sentences, relativity of truth. This does not sit well, to say the least, with a realist account of mathematics, at least to the extent that realism is taken to include rejection of relativism as a component. Secondly (and this problem also arises on anti-realist assumptions) exactly the same criticism of theoretical nihilism can be advanced against this position as against the earlier more conventional views.

True, it would beg the question to assume that the "indeterminist" proponent of the systematic ambiguity theory cannot produce a systematic semantic theory simply because that theory would contain essentially ambiguous or indeterminate terms, such as schematic variables. The demand that we reduce or eliminate the ambiguity at some level will be rejected as illegitimate. The indeterminist might, for example, give a semantics from within a metatheoretic framework of a set theory such as ZF amended by the excision of initial universal quantifiers.(34) The resulting free variables are then to be interpreted not as implicitly universally quantified but rather as "schematic", whatever exactly this might mean.(35) The question arises, though, as to what semantic treatment such a theory would give of an object theory which itself contained schematic variables (e.g., in the most interesting case, where the object theory is the meta-theory)? It is hard to see what formal treatment could be given other than relativising schematic sentences to assignments to schematic variables. If the assignments are functions over some domain which is an object of the metatheory then we have accumulated all sub-domains of the hierarchy of relative interpretations after all and are back with standard model-theoretic semantics; and if we are not back there it is hard to see how we can give an explicit definition of satisfaction and thence of logical consequence. If some alternative semantics of a type already considered--plural quantification, use of second-order logic etc.--is intended then we can ask how the use of schematic variables answers the objections given in previous sections to such alternatives?

It seems clear that the indeterminists reject the project of giving a systematic semantics, one which could help explain the nature of our understanding of mathematical language. Parsons emphasises the primacy of the use of a language over semantic reflection on it (Parsons 1974b, fn. p. 240).(36) If the point is that a community could have a meaningful practice without possessing a systematic theory of that meaningful practice then that is obviously correct. But if the point is further that we, of necessity, belong to a community of that type, incapable of comprehensive systematic reflection on our language, then this is Tarskian nihilism. Truth and meaning are, for the indeterminist, relative to an "I know not what", to a parameter or parameters which we can never hope to conceptualise or to express in any language we are able to acquire; and the indeterminist gives us no reason to suppose that the general semantic characteristics of those proper sub-languages which we can semantically reflect on generalise to the metatheory within which we do our reflecting. These fundamental determinants of meaning and truth must for ever remain beyond our ken. Again, even if Putnam is wrong in thinking this approach is incoherent (1990, p. 15), it seems to me to be represent a form of intellectual despair which we should try very hard to avoid.(37)

10. Indefinite extensibility

Perhaps the nihilistic consequences of an indeterministic view on sets flow not from the position itself but from the assumption of a background framework of classical logic. Such a framework, after all, is rather implausible from an indeterminist perspective. For the admission of propositions lacking determinate truth values (or not assertibly having truth values) renders it very difficult to maintain that classical logic is unrestrictedly valid. Perhaps, then, we should investigate the implications of an indeterminacy view combined with a non-classical logic. One interesting case of such a combination is Dummett's discussion of what he calls indefinitely extensible concepts and his idea that it is intuitionist, not classical logic, which is the correct logic--in mathematics, at any rate (for a similar position see Lear 1977).

Following Russell, Dummett characterises an indefinitely extensible property P as one for which any putative extension E yields a further instance of P not belonging to E.(38) As examples, he gives the property of being an ordinal and the property of being a set. Clearly the theory of extensions presupposed here is not naive set theory with extensions defined as naive sets of instances of properties. For then there would be no question of there being a set which did not belong to the extension of the property set, that is which did not belong to the set of all instances of the property set. Rather extensions seem to be identified with "sets" of property instances in a framework of a fairly standard set theory in which not only ordinals but also sets, due to the presence of an axiom of foundation, perhaps, cannot belong to themselves. On this way of understanding Dummett's argument, we can conclude that any interpretation of "ordinal" or "set" can be recognised, in the metatheory, to fall short of including all ordinals or sets. We can then reinterpret "ordinal" and "set" in terms of more comprehensive concepts, say those used in the metatheory; but (by Godel's second incompleteness theorem, perhaps) this reinterpretation must take place in a yet stronger theory and we are off on a sequence or hierarchy of theories and interpretations, a sequence strongly parallel to the hierarchy of Tarskian truth predicates.(39)

Now if this was all Dummett was prepared to say about indefinitely extensible concepts then Fitch's strictures on the self-refuting nature of hierarchical theories would apply once more. In what theory do we describe this hierarchy of indefinitely extensible concepts whose "extensions" always fall short of completely characterising the notions we are after? In particular, what theory of "extensions" is being used? Certainly, whatever our background set theory, we cannot identity "extension of property P" with "set of objects which have property P (are instances of P-ness)". For then the thesis of indefinite extensibility (beginning of the previous paragraph) as applied to the property of being a set would claim that the set of sets (extension of the property set), is not a set, not an instance of the property set. Nor can an extension be defined as a [set.sub.i] of instances of P, for some specific ordinal i, where the sets subscripted by index i are all and only those things assigned to the extension of "set" at the ith level of Dummett's hierarchy of indefinitely extensible concepts. For if Dummett holds that the stretches of discourse in which he outlines the indefinite extensibility theory are not correctly interpreted by level i of the hierarchy then the identification of "extensions" with the sets/is wrong. If, though, they are correctly interpretable by level i then Dummett is committed, since he holds that the properties of being a set and being non-self-membered have extensions, to the existence of, for example a set/of all [set.sub.i]s. For the extension of the property set is, when we make explicit the subscripts indicating the level of the hierarchy at which the current piece of discourse occurs, the set/of all and only those things which instantiate, at level i of the hierarchy, the property of being a set. Similarly he is committed to a [set.sub.i] containing just those [set.sub.i]s which do not belong/to themselves. Paradox reemerges in its original form.

One could try to claim that context can never be held fixed over any significant stretch of an argument leading to paradox; the interpretation of "set" and "ordinal" shifts from occurrence to occurrence throughout the course of the reasoning leading to the Russell or Burali-Forti paradox, no matter how strenuously one tries to focus one's semantic attention and hold it fixed. This is the position adopted with respect to the notion of truth by Burge (1979). But I find it hard to think of a more desperate response to the paradoxes than this, except perhaps the dialetheic acceptance of true contradictions. Even if true of the unreflective use of "set" or "true", so that they must be thought of as having an implicit "schematic" and nonquantificational index shifting across paradoxical pieces of reasoning, can we plausibly maintain this for more reflective notions such as "set of satisfiers at stage i of `set'"? To deny that the variable "i" here is contextinsensitive and can be generalised on comes perilously close to self-refutation (cf. Priest 1995, p. 170). Moreover the resulting position is clearly another form of semantic nihilism. As with some interpretations of quantum mechanics, in the very attempt to "measure" our concept of "set" or "true" we knock out the old concepts expressed by the terms and replace them, in a "lawless" fashion, by new ones. At no stage have we the resources to capture the nature of our language at that stage.

It is difficult, then, to see what theory could play the role of a positive theory of "extension", if not naive set theory. Thus Dummett writes:

Better than describing the intuitive concept of ordinal number as

having a hazy extension is to describe it as having an increasing

sequence of extensions: what is hazy is the length of the sequence,

which vanishes in the indiscernible distance. (Dummett

1991, pp. 316-7)

Yet "the sequence", and "the length" of that sequence, are singular terms which look very much as if they belong to naive set theory. On the face of it, Dummett's sequence enumerates the naive class of all ordinals, so that "the length" refers to the order type of this class. Moreover, even if Dummett refrains from using singular terms purporting to refer to the class of ordinals or its order type,(40) he still has to generalise over ordinals, intuitively construed. More generally, it is essential to the characterisation of the concept of indefinite extensibility that we quantify over all the extensions assigned to the problematic concepts; only by doing so can we affirm that each such extension can be extended further. (Cf. Dummett's generalisation over "any definite totality", 1991, p. 316.) In such a generalisation, the concept of "extension" cannot be identified with any of the concepts of "set" whose extensions we generalise over.

It is not that Dummett proscribes use of notions other than standard set-theoretic notions whilst self-refutingly adumbrating a hierarchy using non-standard concepts. Even in his first Frege volume a distinction is drawn between different types of totality(41) and this is developed more fully in the second Frege volume (Dummett 1991, pp. 316-20). There he explicitly distinguishes, as indicated, "intuitive totalities" from "definite totalities", and precise, formal concepts from "vague", "hazy" ones such as the length of the class of ordinals, whilst apparently endorsing the legitimacy of using the hazy notions. The distinction between an intuitive totality (or a naive totality, one might say) and a definite one is that bivalence with respect to quantifications over the totality in question holds for definite but not intuitive totalities. It does not follow from this distinction, Dummett emphasises, that "quantification over the intuitive totality of all ordinals is unintelligible" (Dummett 1991, p. 316--cf. 1973, p. 568; also 1994, p. 249). Some such, for instance "Every ordinal has a successor", are determinately true. However, coherent theories dealing with intuitive totalities cannot, Dummett wishes to maintain, be expressed in formal, precise, bivalent languages, citing this as vindication of the intuitionist position.

If formal languages are those for which we can lay down a purely syntactic criterion, perhaps even an effectively decidable criterion, for what it is to be a correct demonstration then I agree with Dummett on the expressive limitations of formal languages. Mathematical demonstrability has never been a purely formal matter nor is the goal of a completely formalised mathematics a rational or attainable one, even in this computerised age. It by no means follows that a constructivist or intuitionist approach to mathematics of the type advanced by Dummett must be adopted.

For one thing, intuitionist logic is not the only well-known system for which excluded middle fails.(42) For another, we cannot simply switch from classical to intuitionist logic and leave all else unchanged---intuitive set theory cannot be naive set theory with intuitionist logic. As Dummett acknowledges in a footnote (1991, fn. 4; see also 1993, pp. 441-2), simply abandoning classical logic alone is insufficient to remove contradiction--naive comprehension is of course intuitionistically inconsistent. So if the theory of indefinitely extensible concepts is not itself to be so indefinite and hazy as to be beyond useful discussion, we need a theory of intuitive sets (or totalities, or whatever one wishes to call them) embodying some general principles, albeit not ones which can be completely regimented in a formal calculus of the standard type.

Nor can we simply take over standard non-naive set theory and switch to intuitionist logic or else use the intuitionist theory of properties, the theory of species. For the latter is every bit as hierarchical as the former--as Heyting puts it:

circular definitions are excluded by the condition that the members

of a species S must be definable independently of the definition

of S; this condition is obvious from the constructive point of

view. It suggests indeed an ordination of species which resembles

the hierarchy of types. (Heyting 1956 p. 39; see also Dummett

1977, pp. 38-9)

The objections to hierarchical theories, however, have the same force when the background logic is constructivist as when classical. So if we wish to generalise over all the definite extensions each of which can be indefinitely extended without ever exhausting the intuitive notions, we need a non-hierarchical theory. Even if Dummett can forswear singular terms referring to "over-large" totalities, something which is not obvious, he still utilises concepts of ordinal and set stronger than those which feature in any of the extensions of standard theories of sets or species generated by the method he sketches; he utilises these concepts when claiming that every definite totality (i.e. definite set) assigned as extension to "set" generates another set, perhaps an "intuitive" one, not belonging to that definite totality.

To conclude, naive set theory seems to be the only feasible candidate for the overarching non-hierarchical theory in which the notion of indefinite extensibility is to be expressed. Granted naive set theory, however, the notion of indefinite extensibility itself will have little work to do, being just another example of the possibility of indefinite iteration of operations such as successor, or the generation of stronger truth or proof predicates, and so on. Indeed, one might well have expected any constructivist attempt to give number theory and set theory a systematic semantics to fail. For constructivists tend to favour the notion of "potential" infinity and deny the existence of "actual infinity". Whatever precisely that means,(43) it would seem to have the implication that mathematical theories deal only with finite objects, though there is an unending supply of such. Hence that unending domain cannot be part of the mathematical theory itself: there can be no mathematical, reflective semantics, for the constructivist. So semantic nihilism would seem to be an inevitable outcome of constructivism.

One should also acknowledge, though, that the potential/actual division occurs just as strongly among standard non-constructivist set theorists, notwithstanding their acceptance of "actual" infinities (accepted if for no other reason than that they are essential for classical analysis---and arguably for mathematical physics in general, though there are dissenters44). Despite the acceptance of actual infinities, the standard theorist thinks of the sequence of transfinite cardinals as a potential, uncompleted infinity in exactly the same way as the constructivist thinks of the sequence of finite numbers. Just like the constructivist, the standard theorist's only hope of coherence rests with semantic nihilism. All attempts to characterise the "domain" (hyper-class, collection or whatever) of all cardinals and more generally to bring the interpretation of the theory as a whole into systematic view meet, we have seen, with disaster. For such a theorist, it would be best not even to whistle quietly in the wind the names of the sets of all cardinals, of all ordinals and of all sets. Her only hope is to plead at each juncture the Fifth Amendment, as Michael Potter has put it (1993, p. 185), or take a vow of Trappist silence (one, though, that few, if any, seem capable of obeying).

11. Conclusion

What conclusions should we draw from all this? Firstly that Dummett's idea that reflective theorising about mathematics will inevitably lead to indeterminacy and necessitate use of a non-classical logic is basically right; but a coherent working out of the idea will require a coherent framework for naive set theory, and that is evidently not an intuitionistic one. A more promising approach, in some respects more radical than constructivism but in others less so, is to abandon some of the structural assumptions of classical logic. Thus there is a natural and fairly sharp division of logical rules into the operational rules, such as the introduction and elimination rules of natural deduction calculi and their sequent versions, and structural rules specific to the proof architecture in question which tell us on what conditions we can chain together applications of operational rules to form proofs of more than minimal length.

What the set-theoretic antinomies show us is that the package of (i) classical operational rules, (ii) classical structural rules and (iii) naive set theory, is trivial and thus hopeless. What I have tried to suggest in this paper is that we have mistakenly fingered the third component as the villain of the piece when in fact it may be the structural rules which are at fault. One option is to accept naive set theory, accept that contradiction follows from it but modify one's structural rules so as to yield a paraconsistent logic which implements the idea that contradictions can be true and indeed necessarily true without trivialisation. As remarked at the outset, in this regard I am on the side of orthodoxy--acceptance of contradiction is to be avoided at all costs.

It needs to be emphasised, then, that acceptance of contradiction is not the only option for those who refuse to abandon naive set theory. Consider once more the example of a standard derivation of Russell's paradox in a Lemmon-style proof-system:
--    (1) [exists]y[inverted A]x (x [is an element
        of] y [equivalence] x [is not an equal to] x)    Axion
2     (2) [inverted A] (x [is an element of r
        [equivalence] x [is not an element of] x)        Hyp.
2     (3) r [is an element of] r [equivalence] r         2 [inverted
        [is not an element of] r                          A] E
2     (4) r [is an element of] r                         Hyp.
2,4   (5) r [is not an element of] r                     3,4 [equi-
                                                          valence] E
2,4   (6) [perpendicular]                                4,5 ~E
2     (7) r [is not an element of] r                     6 ~I
2     (8) r [is an element of] r                         3,7 [equi-
                                                          valence] E
2     (9) P                                              7,8 ~E
--    (10) P                                             1,2,9,
                                                          [exists] E

There are a number of places where contentious structural rules are used, most notably at lines 6, 8, 9. There we apply, in effect (if we think of sequent systems as book-keeping ways of commenting on more basic lower-order systems) a generalised, Cut, or generalised transitivity of entailment, principle of the form:


in which we place no restrictions, syntactic or semantic, on the formulae which may occur in X [intersection] Y.(45) Certainly to add restrictions on transitivity is a radical move,(46) and one which would have to be made in a controlled fashion, if that piling of lemma on top of lemma essential to standard mathematics is to be preserved. Nor could any such restrictions be plausibly thought of as implicit in our actual practice of inference; rather they could only be revisions of that practice, revisions motivated by a desire to make better sense of it overall in the light of the difficulties we seem to land ourselves in whichever way we turn when reasoning about sets and semantics (and perhaps also in some other areas, such as the Sorites paradox). At any rate, whatever the intuitive force of the classical operational rules, it is by no means obvious that the classical structural rules are more intuitively correct than naive comprehension or Frege's Axiom V.

Embarking in a substantial way on this positive programme is well beyond the scope of a single paper: one would have to show that the revisionary logic not only blocked the usual derivations of paradox but was sound in general, with respect to semantics developed in a framework of naive set theory plus the non-classical logic. And much more would be needed besides, for example a demonstration that despite the deep logical restrictions needed to block contradiction, one could still derive an acceptable amount of standard mathematics. In the absence of all this, I cannot reasonably hope in this paper to have convinced readers that naive set theory is indeed innocent. What I have attempted to achieve is an illustration of the inadequacies of at least the most common of the non-naive resolutions of the semantic problem, in particular showing the tendency of such solutions to lead to a dangerously dogmatic defeatism regarding the prospects of an illuminating theory of meaning and understanding. I hope also to have highlighted the possibility that it is the structural principles of classical logic, principles with nothing like the pre-theoretical hold on us of the axioms of naive set theory, which are to blame for the derivation of absurdity. This, I hope, sets out a strong prima facie case for the conclusion that a serious miscarriage of justice may have taken place and a retrial is long overdue.(47)

(1) The general form is to be found in Brady (1983) and Priest and Routley (1989), [sections]3.2.

(2) The dyadic form is [inverted A]R[inverted A]S [inverted A]z({x: Rzx} = {y: Szy} [equivalence] [inverted A]w(Rzw [equivalence] Szw)).

(3) Here I disagree with Hallett (1984), p. 38 where he identifies Russell's Principles of Mathematics as the original source of the naive set theory and likewise with Lavine (1994), pp. 1-3 where he adds Peano to Russell as the originator with Frege a precursor. For some comments on Lavine, and in particular whether Cantor's Inhalt could be construed as naive sets, see Weir (1996, p. 137).

(4) Here and at lines 8 and 9 we tacitly use the classical structural rule allowing us to utilise assumptions--such as those at lines 2 and 4--as many times as we like or, where, as above, sequents are treated as sets, not sequences, of wffs, allowing the same assumption to occur in the antecedents of distinct premisses of an application of a sequent rule.

(5) Thus including the well-known relevant logics E and R. Cf. Read (1988, Sc. 4.3) and Slaney (1989).

(6) Lavine disagrees but only because he thinks naive set theory a non-intuitive invention of Peano or Russell. Cf. fn. 4.

(7) For an early "post-lapsarian" championing of naive set theory see Fitch (1952), [sections]18, pp. 106-11, especially p. 111 and Appendix C, pp. 217-25; also his (1969). A number of logicians have also investigated, from a number of different angles, non-trivial versions of naive set theory, for example, Brady (1983, 1989), Brady and Routley (1989), Hinnion (1994), Restall (1992), and White (1979).

(8) The text has "less" for "more". Thanks to Bob Kirk for preventing me from reproducing the slip.

(9) For the iterative conception see George Boolos (1983) and Charles Parsons (1977). See also Lavine (1994 pp. 5, 135, 143-50). (1) The general form is to be found in Brady (1983) and Priest and Routley (1989), [sections]3.2.

(10) Kreisel (1967, especially pp. 89-93), and Boolos (1985, pp. 339-40). This is also the tack taken by Cartwright (1994, in particular p. 10). For criticism see Etchemendy (1990, pp. 145-48) and, from a somewhat different angle, Read (1994, pp. 254-6).

(11) Shapiro (1991, Sc. 6.3) extends reflection principles to second-order logic but using a theory ZFC2+ stronger than second-order ZFC in a language augmented by the addition of a second-order primitive satisfaction relation which takes first-order predicates in some of its argument places. Truth for the language of second-order ZFC is then definable via the theory ZFC2+ and without any "ontological" expansion--the technique of letting assignments be many-many relations with proper classes in their range enables the ontology of ZFC2+ to be the same as that of ZFC2. However the cost is "ideological" expansion: ascent to a language [L.sub.2.5] which extends standard second-order by adding a second-order relational constant which takes first-order predicate variables and constants as arguments. And of course truth in [L.sub.2.5] is not definable in [L.sub.2.5] via ZFC2+. If the latter is our overall theory of sets, it cannot specify what its own interpretation is. [sections]5 below criticises the appeal to proper classes in the interpretation of set theory.

(12) For the "triumph" of first-order logic over the higher-order and infinitistic systems in which modern logic first emerged, see Moore (1980), (1988); also Shapiro (1991, Ch. 7). One reason for the triumph might be G6del's 1930 completeness theorem for first-order logic which showed that a reasonable system of rules was adequate for the natural semantics for first-order systems. But, as Zermelo in effect urged, why should adequacy vis-a-vis that semantics not tell against firstorder logical systems, given the manifest expressive limitations of first-order semantics as witnessed by e.g. the Lowenheim-Skolem theorems. As Boolos (1975) remarks in a different context, we could argue on those lines even more strongly in favour of propositional logic as "the" logic. Perhaps the triumph owes more to the grip of finitism among leading figures in proof theory of that period than to the philosophical and logical merits of first-order, finitistic, formulations of theories, for these merits seem rarely to have been argued for explicitly (cf. Shapiro 1991, p. 181).

(13) For instance, these links may throw light on strong theorems of infinity.

(14) So T is Axiom V, or else T is the set of instances of naive comprehension plus extensionality, or some other formulation of the theory as outlined in [sections] 2.

(15) Or, similarly, if she treats [element of] as a semantic constant. Whether or not one wishes to class it as a specifically logical constant is more a terminological than a substantive issue, though certainly classing it as non-logical is more in keeping with current usage.

(16) There are set theories with classical background logics in which this holds too: e.g. those of Church and Mitchell for which see Forster (1992, especially Ch. 4). A more natural such theory, arguably, is Oberschelp's (1973). And of course Cantor, who gave us the power set theorem showing how the powerset of any "well-behaved" set S is larger than S, did not believe the theorem applied to his "inconsistent multiplicities": cf. his letter to Jourdain quoted in Lavine (1994, pp. 98-9).

(17) Cf. Kreisel (1967, pp. 90-1); Hellman (1989, p. 54) and Lewis's satire on the NBG approach (1991, pp. 61-71 and especially p. 68).

(18) Here is a contrast with the related technique of Shapiro mentioned in fn. 12. Boolos's procedure will not work for languages containing relational second-order variables (Boolos 1993, p. 218).

(19) Boolos thinks it a reasonable working hypothesis that truth or supervalidity and model-theoretic validity coincide not only in the first-order case but also in the full second-order system in which the semantics is carried out. But he acknowledges this claim appears insusceptible of proof on current axioms (1985, p. 343). Lavine also appeals (1994, pp. 234-5) to schematic definitions in the form of his "class models"--cf. Shoenfield's interpretation (1967, [sections] 4.7)--to solve the semantic problem.

(20) Cf. Boolos's undogmatic suggestion (Boolos 1984, p. 446) that we may simply have to accept the impossibility of semantic closure for the language of set theory, which is thereby no worse off in this respect than the language of semantics.

(21) Cf. the theories mentioned in fn. 17.

(22) For examples of arguments of this type see e.g. Drake (1974, pp. 68, 186); Shoenfield (1967, p. 306).

(23) By truth theory or model theory I intend the vague but workable notion of a theory which functions pretty much as the naive theories were intended to function, by those who thought them classically consistent.

(24) Cf. Hellman (1989, Ch. 2). See also fn. 33 ahead.

(25) Alternatively we could have a hierarchy of primitive, one-place truth predicates, with languages differing solely by which such predicates they contain. But if there is a language in the hierarchy with the expressive power of actual language it must be able to quantify over the total set S of such truth predicates so as to say such things as: no predicate in S applies to wff P; and then such a language, and its extensions, behaves much as L above.

(26) For a similar "superparadox" objection see Priest (1987, [sections] 1.5).

(27) For a sketch of a dialetheic closed semantics see Priest (1987, Ch. 9).

(28) Moreover it would be no mean feat to show that the early edition could include, coherently and without entailing contradiction, "Everything this edition says is true".

(29) Though there may be problems over whether semantic nihilism can even be stated without violating its own precepts. The (not totally precise) claim which the optimist rejects and the nihilist maintains is that there is no general systematic theory of language explaining at the least what the word/world links are which make our sentences true or false (if not what it is to grasp those links). But even the very concept of systematic links between all the terms of our current language and the world is something which some of the nihilist arguments seem to deny we have. Such a nihilist would be in the self-refuting boat of someone who says "I lack the following concept: ..." and then goes on to express the concept.

(30) On the limitations of category theory as a foundation for mathematics, see Bell (1981).

(31) Cf. Parsons (1974a, pp. 218-20); Parsons (1977, pp. 289-97); Lavine (1994, pp. 317-20). However vagueness, at least if the vagueness of "bald" or "fat" is taken as the model, seems hopeless as an account of the indeterminacy of paradoxical mathematical sentences: there is nothing corresponding to the ordering of borderline cases of baldness or fatness, along some fairly well-specified set of dimensions, as more or less bald or fat.

(32) Cf. Lear (1977). This idea can be given a modal reading as in Hellman (1989, Ch. 2). But, as I indicate below, essentially the same criticisms can be made of the modal theory as of the systematic ambiguity theory.

(33) See also Parsons (1974b, fn. p. 240) for the argument that full generality can not be expressed. See Lavine (1994, pp. 317-8). For an analogous point in connection with the Liar Paradox, see Burge (1979, pp. 108, 116-7). Similarly, Hellman eschews a model-theoretic interpretation of his modal structuralist theory, in part for similar reasons: "There is to be no totality of `all possible structures,' nor any union of the domains of all possible structures" (Hellman 1989, p. 59). As he indicates, when he uses a predicate such as "X,f is a full ZF model", it is to be understood as a second-order relational predicate applicable to a first order monadic predicate X together with a dyadic first-order f, and is not to be interpreted in terms of models understood class-theoretically in the standard fashion.

(34) Thus PowerSet becomes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with x schematic, and similarly for Extensionality, Union, Foundation and the instances of the Replacement scheme. One then adds a suitable Substitution Rule for schematic variables to the logic.

(35) More radically one might, with Lavine (1994), adopt a theory such as primitive recursive arithmetic, understood schematically, as the metatheory with which one investigates a recursively axiomatisable theory.

(36) Hellman argues in similar fashion that a modal primitive "must not be thought of as requiring a set-theoretical semantics in order for it to be intelligible" (Hellman 1989, p. 60).

(37) The comparable position with respect to the first-person pronoun is a "subjectivist" view according to which third-personal accounts of the meaning of the pronoun-"I" refers on each occasion of utterance to the utterer and so forth--must inevitably fail to capture something inherently inexpressible. This sort of position is surely unattractive from a naturalist perspective and I suggest the indeterminist position suffers from a similar defect.

(38) Cf. Dummett (1991, p. 317, especially fn. 5). See also Dummett (1963), pp. 194-7, (1973, pp. 532-3), (1993, pp. 441-3,454-5). Those unhappy with talk of properties and their instances can substitute talk of predicates and their satisfiers.

(39) Or of that hierarchy of G6delian proof predicates characterising ever stronger notions of proof which is generated by adding as an axiom at each stage [Alpha] the hitherto undecidable diagonalisation on "unprovably", [Beta] [is less than] [Alpha].

(40) Not an easy a thing to do. Thus Dummett (1994) attempts to rebut Boolos's charge that he trades in such singular terms. But immediately before stating that he has avoided such terms he writes, speaking of objects in the range of variables generalising over ordinals, that "the objects over which the variables range can be well-ordered by magnitude, and their order type will satisfy the criterion for being an ordinal number" (1994, p. 248, my emphasis). He means, of course, the ordertype not of each individual object but of the whole class (set, domain, totality or whatever). True, he is dealing with quantification platonistically construed here, where he argues it always lies to hand to introduce the idea of the totality (nonintuitive presumably) of objects in a given range. But he had claimed to be able to do so without using phrases which, for someone like Boolos, stand for nothing, phrases like "the universal class"; "the order-type of the ordinals" is another such phrase.

(41) Dummett (1973, p. 533) did claim that "there can be no such thing as the domain of all objects"--not simply that there can be no "definite" domain or some such---and again he says of the concept of being non-self-membered, that "we can form no conception of the totality of all objects falling under that concept" (1993, p. 441). On the other hand, he speaks of the totality of all sets, "in the intuitive sense of `set'", being over-large and of the need to distinguish legitimate from illegitimate totalities (1973, p. 516); and he distinguishes a "set" from "a set in the intuitive sense of the term' 1994, p. 248).

(42) For example, there is the strong Kleene logic in which rules are required to preserve the top truth value and logical truths are required to possess it; excluded middle fails in this system though double negation is sound (indeed it also preserves falsity upwards from conclusion to premiss, just as a sound rule ought to do).

(43) For discussion, see Moore (1990, [subsections] 13.4, 15.3); Priest (1995, pp. 136-40).

(44) Who often appeal to the work of Bishop and Bridges; see, for example, Bridges (1987).

(45) In an alternative sequent proof architecture in which antecedents are sequences of wffs, not sets, the contentious move, one blocked in linear logics, is to permit unrestricted contraction.

(46) The move is made by Tennant (1987, Ch. 17) and Smiley (1959, pp. 233-4 and [sections] 2), (following on from work by Geach and von Wright). However both these writers specify criteria which are, or are equivalent to, purely syntactic criteria, for the failure of generalised transitivity, and I believe such criteria have no chance of success in the project of taming naive set theory. Dummett considers but rejects abandonment of transitivity of entailment (1975, p. 252). Restricting the general form of Cut does not necessarily mean abandoning the special case: ifa ?? B and B ?? CthenA ?? C.

(47) Thanks, for critical comments and suggestions, are due to audiences at the universities of Nottingham, St. Andrews and Gent who heard various parts of earlier versions of the paper and to Marcus Giaquinto, Jonathan Lowe, Graham Priest and, especially, Stewart Shapiro.

Department of Philosophy The Queen's University of Belfast Belfast BT7 INN


Bell, J. L. 1981: "Category Theory and the Foundations of Mathematics". British Journal for the Philosophy of Science, 32, pp. 349-58.

Benaceraff, Paul and Putnam, Hilary (eds.) 1983: Philosophy of Mathematics: Selected Readings, second edition. Cambridge: Cambridge University Press.

Boolos, George 1971: "The Iterative Concept of Set", in Benaceraff and Putnam 1983, pp. 486-502. Originally published in 1971 in The Journal of Philosophy, 68.

-- 1975: "On second-order logic". The Journal of Philosophy, 72, pp. 509-27.

-- 1984: "To be is to be the value of a bound variable (or to be some values of some variables)". The Journal of Philosophy, 81, pp. 430-49.

-- 1985: "Nominalist Platonism". The Philosophical Review, 94, pp. 327-44.

-- 1993: "Basic Law V". Proceedings of the Aristotelian Society, Supplementary Volume 67, pp. 213-33.

Brady, Ross 1983: "The Simple Consistency of a Set Theory Based on the Logic CSQ". Notre Dame Journal of Formal Logic, 24, pp. 431-49.

-- 1989: "The Non-Triviality of Dialectical Set Theory", in Priest, Routley and Norman 1989, pp. 437-70.

Brady, Ross and Routley, Richard 1989: "The Non-Triviality of Extensional Dialectical Set Theory", in Priest, Routley and Norman 1989, pp. 415-36.

Bridges, Douglas 1987: Varieties of Constructive Mathematics. Cambridge: Cambridge University Press.

Burge, Tyler 1979: "Semantical Paradox", in Martin 1984, pp. 83-117. Originally published in 1979 in The Journal of Philosophy, 76.

Burgess, John 1983: "Common Sense and `Relevance'". Notre Dame Journal of Formal Logic, 24, pp. 41-53.

Cartwright, Richard 1994: "Speaking of Everything". Nous, 28, pp. 1-20.

Drake, Frank 1974: Set Theory: An Introduction to Large Cardinals. Amsterdam: North Holland.

Dummett, Michael 1963: "The Philosophical Significance of Godel's Theorem", in his Truth and Other Enigmas. London: Duckworth, 1978, pp. 186-201. Originally published in 1963 in Ratio, 5.

-- 1973: Frege: Philosophy of Language. London: Duckworth, second edition, 1981.

-- 1975: "Wang's Paradox", in his Truth and Other Enigmas, pp. 248-68. Originally published in 1975 in Synthese, 30.

-- 1991: Frege: Philosophy of Mathematics. London: Duckworth

-- 1993: "What is Mathematics about?", in his The Seas of Language. Oxford: Clarendon, 1993, pp. 429-45. Also published in 1994 in Alexander George (ed.) Mathematics and Mind. Oxford: Oxford University Press.

-- 1994: "Chairman's Address: Basic Law V". Proceedings of the Aristotelian Society, New Series 94, pp. 243-51.

Dummett, Michael with Minio, Robert 1977: Elements of Intuitionism. Oxford: Clarendon.

Etchemendy, John 1990: The Concept of Logical Consequence. Cambridge, MA: Harvard University Press.

Fitch, Frederic 1952: Symbolic Logic. New York: The Ronald Press Company.

-- 1969: "A Method for avoiding the Curry Paradox", in Nicholas Rescher (ed.) Essays in Honor of Carl G. Hempel. Dordrecht: Reidel, pp. 255-65.

Forster, T. E. 1992: Set Theory with a Universal Set. Oxford: Clarendon.

Godel, Kurt 1946: "Russell's Mathematical Logic", in P. Schilpp (ed.) The Philosophy of Bertrand Russell. Evanston, Illinois: The Library of Living Philosophers, pp. 125-53.

Hallett, Michael 1984: Cantorian Set Theory and Limitation of Size. Oxford: Clarendon.

Hellman, Geoffrey 1989: Mathematics without Modality. Oxford: Clarendon.

Heyting, A 1956: Intuitionism: An Introduction. Amsterdam: North Holland.

Hinnion, R 1994: "Naive Set Theory with Extensionality in Partial Logic and in Paradoxical Logic". Notre Dame Journal of Formal Logic, 35 pp. 15-40.

Hintikka, J. (ed.) 1969: The Philosophy of Mathematics. Oxford: Oxford University Press.

Kreisel, G. 1967: "Informal Rigour and Completeness Proofs", in Hintikka 1969, pp. 78-94. Originally published in 1967 in Imre Lakatos (ed.) Problems in the Philosophy of Mathematics. Amsterdam: North Holland. pp. 138-57.

-- 1968: "A Survey of Proof Theory". Journal of Symbolic Logic, 33, pp. 321-88.

Kripke, Saul: 1975: "Outline of a Theory of Truth", in Martin 1984, pp. 53-82. Originally published in 1975 in The Journal of Philosophy, 72.

Lavine, Shaughan 1994: Understanding the Infinite. Cambridge, MA: Harvard University Press.

Lear, J. 1977: "Set and Semantics". The Journal of Philosophy, 74, pp. 86-102.

Lewis, David 1991:Parts of Classes. Oxford: Blackwell.

Locke, John 1689: An Essay Concerning Human Understanding, ed. P. Nidditch. Oxford: Clarendon, 1975.

Martin, R. L. (ed.) 1984: Recent Essays on Truth and the Liar Paradox. Oxford: Clarendon.

Martin, R. L. and Woodruff, P. W. 1975: "On Representing `True-in-L' in L", in Martin 1984, pp. 47-52. Originally published in 1975 in Philosophia, 5.

McGee, Vann 1991: Truth, Vagueness and Paradox. Indianapolis: Hackett.

Moore, Adrian 1990: The Infinite. London: Routledge.

Moore, Gregory H. 1980: "Beyond First-Order Logic: The Historical Interplay between Mathematical Logic and Axiomatic Set Theory". History and Philosophy of Logic, 1, pp. 95-137.

-- 1988: "The Emergence of First-Order Logic", in History and Philosophy of Modern Mathematics. Minnesota Studies in the Philosophy of Science No. 11., eds. W. Aspray and P. Kitcher. Minneapolis: University of Minnesota Press, pp. 95-135.

Oberschelp, Arnold 1973: Set Theory Over Classes, in Dissertationes Mathematicae, 106. Warsaw: Panstwowe Wydawnictwo Naukowe.

Oxford English Dictionary 1989: Volume III, second edition. Oxford: Clarendon.

Parsons, Charles 1974a: "Sets and Classes", in his 1983, pp. 209-21. Originally published in 1974 in Nous, 8.

-- 1974b: "The Liar Paradox", in Parsons 1983, pp. 221-67. Originally published in The Journal of Philosophical Logic, 3.

-- 1977: "What is the iterative conception of set?", in Parsons 1983, pp. 268-97. Originally published in 1977 in R. E. Butts and J. Hintikka (eds.) Logic, Foundations of Mathematics, and Computability Theory, Dordrecht: Reidel, pp. 335-67.

-- 1983: Mathematics in Philosophy: Selected Essays. Ithaca: Comell University Press, 1983,

Parsons, Terence 1987: "On the Consistency of the First-Order Portion of Frege's Logical System". Notre Dame Journal of Symbolic Logic, 28, pp. 161-8.

Potter, Michael 1993: "Iterative Set Theory". Philosophical Quarterly, 43, pp. 127-93.

Priest, Graham 1987: In Contradiction. Dordrecht: Nijhoff.

-- 1994: "Review of V. McGee's Truth, Vagueness and Paradox". Mind, 103, pp. 387-91.

-- 1995: Beyond the Limits of Thought. Cambridge: Cambridge University Press.

Priest, G. and Routley, R. 1989: "Applications of Paraconsistent Logic", in Priest, Routley and Norman 1989, pp. 367-93.

Priest, G., Routley, R and Norman, J. (eds.) 1989: Paraconsistent Logic: Essays on the Inconsistent. Munchen: Philosophia Verlag.

Putnam, Hilary 1990: "Realism with a Human Face", in J. Conant (ed.) Realism with a Human Face. Cambridge, MA: Harvard University Press, 1990 pp. 3-17.

Read, Stephen 1988: Relevant Logic. Oxford: Blackwell.

-- 1994: "Formal and Material Consequence". Journal of Philosophical Logic, 23, pp. 247-65.

Restall, G. 1992: "A Note on Naive Set Theory in LP". Notre Dame Journal of Formal Logic, 33, pp. 422-32.

Russell, Bertrand 1901: "Mathematics and the Metaphysicians", in his Mysticism and Logic. London: Longmans Green, 1918, pp. 74-96. Originally published in 1901 as "Recent Work in Philosophy of Mathematics" in The International Monthly.

Shapiro, Stewart 1991: Foundations without Foundationalism: A Case for Second-Order Logic. Oxford: Clarendon.

Shoenfield, J. R. 1967: Mathematical Logic. Reading, MA: AddisonWesley.

Slaney, J. K. 1989: "RWX is not Curry Paraconsistent", in Priest, Routley and Norman 1989, pp. 472-80.

Smiley, T. 1959: "Entailment and Deducibility", in Proceedings of the Aristotelian Society, 59, pp. 233-54.

Tennant, Neil 1987: Anti-Realism and Logic. Oxford: Clarendon.

Weir, Alan 1996: "Review of Shaughan Lavine, Understanding the Infinite". Philosophical Books, 37, pp. 136-9.

White, Richard 1979: "The Consistency of the Axiom of Comprehension in the Infinite Valued Logic of Lukasiewicz". The Journal of Philosophical Logic, 6, pp. 503-34.
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