Truth, indefinite extensibility, and Fitch\'s paradox

1 TRUTH, INDEFINITE EXTENSIBILITY, AND FITCH’S PARADOX* José Luis Bermúdez In J. Salerno (Ed.), New Essays on the Knowability Paradox (OUP, 2009) pp. 76-90

Fitch’s original presentation in Fitch 1963 of the line of argument that has come to be known as Fitch’s paradox begins with the notion of a truth class of propositions. A class α of propositions is a truth class just if, as a matter of necessity, every member of α is true. That is, (1)

∀p

[p ∈ α ⇒ p]

Suppose that α is a truth class closed under conjunction elimination and consider the proposition (2)

p & ¬(p ∈ α)

asserting that p is a true proposition that is not a member of α. Assume, for reductio, that (2) is included in α: (3)

[p & ¬(p ∈ α)] ∈ α

Since α is closed under conjunction elimination we have (4)

p∈α

(5)

[¬(p ∈ α)] ∈ α.

and

Since α is a truth class, (5) gives (6)

¬ (p ∈ α)

which contradicts (4) and shows that, as a matter of necessity, (3) cannot be true. That is (7)

∀p ¬ ◊ [(p ∧ ¬(p ∈ α)) ∈ α]

This is Fitch’s Theorem 2. Now, let α be the class of known truths, where q is a member of the class of known truths just if there is a time at which q is known by somebody. Plainly α is a truth class, so that (1) holds. On *

Thanks to an anonymous referee for comments on an earlier draft.

2 this interpretation (2) states that p is an unknown truth – i.e. that there is no time at which someone knows p. Suppose we assume, as seems highly plausible, that there is at least one unknown truth. Let that be p. It follows that it is true that [p & ¬ (p ∈ α)], which is our (2). But Theorem 2 shows that it is contradictory to suppose that there is a time at which someone knows that [p & ¬ (p ∈ α)]. So, not only is there at least one truth that is unknown, but at least one proposition that is unknowable. What has come to be known as Fitch’s paradox derives essentially from Theorem 5, which shows that, provided we accept the existence of an unknown truth, it cannot be the case that all truths are knowable. This is supposed to be paradoxical because there are well-established philosophical positions that maintain precisely the claim that Fitch shows to be incoherent, namely, that it is true both that there is at least one unknown truth and that all truths are knowable. Any form of verificationism or semantic anti-realism appears to be committed to the general principle that all truths are knowable, while no anti-realist or verificationist is likely to accept that all truths are known. However, accepting that all truths are known seems to be the only alternative to denying that all truths are knowable. There is a familiar ab homine response to Fitch’s paradox. It has been pointed out by a number of authors (originally in Williamson 1982) that the argument from the knowability principle that all truths are knowable to the omniscience principle that all truths are known is not intuitionistically valid. Suppose we formulate the knowability principle as (8)

∀p [p ⇒ ◊ (p ∈ α)].

We can substitute the assumption that there is at least one unknown truth into (8) to give (9)

∀p [[p ∧ ¬(p ∈ α)] ⇒ ◊ ([p ∧ ¬(p ∈ α)] ∈ α)].

We recall Theorem 2 (7)

∀p ¬ ◊ [(p ∧ ¬(p ∈ α)) ∈ α]

Trivially, (7) and (9) jointly yield (10)

∀p ¬ [p ∧ ¬(p ∈ α)].

The inference from (10) to the omniscience principle (11)

∀p [p ⇒ (p ∈ α)]

is classically, but not intuitionistically, valid. From an intuitionistic point of view we are entitled only to move from (10) to (12)

∀p [p ⇒ ¬¬(p ∈ α)].

Nonetheless, as it stands this response is hardly satisfying. Although the appeal to intuitionistic logic may well block the move from the knowability principle to the omniscience principle, one

3 can plausibly ask why we should adopt an intuitionistic logic at all. It is true that anti-realists such as Dummett have argued that anti-realism stands or falls with an intuitionistic revision of classical logic. But at the very least, if the appeal to intuitionism is not to be question-begging, we need some independent reason for thinking that classical logic should be revised in the way the intuititionst suggests. The aim of this paper is to take a step back from the details of Fitch’s argument and the particular rules of inference in which it depends in order to explore a line of argument that holds the promise both of undercutting Fitch’s enterprise as a whole (as opposed to simply the deployment of Theorem 5 against anti-realism) and, as a corollary, of explaining why intuitionistic logic is appropriate in this context. This argument has its roots in the notion of indefinite extensibility, as discussed by Michael Dummett in a number of writings (most extensively in Ch. 24 of Dummett 1990). Dummett uses the argument to try to motivate antirealism about mathematics. In particular, he deploys the putative indefinite extensibility of such concepts as set, natural number, and real number to argue for the rejection of classical logic in the relevant domains. Only an intuitionistic logic, he thinks, can do justice to indefinite extensibility. The problem arises when we try to quantify over indefinitely extensible domains. Quantification over indefinitely extensible domains does not always, Dummett thinks, yield statements with a determinate truth value. When we make an existential quantification over an indefinitely extensible domain what we are really doing is claiming to be able to cite an instance, and when we make a universal quantification we are claiming to have an effective operation that is universally applicable. And this requires that the quantifiers be understood intuitionistically rather than classically. How might the notion of indefinite extensibility be applied to the issues about knowability raised by Fitch? Suppose it is the case that the concepts proposition and true proposition are indefinitely extensible, so that there is no definite totality of (true) propositions of which we have a definite grasp. If Dummett’s claims about indefinitely extensible concepts are along the right lines then at the very least we need to inquire into the status of the universal quantifications that are at the heart of Fitch’s reasoning. What is the status of the claim that all truths are knowable, which is required to derive the so-called paradox from Theorem 5? And, for that matter, what is the status of the claim that defines the notion of a truth class? It may turn out that the indefinite extensibility of the concept proposition renders these universal quantifications problematic in ways that blunt the force of Fitch’s work and the conclusions that have been drawn from it. II In order to explore this terrain, however, we need to begin by clarifying the basic notion of indefinite extensibility and the connection that Dummett sees between indefinite extensibility and intuitionistic logic. Dummett gives the following characterization of an indefinitely extensible concept in ‘What is mathematics about?’. An indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under that concept, we can, by

4 reference to that totality, characterize a larger totality all of whose members fall under it. (Dummett 1996, p. 441) Elsewhere he mentions as an antecedent Russell’s diagnosis of the set-theoretic paradoxes in terms of what he (Russell) terms self-reproductive classes. Russell states: The paradoxes result from the fact that there are what we may term selfreproductive processes and classes. That is, there are some properties such that, given any class of terms having such a property, we can always define a new term also having the same property. (Russell 1906, p. 144) Indefinite extensibility is a property of concepts, while it is classes that are self-reproductive. In both cases the problematic phenomena emerge from features of (certain) infinite totalities. The totalities of which we have indefinitely extensible concepts are all infinite and the properties that generate self-reproductive classes are properties defining infinite collections. For Dummett, the indefinitely extensible concepts include the concepts set, natural number, real number, and ordinal. Russell’s list of self-reproductive classes would no doubt be similar, although I doubt that he would have counted the property of being a natural number as generating a selfreproductive class. The fact that, for Dummett, indefinitely extensible concepts invariably characterize infinite totalities does not mean either that indefinite extensibility is really a property of infinite totalities (so that we can distinguish definite totalities from indefinitely extensible totalities) or that every concept that characterizes an infinite totality is thereby indefinitely extensible. We can appreciate both these points, together with the general definition of indefinite extensibility, through examples. One example, which Dummett discusses frequently, is the concept of an ordinal number. One interesting feature of the indefinite extensibility of the concept ordinal is that it reveals itself in paradox (in this case the Burali-Forti paradox). Suppose that there is a set α of all ordinal numbers. This set is transitive (since every member of it is an ordinal and every member of an ordinal number is itself an ordinal number) and it is well-ordered by ∈. Since every transitive set well-ordered by ∈ is an ordinal number (Enderton 1977, p. 191), it follows that α is an ordinal number. But then α would be a member of itself, which no ordinal can be. What generates the paradox is the simple fact that any collection of ordinal numbers gives rise to an ordinal number that is not in that collection (viz. the order-type of the collection). This fits Dummett’s description perfectly. For any collection of ordinals α we can characterize a larger collection, which is the union of α and the order-type of α. The concept set is itself indefinitely extensible. This can be seen in a number of ways. For any given collection β of sets we can characterize a larger collection γ ⊃ β, all of whose members are sets. This is ℘(β), the power set of β – since, by Cantor’s theorem, card (℘(β)) = 2 card ( β ) . But the indefinite extensibility of the concept set can be shown with far less machinery. For any set δ we can construct a set that is not a member of δ. Let A = {x ∈ δ : ¬ (x ∈ x)}. By construction we have A ∈ A ⇔ A ∈ δ ∧ ¬ (A ∈ A). Hence, if A ∈ δ, then we have the obviously contradictory A ∈ A ⇔ ¬ (A ∈ A). So we can characterize our larger totality by taking δ € ∪ {A} – in fact, in a set theory with no self-membered sets, δ = A and the larger totality is δ ∪ {δ}. This is sufficient to show that assuming a set containing all sets leads to paradox. But indefinite extensibility does not entail or require paradox. The indefinite extensibility of the concept real number is revealed by Cantor’s proof of the non-denumerability of the set of real numbers, but there is nothing paradoxical about the fact that the real numbers cannot be put into a one-one correspondence with the set of natural numbers. What Cantor’s proof shows is that any

5 putative enumeration of the set of real numbers will yield a real number that does not feature in the enumeration. Adding that new real number to the original enumeration will give the “larger totality”, all of whose members fall under the concept real number. In fact, according to Dummett, the domains of all fundamental mathematical theories are given by indefinitely extensible concepts because, in a claim that has puzzled many commentators, he holds that the concept natural number is indefinitely extensible.1 The indefinite extensibility of the concept natural number derives from what Dummett terms the intrinsic infinity of the totality of natural numbers. The totality of natural numbers is intrinsically infinite because, whatever totality of natural numbers we are given, we always have a means of finding another element of the totality – an element characterized in terms of the elements we already have. Indefinitely extensible concepts are problematic, according to Dummett, because their extensions do not form definite totalities. This is not because their extensions are in some sense hazy or vague. There is no vagueness in the concept set. There are no entities that we would place on the borderline between things that are sets and things that are not sets. Nor is there any indeterminacy about what is to count as an ordinal number or a real number. The problem comes, Dummett thinks, because it is, strictly speaking, misleading to think of them having extensions in any straightforward sense at all. As he evocatively puts it, indefinitely extensible concepts have “an increasing sequence of extensions: what is hazy is the length of the sequence, which vanishes in the indiscernible distance” (Dummett 1990, p. 317). Each member of the sequence (each putative extension of the concept natural number or real number is perfectly definite. But the sequence can be indefinitely extended. This means that we do not have determinate conceptions of the relevant domains of quantification for statements about objects falling under those concepts. In fact we cannot have determinate conceptions of the totality of mathematical objects (be they sets, ordinal numbers, or real numbers) over which we are quantifying – as soon as we try to form such a determinate conception we are led inexorably to the conception of a totality that is a superset of the totality with which we began. The problem is not to be avoided by familiar strategies such as the distinction between sets and proper classes. If, as in von Neumann-Bernays set theory, we allow there to be collections that are not sets (because they cannot be members of any collection), then this gives us the means to name the extensions of indefinitely extensible concepts. We are in a position to include terms such as “On” (denoting the proper class of all ordinals) in our set theory, but the totalities thereby denoted are no less indefinitely extensible. The power to name the extension of an indefinitely extensible concept can hardly be thought to eliminate its indefinite extensibility. In any event, we do not need proper classes to define the extension of the concept real number, which is a perfectly respectable set (on one way of constructing the real numbers it is the set of Dedekind cuts) – and we already have the name “ω” for the extension of the concept natural number, indefinitely extensible concept though it is (according to Dummett). We can now see how there can be concepts of infinite totalities that are not indefinitely extensible – and, indeed, how the same infinite totality can be characterized both by an 1

Dummett’s thinking on this developed significantly between ‘The philosophical significance of Gödel’s theorem’ (1978) and Frege’s Philosophy of Mathematics (1991). In the earlier article Dummett stopped short of claiming that the concept natural number is indefinitely extensible. There he placed the burden of indefinite extensibility with respect to our understanding of the natural numbers at the door of the Gödel phenomenon. What he says there is that the concept of a property well-defined over the natural numbers is indefinitely extensible. The argument in the text follows the presentation in the later book at pp. 318-319.

6 indefinitely extensible concept and by a perfectly definite one. The set of natural numbers is a good example. We can consider the set of natural numbers either as the extension of the concept natural number or as the extension of the concept member of the first limit ordinal (since ω, the first limit ordinal, has as members all the finite ordinals). The first way of thinking about the set of natural numbers yields an indefinitely extensible concept, for the reasons sketched out earlier. The second way does not, however. It is certainly true that if I form a conception of the totality of members of the first limit ordinal then I can characterize a larger totality. I can, in the standard manner, extend the totality by taking the union of all the members of the totality. But the number that I thereby generate, ω, is not the extension of the concept member of the first limit ordinal. It is the extension of the concept member of the successor of the first limit ordinal. Dummett’s interest, then, is not with infinite totalities per se, but rather with infinite totalities given by indefinitely extensible concepts. These include, he thinks, the domains of the basic mathematical theories, such as number theory and analysis. The fact that these domains are given by indefinitely extensible concepts has deep implications for our understanding of those mathematical domains. Of course, we do manage to quantify meaningfully over mathematical domains that are given by indefinitely extensible concepts – and we do, correspondingly, have some sort of a grasp of the relevant domains of quantification. But this is very different from our grasp of totalities given by definite concepts. In both cases we have, as mentioned earlier, clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept – and for determining when an object falling under the concept but given in one way is identical to an object given in another way. But only for definite concepts is this enough to fix a determinate totality as the extension of the concept – and hence enough to give us a clear understanding of the extension of the concept. For indefinitely extensible concepts we need something more. We need, first, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle of extendibility that shows us how the domain is to be extended beyond the canonical base. In the case of the indefinitely extensible concept natural number the principle of extendibility is the fact that every number has a successor. Things are slightly more complicated for the concept ordinal number. We have, of course, the analogous principle that the successor of every ordinal number is an ordinal number, but we also have the further principle that, if A is a set of ordinals, then the least upper bound of A is also an ordinal. The first principle gives us the successor ordinals, while the second gives the limit ordinals. The crucial step in Dummett’s argument, and the one that has baffled most commentators (e.g. Clark 1998, 61 and see Potter 2004, 29-30 for further references), is the argument from the indefinite extensibility of key mathematical concepts to the rejection of classical quantification. It is, Dummett maintains, quite simply not the case that every quantification over a mathematical domain given by an indefinitely extensible concept has a determinate truth-value. In trying to understand this we would do well to begin with those quantifications that Dummett does think acceptable. Any definite totality of ordinals must therefore be so circumscribed as to foreswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognize as being an ordinal. It does not follow that quantification over the intuitive totality of all ordinals is unintelligible. A universally quantified statement that would be true in any definite totality of ordinals must be admitted as true of all ordinals whatever, and there is a plethora of such statements, beginning with “every ordinal has a successor”. Equally, any statement asserting the existence of an ordinal can be understood, without prior circumscription of the domain of quantification, as vindicated by the existence of an instance, no matter how large. (Dummett 1990, 316)

7 It must be recognized that here, as in many places, Dummett is talking about what it is to understand particular statements – as opposed, for example, to what might make them true. The question of what makes a universal quantification over all ordinals true has a simple and uninformative answer, namely, that the statement hold true of every ordinal. The question of understanding is, unfortunately, rather trickier – although of course we cannot understand a statement without understanding what it would be for that statement to be true. What we understand when we understand a universal quantification over all the ordinals is the fact that the statement holds true of every definite totality of ordinals – a fact that we grasp by grasping that the statement holds true of any arbitrarily chosen collection of ordinals. Dummett is somewhat elliptical here, but we can reconstruct his reasoning with the example that he himself gives – the statement that every ordinal has a successor. Let α be an arbitrary ordinal number and α+ the successor of α. By definition, α+ = α ∪ {α}. The statement says, then, that α ∪ {α} is an ordinal number. Plainly α ∪ {α} is a definite totality of ordinals (composed of α together with all the members of the members of. . . α). So, for Dummett, what we understand when we understand the statement that every ordinal has a successor is the claim that every definite totality of ordinals of the form α ∪ {α} is an ordinal. This claim is perfectly intelligible, Dummett claims (on my reconstruction), because we can understand it as the assertion that there is a procedure for showing that any such definite totality is an ordinal (namely, by noting that α+is a transitive set all of whose members are ordinals). We can, I think, put Dummett’s point in a more general way as follows. When we are dealing with a universal quantification of the form ∀x øx claimed to hold over an indefinitely extensible totality, there is (by assumption) no definite domain of which we can say ∀x øx is true just if ø holds for every object in the domain. Instead, there is an indefinite sequence of domains and ∀x øx is true just if ø holds of every object in every domain. But of course the indefinite sequence is not itself a definite totality, which means that we cannot understand this implicit quantification over the indefinite sequence of domains in the standard manner. What we are really doing when we assert ∀x øx is asserting that the fact that ø holds of every object in a given domain is transmitted across the principle of extendibility that creates a new and more inclusive totality from any given definite totality. But what is it to make such a claim? According to Dummett, when we claim that universal ø-ness is transmitted across the principle of extendibility what we are really claiming is that there is a way of showing that, if ø holds of all the members of any particular definite totality in the sequence, then it holds of all the members of the totality to which that definite totality might be extended Matters are somewhat obscured here by the fact that our example is itself one of the principles of extendibility governing ordinal numbers, but we can still see what is going on by considering the other principle of extendibility. Let ω be the least upper bound of the finite ordinals. Our second principle of extendibility tells us that ω is itself an ordinal. Part of what is asserted when we assert that ∀x ∈ On ∃y (y = x+) is that the very same means by which we show that any finite ordinal has a successor can be extended to the infinite ordinal that is the least upper bound of the set of finite ordinals. And this in fact is the case. The very same line of reasoning that shows that α ∪ {α} is an ordinal when α is a finite ordinal will equally show that ω ∪ {ω} is an ordinal. An opponent of Dummett will most likely object at this point that Dummett is confusing proof and truth. On this view, what we assert when we assert ∀x ∈ On ∃y (y = x+) is simply that every ordinal, be it zero, a successor ordinal, or a limit ordinal, has a successor. Although a proof of this claim will no doubt have to cite just such an operation, its truth depends simply upon there being a successor for every ordinal. We should understand assertion in terms of truth conditions, not in terms of the operations discovery of which will convince us that the truth condition holds. Here we come to the nub of the issue, because Dummett’s point (as I understand it) is precisely that we cannot grasp the truth conditions for quantifications over indefinitely extensible totalities except in terms of the type of transmissibility sketched out in the previous paragraph. And we

8 can only grasp the possibility of such transmissibility through the idea that there is something that secures it – there is no such thing as transmissibility simpliciter. Of course, there would be no need for this were we dealing with definite totalities, where the truth conditions for universally quantified statements can be understood in the normal manner. But we cannot treat domains defined by indefinitely extensible concepts as if they were definite totalities. Dummett’s position, then, is that the truth conditions for universal quantifications over domains given by indefinitely extensible concepts must be understood in terms of operations that secure transmissibility in the manner discussed. To assert such a universally quantified statement is to assert that such operations exist. But this has the inevitable consequence that we must abandon classical logic. From the fact, for example, that it is not the case that it is not the case that every x in some indefinitely extensible domain is F it by no means follows that there is an operation that will secure the transmissibility of ø-ness throughout the increasing sequence of extendible totalities. We might be able, for example, to give a reductio of the thesis that ∀x Fx can be reduced to absurdity. But this would hardly give us the required operation. Nor will the standard quantifier interchange rule ¬∀x ¬ Fx ⇒ ∃x Fx be valid. Even if it is absurd to suppose that there is an operation securing the transmissibility of F not holding in an appropriate domain, this by no means provides an instance of something in the domain that is F. Dummett is surely right that if the truth conditions of universal quantifications over indefinitely extensible totalities make ineliminable reference to operations securing transmissibility, then we cannot understand the logic of such statements classically.

III We return now to Fitch’s paradox. The excursion into the philosophy of mathematics in the previous section has shown that it is possible to argue (with some plausibility, in my opinion) for the thesis that quantification over mathematical domains given by indefinitely extensible concepts should be understood intuitionistically rather than classically. Plainly, applying this to Fitch’s paradox depends upon construing Fitch’s paradox as making ineliminable reference to indefinitely extensible totalities in a way that will support the type of argument canvassed in the previous section. Exploring whether this is indeed the case is the task of this section. But suppose for the moment that we can apply a Dummett’style argument in this domain. There are two ways in which this holds promise for dealing with Fitch’s paradox. Most obviously, as we saw earlier, the omniscience principle can only be derived by an inference that is classically but not intuitionistically valid. If it can be shown that we are dealing with an indefinitely extensible totality over which quantification must be understood intuitionistically then we have a plausible case for denying the validity of this inference. The appeal to intuitionistic logic becomes more than simply a technical fix. But there is a more subtle way in which an argument from indefinite extensibility might get a grip here. The logic governing statements that quantify universally over indefinitely extensible domains is intuitionistic because those quantifications have to be understood in a constructivist manner – that is, in a manner that appeals to the existence of effective operations securing transmissibility across increasing sequences of domains. To assert a universal quantification is essentially to assert that there is such an operation. But then it is very natural to wonder whether a prudent anti-realist really ought to commit themselves to the knowability principle in the form that we have given it (that is, as the universal quantification (8)). In any event, let us begin at the beginning. Are there good reasons for thinking that the collection of all propositions forms an indefinitely extensible totality? A relatively straightforward argument seems to show that there can be no set S of all propositions (cf Grim 1991, Ch. 4). We take a proposition to be an

9 abstract entity, such that S can be non-denumerably infinite, and we assume that S contains the proposition corresponding to the sentence “0 = 0”. Since S is a set it has a cardinality κ. Consider the power set ℘(S). Any arbitrary member si of ℘(S) is a set of propositions. As such there is a proposition corresponding to the sentence “0=0 ∈ si. Let that proposition be pi. Since S is the set of all propositions a subset axiom permits us to form the set P of all such pi. Since P can be put into one-one correspondence with ℘(S) we have card (P) = card (℘(S)) = 2 > κ = card (S). Since it is impossible for a set to have a subset of greater cardinality there can be no set of all propositions. κ

Of course, the argument just sketched out contains many hostages to fortune. Without a precise understanding of what a proposition is it is hard to know how to understand sets of propositions. Nor do we have a proper definition of set P. But we can put these problems to one side. We have enough to go on to see how the case might be made for the indefinite extensibility of the concept proposition. Certainly, Dummett’s definition seems to be satisfied. Given any definite totality S of propositions we can define a totality S* = S ∪ P such that S ⊂ S* and S* is itself a totality of propositions. The real question is not whether the concept proposition is indefinitely extensible in this sense, but whether it is indefinitely extensible in a way that permits the type of argument for intuitionistic logic sketched out in the previous section? That line of argument rests upon certain claims about what it is to grasp the truth conditions of totalities given by indefinitely extensible concepts. The argument, in essence, is that, because we are not dealing with a definite totality, universal quantification over an indefinitely extensible totality must be understood in terms of principles of transmissibility that secure the holding of the relevant property across an increasing sequence of domains. We know, when we are dealing with indefinitely extensible concepts in the mathematical sphere, that any given definite totality of a given type of object will generate a larger totality of the same type that includes it. So, what we assert when we assert some statement to be universally true within the domain given by such an indefinitely extensible concept is that, if the statement holds true of any given definite totality, it will hold true of the larger totality to which the original totality can be expanded – and so on through the increasing sequence of domains. It is for this reason, Dummett thinks, that universal quantification over indefinitely extensible domains incorporates ineliminable commitment to the existence of effective operations, thereby requiring an intuitionistic logic. We saw how this way of thinking about universal quantification makes sense in the context of the ordinals. But is it mandated by quantification over the intuitive totality of propositions? In one sense it is hard to see how it could not be. After all, the argument just canvassed shows that there is no definite totality of all ordinals and so, even though we are assuming that we have clear criteria of identity and individuation for propositions (which may, of course, be a vague hope), we cannot assume that those criteria will determine a truth-value within that definite totality. And so one might well feel justified in arguing with Dummett that we can only make sense of the truth conditions for statements quantifying over all propositions in terms of operations that secure the transmissibility of the relevant property. But this does not really get to the heart of the matter. What we really want to know is what those operations would look like. In the case of quantification over the ordinals we saw an example of how transmissibility across principles of extendibility might be achieved. In order to see how something comparable might work in the case of quantification over all propositions we need a clear idea of what the relevant principle of extendibility might be. We need to know what, in the case of propositions, plays the role that is played for ordinals by the twin principles that every ordinal has a successor that is an ordinal and that the least upper bound of any collection of ordinals is itself an ordinal. Let us revert to Dummett’s original characterization of how we grasp domains given by indefinitely extensible concepts. A necessary but not sufficient condition is that we have clear and unequivocal criteria for determining, of any particular object, whether it falls under the relevant concept – and for determining when an object falling under the concept but given in one way is identical to an object given in another way. When we are dealing with indefinitely extensible concepts we also need, first, a clear collection of objects that canonically satisfy the relevant criteria, and, second, a principle (or principles)

10 of extendibility that shows us how the domain is to be extended beyond the canonical base. The domain given by an indefinitely extensible concept is closed under the relevant principle(s) of extendibility. It is straightforward to apply this general model to quantification over all propositions. We begin with the distinction between simple and compound propositions, where a simple proposition is one that does not contain any quantifiers or truth-functional connectives and a compound proposition is constructed from simple propositions with quantifiers and/or truth-functional connectives. Plainly the totality of all propositions is the closure of the set of simple propositions under the operations of conjunction, disjunction, and so on – just as the totality of all ordinals is the closure of the empty set under the successor and limit operations. So, we may conclude that the principles of extendibility for the totality of propositions are, in effect, the rules governing the truth-functional connectives and quantifiers. We can now see what form must be taken by the principles of transmissibility for quantification over the totality of all propositions. The principles of transmissibility for a universal quantification of the form ∀p Fp must show that the property of F-ness is transmitted across the logical operations whose closure gives the totality of propositions. By the same token, the truth condition for the statement ∀p Fp is, in essence, that all simple propositions are F and that F-ness is transmitted in the appropriate manner across the relevant logical operations. To assert that ∀p Fp is to claim that F-ness is transmitted, and it is this claim that one understands when one understands the statement that ∀p Fp. It appears, therefore, that the concept proposition qualifies as indefinitely extensible by Dummett’s lights, even though it is not (in any obvious sense) mathematical. How does this affect the reasoning that leads to Fitch’s paradox?

IV We note first that the set of simple propositions forms a definite totality. We can run arguments such as that canvassed above on many different proper sub-totalities of the totality of all propositions. If we assume, for example, that “0 = 0 ∈ si” expresses a proposition of mathematics (where si is a set of mathematical truths), then a parallel line of reasoning shows that the totality of mathematical propositions is an indefinitely extensible totality – and hence that the concept mathematical proposition is an indefinitely extensible concept. No such argument can work, however, for the totality of simple propositions, since the propositions that are required to run the argument are not simple propositions. We might think, for example, of subsets of the totality of simple propositions in terms of conjunctions of simple propositions – but the proposition that “0 = 0” is a member of a conjunction of simple propositions is not itself a simple propositions. Since the set of simple propositions is a definite totality we can quantify over it unproblematically. We can say, for example, that every simple proposition is knowable. This might be formulated using restricted quantification as follows (where S is the set of simple propositions): (13)

∀p ∈ S [p ⇒ ◊ (p ∈ α)]).

By Dummett’s lights this principle has a determinate truth value and can be asserted and understood in the standard manner. The antecedent effectively restricts us to the definite totality of simple propositions. The same does not hold, however, for what we can term the unrestricted knowability principle – i.e. (8)

∀p [p ⇒ ◊ (p ∈ α)]

11 Here the quantifier ranges without restriction over an indefinitely extensible totality. By the earlier arguments it appears that the truth condition for (8) rests upon the transmissibility of the property of knowability across the principles of extendibility that collectively yield the indefinitely extensible totality of all propositions. We can certainly take as our “base” in spelling out how this might work the restricted principle of knowability for simple propositions. But the real work in spelling out the truth condition for (8) comes with what we might term the transmissibility principles – those principles that ensure that the totality of knowable propositions extends and expands at exactly the same rate as the totality of propositions. Since the extendibility principles for the totality of propositions are precisely those governing the truth-functional connectives and the quantifiers, we can expect the transmissibility principles to track the principles laying down the truth-conditions for compound propositions. In fact, it is possible to argue that the transmissibility principles just are the principles specifying truthconditions for compound propositions. If we concede that intuitionistic logic is the appropriate logic for indefinitely extensible totalities then the truth conditions for compound propositions will have to be given in intuitionistic terms. They will, in fact, take something like the following form, where in each clause the connective/quantifier/operator on the right is to be understood intuitionistically: (14) (15) (16) (17) (18) (19) (20) (21)

“not ø” is true ⇔ ¬ ø “ø or ψ” is true ⇔ ø ∨ ψ “if ø then ψ” is true ⇔ (ø ⇒ ψ) “ø and ψ” is true ⇔ ø ∧ ψ “For some x ø” is true ⇔ ∃x ø “For all x ø” is true ⇔ ∀x ø “Possibly ø” is true ⇔ ◊ ø “Necessarily ø” is true ⇔ o ϕ

The key point here is that, precisely because the connectives/quantifiers/operators are being understood intuitionistically, principles (14) through (21) secure the transmissibility of the property of knowability across the indefinitely extensible totality of all (true) propositions. Suppose we take an implication of the form “if ø then ψ’ where ø but not ψ is a simple proposition. If we understand “⇒” intuitionistically then “if ø then ψ” is true iff there is some form of procedure that will transform a proof of ø into a proof of ψ (we assume that we can formulate an analog of the proofconditional interpretation of intuitionistic logic when we find ourselves outside the domain of mathematical proof). If we have such a procedure then the conditional is knowable and, by assumption, ø is knowable. Hence ψ is knowable. Pari passu for the other logical constants – and, of course, for quantification (since “∀x” and “∃x” make ineliminable reference to effective operations when understood intuitionistically). In fact, Dummett himself proposes (Dummett 2001) that the correct response to Fitch’s paradox is an inductively specified theory of truth of the type given by principles (14) through (19), coupled with a basic principle akin to our (13) to the effect that simple propositions are knowable. Although Dummett (in what can only be described as a rather elliptical paper) does not put the matter in quite these terms, the inductive specification secures the knowability of the totality of all true propositions without a global quantification such as (8) – and hence without allowing the type of substitution that yields Fitch’s paradox. We can see how this works as follows. Suppose that p is a true simple proposition that has never been and never will be known. From (13) we still have that [p ∧ ◊ (p ∈ α)]. But this, after all, simply tells us what we already knew, which is that p is a true simple proposition that can be known at some time. Since [p ∧ ◊ (p ∈ α)] is a compound proposition we cannot substitute it back into (13) to yield the paradox. The basic point, then, is that (8) is not the correct way to express the basic principle that all truths are knowable. Because we are dealing with a domain given by an indefinitely extensible concept our expression of the basic principle needs to reflect the principles of extendibility in terms of which we

12 grasp that domain. In this case these are the principles governing the logical constants and quantifiers. By spelling out those principles in an intuitionistic metalanguage, as in principles (14) through (19) we effectively give expression to the basic principle that all truths are knowable without stating it explicitly – and so without either compromising the indefinite extensibility of the concept proposition or opening the door to Fitch’s paradox. The final step in the argument is plainly in view in Dummett 2001 where he sets out the inductive specification as a way of blocking Fitch’s paradox. Few commentators, however, have seen the rationale for the inductive specification (and Dummett himself has nothing to say about it). This paper has tried to make that rationale explicit. As I have argued, the motivation for the inductive specification is to be found in the indefinite extensibility of the concept proposition. The inductively specified theory of truth gives the principles of extendibility that we must grasp if we are to have a grasp of the indefinitely extensible totality of all propositions. If Dummett is right, moreover, that the correct logic to use over a domain given by an indefinitely extensible concept is intuitionistic, then the inductive specification must itself be intuitionistic, which is sufficient to ensure that the principles of extendibility are also principles of transmissibility in the sense we have discussed.

References Clark, P. 1998. Dummett’s argument for the indefinite extensibility of set and of real number. In J. Brandl and P. M. Sullivan (Eds.), New Essays on the Philosophy of Michael Dummett. Amsterdam: Rodopi. Dummett, M. 1990. Frege: Philosophy of Mathematics. London: Duckworth. – 1996. What is mathematics about? In The Seas of Language. Oxford. Oxford University Press. – 2001. Victor’s error, Analysis 61, 1-2. Enderton, H. B. 1977. Elements of Set Theory. New York: Academic Press. Fitch, F. B. 1963. A logical analysis of some value concepts, Journal of Symbolic Logic 28, 135 –142. Grim, P. 1991. The Incomplete Universe: Totality, Knowledge, and Truth. Cambridge MA: MIT Press. Potter, M. 2004. Set Theory and its Philosophy. Oxford: OUP. Russell, B. 1906. On some difficulties in the theory of transfinite numbers and order types. Reprinted in B. Russell, Essays in Analysis. London: Allen and Unwin. Williamson, T. 1982. Intuitionism disproved, Analysis 47, 154-158.

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