Chapter I Anti-realism and Knowability∗ Julien Murzi March 12, 2008

Abstract I introduce a number of epistemic conceptions of truth and argue that possibilist ones look prima facie more promising than their actualist counterparts. I then present the Paradox of Knowability and discuss its metaphysical nature. I side with Tim Williamson and argue, contra Jon Kvanvig, that the proof discovered by Church-Fitch is best understood as a counterexample to Weak Verificationism.

The predicate “true” is harmlessly used in our everyday language. “It is true that the water is really cold”, “Not everything Monica said was true”, “Is that true?”. These are all common uses of our truth predicate. Yet, “true” raises a number of philosophical issues whose origin at least as old as philosophy is. What is truth? Which entities, if any, are capable of being true or false? Is truth conceptually linked to our epistemic capacities? This latter question will be our primary concern in this chapter (and, indeed, in this study). The attempt of clarifying the relation between human thought and reality is, in Crispin Wright’s words, “constitutive of metaphysics”.1 The problem may be framed as the problem of what makes our statements true of false. Should we identify truth-makers with facts, or states of affairs, that are ∗

Many thanks to Cesare Cozzo, Luca Incurvati, Carrie Jenkins, Jonathan Kvanvig, Joe Salerno and Tim Williamson. 1 Wright (1993), p. 1.

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independent from our epistemic capacities? Or, rather, should we identify them with the arguments by means of which we justify such statements? Our starting point will be the broadly anti-realist conception of truth defended by, among others, Immanuel Kant, Charles Sanders Peirce, William James, Ian Brouwer, Arendt Heyting, Michael Dummett, Hilary Putnam, Dag Prawitz, Crispin Wright, Neil Tennant. Each of these authors (or temporal stages thereof) has argued that truth epistemically constrained, in the sense that it cannot transcend, even in principle, our epistemic capacities. Something like (EC) φ → Fφ, these authors argue, must hold as a matter of metaphysical necessity (where F is an epistemic predicate of some sort). As we shall see, the prima facie most plausible interpretation of F is the property of being possibly known. This commits anti-realists to what we have labelled the knowability principle: (KP) If φ, then it is possible to know that φ. Now KP faces a number of difficulties. Some of them are semantic in character. Is it possible, for instance, to construe a systematic theory of meaning with a truth predicate satisfying KP? Other difficulties are of a more direct kind. The Paradox of Knowability is one of them. The argument locates a tension between the idea that all truths are knowable and the apparently innocent claim that some truths are, as a contingent matter of fact, forever unknown. If the Church-Fitch reasoning is correct, these two claims are logically incompatible, on a sufficiently broad understanding of the term. Such a result cannot but be surprising. The existence of forever unknown truth, we might want to think, should not preclude the truth of KP. Yet, if the argument is correct, we must put aside our surprise and accept the result. Otherwise, one should locate a mistake in the derivation of the Paradox. This latter option might prove to be no easy task, however: the argument is quite simple and each of its steps is relatively unobjectionable. 2

We shall proceed as follows. Section 1 presents semantic realism and briefly introduces some of the problems it may face. Sections 2 and 3 examine a number of epistemic conceptions of truth. Section 4 suggests that antirealists have good reasons to identify their epistemic constraint on truth with the knowability principle. Section 5 presents the Paradox of Knowability. Section 6 argues that the Paradox raises no logical problem, contrary to what Jon Kvanvig has recently argued. Section 7 offers some conclusive considerations.

1

Realism and Potential Unknowability

A variety of metaphysical doctrines find a more or less comfortable place under the umbrella of the label ‘realism’. We cannot consider them all here, and we shall not even attempt to do so. We shall limit ourselves to saying that, in general, realism may be thought of as the view that reality is independent from us. Now, independency comes in degrees. Ontological independency, for instance, is consistent with semantic anti-realism. We can perfectly conceive of a class of statements obeying EC and yet describing an ontologically independent reality. As we concei8ve of it, semantic realism requires epistemic independence. We may characterize it as the thesis that some contents—sentences, propositions, statements—may be understandable and yet un-F-able (or un-F-ed, for that matters) in principle. As Crispin Wright puts it: One way of making this conception more concrete is [. . . ] to hold that the world may be determinate in thinkable, describable, but unascertainable ways. (Wright, 1993, p. 3) We may thus define semantic-realism as follows: (SR) Necessarily, it is possible that there be understandable contents whose truth-conditions are satisfied (if at all) by an epistemically inaccessible portion of reality. 3

Sentences such as “Goldbach’s conjecture is true”, “There exist infinitely many binary starts”, “There is no intelligent extraterrestrial life”, “2ℵ0 = ℵ1 ” are, or at least can be, examples of similar statements.

1.1

Modesty and Presumption

Semantic realists face objections from at least two different scores. As Wright writes in the introduction of his Realism, Meaning and Truth, realism is “a mixture of modesty and presumption” (p. 1). The realist is modest when she concedes that reality is not our creation, and that our thought and our knowledge can—at best—trace a map, more or less accurate and more or less complete (the metaphor is, of course, Frege’s). She is presumptuous when she affirms that, at least some times, we effectively know such an independent reality. The two main lines of attack to realism—scepticism and idealism— move precisely along these two tracks. The sceptic mainly targets the realist’s presumption. Although she agrees with the realist in believing that “our investigative efforts confront an autonomous world” (Wright, 1993, p. 2), the sceptic doubts that the very possibility of knowing that world is open to us. If the world is independent of us, the sceptic worries, then we cannot have any guarantee that our best beliefs and our best judgements successfully represent it as it really is. On the other hand, the idealist questions the modest aspect of the realist package. If scepticism is to be avoided and if (as we shall see in a moment) communication and understanding are not to be left in a mystery, then—the idealist argues—reality cannot be independent (or, at least, completely independent) from us.

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1.2

Dummett’s Manifestation Challenge

It is worth sketching one of the most influential lines of arguments against semantic realism: Michael Dummett’s so-called manifestation challenge.2 Both the semantic realist and her anti-realist opponent start from the Fregean assumption that the meaning of a sentence is given by with its truth-conditions. Realists, however, contend that truth conditions are potentially evidencetranscendent. They may obtain, or fail to obtain, independently of our capacity to ascertain that they do. To this, anti-realists object that, on the Fregean assumption, if truth-conditions are evident-transcendent, understanding— i.e. knowledge of truth-conditions—becomes utterly mysterious. For, they say, understanding must be manifestable in use. But how could knowledge of evidence-transcendent truth-conditions be manifested in our linguistic practice? Unless the realist provides an answer to this question, anti-realists conclude, the realist’s contention that some sentences have potentially evidencetranscendent truth-conditions cannot be taken on board. If correct at all, the argument invites us to the conclusion that, if we want to have a coherent account of meaning and understanding, we must rule out—as a matter of conceptual necessity—the possibility that truth may be epistemically unconstrained.

2

Truth, Justification and Idealized Assertibility

Let us grant the semantic anti-realist that realism cannot be true, and that EC must hold. Then the question arises: how truth is to be epistemically characterized, more exactly? Which epistemic predicate F can best suit the purpose of semantic anti-realism? 2

See, e.g., The Philosophical Basis of Intuitionistic Logic, in Dummett (1978, p. 225).

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2.1

All Truths are Justifiably Believed?

A first option may be that of identifying F with the property of being justified in believing. This yields the following interpretation of EC: (ECJBn ) φ → J B n φ, where J B n φ reads ‘someone presently has a justified belief that φ. 2.1.1

Stability

It should be immediately clear that a similar proposal cannot be made at work. To begin with, the notion of actually possessing a justified belief and the notion of truth seem to have radically different properties, so that the project of imposing the former as a necessary condition for the latter does not appear to be a very promising one. As Hilary Putnam observes, truth seems to be a stable property of its bearers.3 If true at all, an utterance, or the proposition (statement) it expresses, is always true.4 On the face of it, the property of having a justified belief is not stable in time. Indeed, justifications can be lost for a number of reasons. This suggests the following improvement of ECJBn : (ECJB ) φ → J Bφ, where J Bφ reads ‘someone at some time justifiably believes that φ’. This latter proposal partially respects the stability constraint. On such an account, propositions become true (or false), given that, if truth requires justification, a proposition φ cannot be true before someone has a justified belief that φ. Still, on the revised account, once truth and falsity are acquired, they cannot be lost. 3

This is of course a controversial assumption. Temporalists such as the Stoics, David Kaplan or Berit Brogaad have claimed that proposition may change their truth-values over time. See Kaplan (1989) and Brogaard (forthcoming). 4 See Putnam (1981, p. 55).

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2.1.2

Matchboxes and Forever Unjustified Truths

The modified account is still problematic, however. First of all, ECJB is still a form of Idealism. The account still has it that truth—even arithmetical truth—is dependent on what we actually do. If someone at some time proves that φ, then φ is true. If nobody will ever bother to check, then φ is not true. But this is clearly unacceptable. Secondly, as we saw in the Introduction, it is possible to prove that there are propositions that are presently unjustified and nevertheless true. Let p be any decidable sentence, say, “the number of matches in this matchbox is greater than 20”. Since p is decidable, Bivalence applies: either p or its negation is true.5 Now suppose that all the evidence available for p has been destroyed—because, say, we threw our matchbox into the fire before counting the matches. Hence, one between “p and nobody ever will be justified in believing that p” and “¬p and nobody ever will be justified in believing that ¬p” is true. By disjunction elimination, it follows that there is a statement of the form φ ∧ ¬J Bφ.6 A much quoted stanza by Gray provides similar counterexamples: Full many a gem of purest ray serene, The dark unfathom’d caves of ocean bear: Full many a flower is born to blush unseen, And waste its sweetness on the desert air. Since an infinite number of similar examples can be easily constructed, it is safe to say that there exist an infinite number of true but (presumably) forever unjustified propositions. This suggests a modal, or possibilist, interpretation of ECA . Cristoph Kelp and Duncan Pritchard have recently proposed one.7 It is to their proposal that we now turn. 5

We take it that a statement is decidable just in case either it or its negation are knowable (or, more generally, have the property F). See infra, Chapter 2, Section 3. 6 The example is Cesare Cozzo’s. See (Cozzo, 1994, p. 72). 7 See Kelp and Pritchard (forthcoming).

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2.2 2.2.1

All Truths are Justifiably Believable? Kelp and Pritchard’s Proposal

Cristoph Kelp and Duncan Pritchard have recently suggested that anti-realist ought to adopt the following interpretation of EC: (EC♦JB ) φ → ♦J Bφ. A statement is true, EC♦JB only if it is possible that it be justifiably believed. Crucially, Kelp and Pritchard argue, the proposal has the virtue of non incurring into the problem raised by Church-Fitch. As we shall see, the Church-Fitch reasoning requires that the epistemic predicate involved in the statement that all truths are F-able be factive and closed under conjunction elimination. That is, Fφ must entail φ and F(φ ∧ ψ) must entail Fφ ∧ Fφ. Justified belief satisies the latter requirement but not the former. Indeed, my justified belief that my car is where I parked it this morning does not entail that my car is where I parked it yesterday: my car could have been stolen, or removed, for all I know. As a result, from the assumption that a proposition of the form φ ∧ ¬J Bφ holds we can only derive (∗ ) J Bφ ∧ J B¬J Bφ.8 However, unless the following disbelief principle: (DP) J B¬J B → ¬J Bφ is assumed, no contradiction can be derived from (∗ ). Is DP true? It states that, if an agent justifiably believes that she is not justified in believing that φ, then she is not justified in believing that φ. For one, the principle might be challenged on Freudian grounds. For another, 8

Proof: Assume that J B(φ ∧ ¬J Bφ). Then, by J B(φ ∧ φ) → (J Bφ ∧ J Bφ), J Bφ ∧ J B¬J φ. 

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as Kelp and Pritchard point out, familiar Kripkean counterexamples may be right behind the corner.9 So let us grant, following Kelp and Pritchard, that DP is false. There thus may be a technical reason for imposing the requirement that every truth be possibly justifiably believed. The question arises, however, whether EC♦JB is sufficient for the anti-realist’s purposes. 2.2.2

Objections to Kelp and Pritchard

Pace Kelp and Pritchard, we doubt that EC♦JB is faithful to the spirit of semantic anti-realism. Furthermore, we suggest, it carries problems if its own. Let us focus on mathematical discourse. In Kelp and Pritchard’s view, what the anti-realist ought to require for the truth of a mathematical statement is merely that, if a mathematical statement is true, it should be possible for someone at some time to have a justified belief in its truth. Now the standard notion of mathematical justification is, of course, that of a proof. However, the notion of a proof is factive: if someone has or will have a proof that φ, then it is true that φ. If the notion of mathematical justification were cashed out in terms of the notion of a proof, therefore, an analogue of the Paradox of Knowability would go through just as well.10 What Kelp and Pritchard must have in mind, thus, is a non-factive and proof-independent notion of justification. Let us assume for the sake of argument that such a notion of justification may find a legitimate use within mathematical discourse. For instance, one might claim that a belief in Goldbach’s Conjecture may be justified even in absence of a proof of it (perhaps on the grounds that no counterexample has been found so far, and that the Conjecture has been tested up to very large values of natural numbers). Then, Kelp and 9

See Kripke (1979). Assume that p is a true mathematical statement that nobody will ever prove. Then, on the assumption that all mathematical truths are provable, a contradiction can be derived if it is further assumed that provability is not only factive, but also distributes over conjunction. 10

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Pritchard’s anti-realist would have to countenance possible scenarios such as the following: Goldbach’s conjecture is true and justifiably believed and yet neither its truth nor its negation is provable. But this surely cannot be, from an anti-realist standpoint. If valid at all, Dummett’s challenges rule out—as a matter of conceptual impossibility—that the truth-conditions of a mathematical statement obtain without our being in a position to have a proof that they do. For arguably, we understand a mathematical statement only when we have an idea of what would count as a proof—not as a fallible justification—of it. If true but unprovable statements are allowed, as Kelp and Pritchard implicitly suggest, we are thus left with no anti-realist account of understanding. Finally, if the notion of a justified belief involved withing EC♦JB is not factive, it might be argued that, for some p, it is possible that, actually, J Bφ and J B¬φ.11 If this is correct, then there is a slippery slope from semantic anti-realism to Dialethism—the view that there are true contradictions. Pace Jon Cogburn,12 however, we dismiss this latter possibility.

2.3

Idealized Assertibility

Actual justification and possible justification proved to be inadequate candidates for a satisfactory interpretation of F. Semantic anti-realists have still many options left, however. Here we shall consider some epistemic conceptions of truth that define truth in terms of the property of being warrantedly assertibile.13 11

As we shall see in due course, the same problem does not arise with other non-factive epistemic predicates, such as being possibly known. While it is possible to know that p and it is possible to know that ¬p, it is surely not possible—insofar as knowledge is factive—to know both p and ¬p. 12 See Cogburn (2004a). 13 In his The Folly of Trying to Define Knowledge, Michael Blome-Tillmann has recently argued that warrant is just knowledge (see Blome-Tillmann (2007)). Here we shall refer to a non-factive notion of a warrant. The reader may alternatively read ‘warrantedly’ as ‘correctly’ or ‘justifiably’.

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2.3.1

Wright on Truth and Warranted Assertibility

It might be reasonably thought that the property of being warrantedly assertible is a non-starter. For one, warranted assertibility comes in degrees (think of utterances of “It is sunny today” in contexts c1 , . . . , cn such that it is sunny in c1 , and there are no clouds, it is sunny in c2 , and there is one cloud, and so on. Then, some utterances of “It is sunny today” will be more assertible than others). On the face of it, as Frege once put it, truth is not a matter of degrees.14 For another, as Crispin Wright has shown,15 if the Equvalence Schema (ES) φ ↔ T φ or the Disquotation Schema (DS) ‘φ’ is true if and only if φ16 hold good (where ‘T ’ is a truth predicate), the property of being must diverge in extension from that of being warrantedly assertible. To see why it is so, we first prove that, if T φ ↔ Aφ, then A¬φ ↔ ¬Aφ. Proof : Assume for reductio that T φ just in case Aφ. Then, by ES, derive both ¬φ ↔ T ¬φ (by substition of φ for ¬φ) and ¬φ ↔ ¬T φ (owing to the fact that ` (φ ↔ ψ) ↔ (¬φ ↔ ¬φ)). Then, by the transitivity of ‘↔0 , derive T ¬φ ↔ ¬T φ. From our initial assumption, we may substitute A for T in this latter result. Hence, A¬φ ↔ ¬Aφ.  As Wright points out, this latter biconditional fails right-to-left. From the fact that it is not warrantedly assertible that φ it does not follow that its negation is warrantedly assertible. For there might well be states of information that are neutral with respect to φ. Hence, we must discharge our 14

See Frege (1956), p. 60. See Wright (1992), Chapter II. 16 ES applies to propositions (statements); DS applies to sentences. 15

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intitial assumption. As Wright puts it, truth and warranted assertibility must “diverge in extension” (Wright, 1992, Chapter II). How to solve the problem? A better idea might be that of resorting to an idealized notion of warranted assertibility. We consider three such proposals. 2.3.2

Peircian Biconditionals

A number of commentators have attributed to Charles Sanders Peirce the view that true statements are those that, at the end of enquiry, researchers would believe to true.Although this is very likely to be an unfaithful reconstruction of Peirce’s thought, the idea is nevertheless worth examining. The thought is that there exists an ideal situation Q in which, as Hilary Putnam puts it, “the community would be in a position to confirm every true statement (and disconfirm the false ones)”.17 In symbols: (Peirce) ∃Q∀p(p ↔ (Q € Ap), where the quantifiers ∃Q and ∀p range, respectively, over ideal epistemic situations and propositional entities of some sort, € is the standard counterfactual conditional and ‘A’ reads ‘it is warrantedly assertible that’.18 Peirce faces at least two problems. Firstly, as Richard Rorty has pointed out, it is quite dubious that our descendants will say, some day, “Inquiry is now finished!”.19 Secondly, we have no guarantee that the very same ideal situation can guarantee the warranted assertibility of every true statement. A set of epistemic conditions may be ideal for some statement and—at the same time—fail to be ideal for some other statement. Semantic anti-realists, it would seem, had better relativize Q to every single statement for which semantic anti-realism is proposed. Ideal epistemic condition should be, in 17 18

Putnam (1990, p. viii). We adopt quantification over sentence position for distinguishing Peirce from P80 be-

low. 19

See Rorty (1984), p. 7.

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Crispin Wright’s term, topic-specific.20 2.3.3

Putnam80 ’s Biconditionals

This latter idea have been famously defended, in the 80’s, by Hilary Putnam.21 Although Putnam80 never explicitely provided necessary and sufficient conditions for truth, a number of commentators, including Crispin Wright and Wolfgang K¨ unne,22 have attributed to Putnam80 the view that a statement is true just in case, as Putnam himself puts it, “it could be justified were epistemic conditions be good enough”23 . Now, such a proposal does not seem to be different from Peirce. Yet there is indeed a difference between this latter biconditional and the view that is currently attributed to Putnam80 , namely the order of the two quantifiers ∃Q and ∃p. In particular, whereas Peirce is of the form ∃∀, the correct formalization of what seems to be Putnam80 ’s view is of the form ∀∃, as its formalization clearly reveals: (P80A ) ∀p∃Q(p ↔ (Q € Ap)). For every statement p, P80A says, there exist ideal epistemic conditions Q such that p holds just in case, if Q obtained, then p would be assertible.24 According to Crispin Wright, conceptions of truth along the lines of P80A fail to deliver an anti-realistically acceptable account of truth. The problem, Wright argues, is that they seem to impose on truth an unacceptable requirement of completeness. If P80A were to hold, Wright suggests, every statement or its negation would be assertible—and then true—in epistemically ideal circumstances. The reason why it is so, Wright contends, is that whenever 20

See Wright (2000, p. XXX). Putnam’s idea have now changed (See e.g. Putnam (1990)). We shall refer to the internalist Putnam as Putnam80 . 22 See Wright (2000, p. XXX) and K¨ unne (2003, p. XXX). 23 Putnam (1990), p. vii. 24 As we shall see in Chapter 3, conceptions of truths of the form of P80A run into Fitchlike paradoxes, as it has been shown, e.g., by Plantinga (1982). We shall put aside thesis issues here. 21

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in epistemically ideal circumstances it is not warrantedly assertible that p, then the conditions are ideal for warrantedly asserting its negation ¬p. On the face of it, in Wright’s view, epistemically ideal conditions can be, for some p, neutral with respect to both p and its negation ¬p. Hence, Wright concludes, P80A is false. As he puts it: Surely it is not true [. . . ] that [every statement] is decidable— confirmable or disconfirmable—under ideal epistemic circumstances (Wright, 1992, p. 40). Putnam’s biconditional, Wright concludes, contains an a priori mistake. Now, it is not obvious that we have no guarantee that it is not the case that every statement or its negation would be warrantedly assertible under epistemically ideal circumstances. Furthermore, as Wright himself, acknowledges, the problem is more general and arises for EC itself, given that, by contraposition on EC itself, ¬Fφ entails ¬φ. The problem, if there is one, is thus not attributable to P80A as such.25 Christopher Peacocke raises a perhaps more serious concern for Putnam80 ’s proposed elucidation of truth.26 The property of being assertible if epistemic conditions were topic-specifically ideal, Peacocke points out, does not collect across conjunction.27 Call the aforementioned property AI. Then, in general, if AIφ and AIψ, it does not follow that AI(φ ∧ ψ). As we observed earlier, epistemic conditions that are ideal for appraising the truth of φ may not be good enough for appraising the truth of ψ. On the face of it, Peacocke points out, the truth predicate does indeed collect across conjunction (if T φ and T ψ, then obviously T (φ ∧ ψ)). Hence, truth cannot be identified with the property of being assertible if epistemic conditions were topic-specifically ideal. 25

We shall come back on this issue in detail in Chapter 1, Sections I and II. See Peacocke (1999), p. 30. 27 See also K¨ unne (2003), p. 422. 26

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2.3.4

Superassertibility

We find in the literature at least one assertability-based alternative to Peirce and P80A : Crispin Wright’s suggested account of truth as superassertibility. Wright’s idea is that F ought to be construed in terms of actually accessible epistemic states. Whenever EC holds good, Wright suggests, truth is to be identified with the property of being superassertible. As he puts it: Matters may turn out interestingly differently, however, if the idealisation assumes the form proposed in the notion of superassertibility. Superassertibility is the property not of being assertible in some ideal—perhaps limiting—state of information, but of being assertible in some ordinary, accessible state of information and then remaining so no matter what additions or improvements are made to it. When superassertibility for a given class of statements is taken to be truth, then truth is held to consist not in assertibility at some ideal limit of information gathering but in enduring assertibility over indefinite improvements. (Wright, 2006, p. XXX) A statement φ is true, Wright proposes, just in case there exists a state of information i such that (i) i is actually accessible, (ii) φ is warrantedly assertible in i and φ remains assertible however i is enlarged upon or improved. More formally: (SuperA) φ ↔ ∃x(x is an accessible state of information ∧ x is warrantedly assertible in x ∧ x remains warrantedly assertible no matter how x is enlarged upon or improved).28 2.3.5

Potential Concerns

Wright’s biconditional runs into at least two potential objections. Firstly, SuperA does not appear to respect the stability constraint we mentioned 28

We adopt Wolfgand K¨ unne own statement of SuperA. See K¨ unne (2003, p. 418).

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earlier on in Section 2.1. Indeed, actually accessible states of information are such that information may be (irremediably) lost. Therefore, at least on a literal interpretation, SuperA fails to do justice of a number of intuitively true statements. Furthermore, authors such as John Skorupski have argued that SuperA is undermined by potential counterexamples. As Skorupski puts it: There is an important difficulty which Wright ignores [. . . ]: a true statement may not be superassertible. Suppose you are bored by my conversation. Suppose I see your eyes glazing, suppressed yawns, etc. — I am warranted in asserting that you are bored. But further information may defeat that warrant: I discover that you have been kept awake for the last forty-eight hours. (What I don’t know is that you are taking pills to counteract the effects of lack of sleep—so that’s not why you are suppressing yawns.) (Skorupski, 1988, p. 522) Now, we doubt that—on a sufficiently strong sense of ‘warrantedly assertible’— Skorupski’s discovery that, say, Jones has been kept awake for forty-eight hours defeats his warrant for his assertion that Jones is bored. Yet, it is true that both Skorupski’s point and the stability problem mentioned above call for a somehow already idealized notion of warranted assertibility. Wright’s original idea that superassertibiliy is built upon the property of being actually warrantedly assertible is then shaken. Of course, this is no conclusive evidence against Wright’s proposal. Still, the difficulties faced by Perice, P80A and SuperA might suggest that the property F we are attempting to characterize had better not be defined in terms of the property of being warrantedly assertible—however idealized the latter may be. A stronger epistemic notion may perhaps be called for.

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2.3.6

Four Constraints

Our attempts to provide a precise formulation of the idea that truth ought to be defined in terms of assertibility has encountered a number of difficulties. Our inquiry so far has nevertheless proved useful, however. We have identified at least four constraints that any epistemic predicate F satisfying EC should respect. Specifically, F should: (a) Diverge in extension from warranted assertibility (Wright); (b) Be stable in time (Putnam); (c) Not be gradable (Frege); (d) Collect across conjunction (Peacocke). In what follows, we shall consider whether an epistemic predicated construed upon the notion of knowledge may satisfy these constraints.

3

Truth and Knowability

We consider here a number of attempts of defining truth in term of knowledge or knowability.

3.1

Strong Verificationisms

As it is in the case of justification, the attempt of identifying F either with the property of being presently known or with the property of being known by someone at some time appear to be beyond the pale. Let us first consider the hypothesis that F were to be identified with present knowledge. We call this proposal exeedingly strong verificationism: (ESVER) φ → Kn φ,

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where ‘Kn ’ reads ‘it is presently known that’. The obvious problem, here, is that by contraposition of ESVER, every presently unknown statement is to be regarded as false. This, however, is an rather strong form of idealism. It makes reality dependent upon present knowledge. But this is utterly implausible. Suppose nobody presently known that there is a dog walking behind the wall. Then, according to ESVER there is no dog behind the wall. We dismiss ESVER without further argument. Consider now the hypothesis that F is to be identified with the property of being known by someone at some time. Following Tim Williamson,29 we call this latter proposal strong verificationism: (SVER) φ → Kφ. Again, SVER seems unacceptable. For consider our matchbox example in Section 2.1.2. The example can be easily applied to knowledge, thereby yielding the result that SVER is provably false, by unexceptionable intuitionistic means. We therefore cannot but agree with Dummett: “surely, we cannot equate truth with being recognized [. . . ] as true” (Dummett, 1993a, p. 446).

3.2

The Knowability Principle

The problems afflicting ESVER and SVER tell in favour of a possibilist interpretation of EC. As we already mentioned in the Introcuction, the knowability principle: (KP) If φ, then it is possible to know that φ, may just be what the anti-realist is looking for. Indeed, the resulting interpretation of F, ‘possibly known by someone at some time’ arguably satisfies each of the aforementioned desiderata: it diverges in extension from warranted assertibility (for some p, p may be is possibly known but not warrantedly assertible), it is stable in time (if φ is possibly known at some time, then 29

See Williamson (2000, Chapter 12).

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it is possibly known at every time), it is not gradable (for any two statements p and q, it does not make much sense to say that p is more (less) knowable than q) and, prima facie, it collects across conjunction (at least intuitively, if both φ and ψ are possibly known, then the conjunction φ∧ψ is itself possibly known).30 We shall assume hereafter that, at least prima facie, KP represents the most plausible interpretation of EC, as witnessed by the fact that eminent anti-realist philosophers such as Michael Dummett, Dag Prawitz, Neil Tennant and Crispin Wright have all subscribed—at least at some point—to some version of it.31 3.2.1

Standard Weak Verificationism

How should we logically frame KP? On its most common logical interpretation, KP reads: (KP1 ) If φ at w, then there exists an x such that wRx and, at x, someone knows at some time that, at x, φ, where w and x are possibly distinct circumstances of evaluations, or worlds, and R is an accessibility relation among circumstances or worlds. In symbols: (WVER) φ → ♦Kφ. 3.2.2

Edgingtonian Knowability

KP admits of a second reading, however: (KP2 ) If φ at w, then there exists an x such that wRx and, at x, someone knows at some time that, at w, φ.32 30

As we shall see in what follows, the Church-Fitch paradox provides a counterexample to this latter claim. 31 See e.g. Dummett (1976), Tennant (1997) and Wright (1992), Prawitz (1998). 32 See Edgington (1985).

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For the time being, we shall put aside the second reading (we shall come back to this point in Chapter 4) and focus on KP’s most orthodox interpretation: weak verificationism.

4

The Paradox of Knowability

As we already anticipated in the Introduction, a well-known argument, first published by Frederic Fitch, purports to show that semantic anti-realism, the view that all truths are knowable, collapses into a na¨ıve form of idealism, according to which all truths will be known by someone at some time. This is the Knowability Paradox. The original proof published by Fitch, whose bulk is originally due to Alonzo Church,33 establishes the following theorem: (T5) If there is a proposition which nobody knows (or has known or will know) to be true, then there is a proposition which nobody can know to be true (Fitch, 1963, p. 139). Formally: (T5) ∃p(p ∧ ¬Kp) → ∃p(p ∧ ¬♦Kp). The Knowability Paradox is contrapositive of T5: (CT5) If all truths are knowable, then all truths are known. Since the consequent of CT5 is clearly false, the anti-realist claim that (KP) All truths are knowable is threatened. 33

Fitch credits the bulk of the proof to an anonymous referee of an unpublished paper of 1945. We now know that the referee in question was Alonzo Church. We may thus alternatively label the proof The Church-Fitch Paradox. See Appendix I and Appendix II below.

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4.1

The Proof

Call the most straightforward formalization of the antecedent of CT5 weak verificationism: (WVER) ∀p(p → ♦Kp)34 And call the formalization of its consequent strong verificationism: (SVER) ∀p(p → Kp) As usual, the knowledge operator K reads ‘it is known by someone at some time’. The Paradox of Knowability requires that knowledge be necessarily factive and closed under conjunction elimination. In symbols: (FACT)  ∀p(Kp → p); (DIST)  ∀p∀q K((p ∧ q) → (Kp ∧ Kq)).35 The proof further requires that provable formulas are necessary. In symbols: (Necessitation) If ` φ, then φ. The proof may be presented in two steps. We first prove that truths of the form p ∧ ¬Kp are unknowable: (1) (2) (3) (4) (5) (6)

K(p ∧ ¬Kp) Assumption for ¬-I Kp ∧ K¬Kp 1, DIST Kp ∧ ¬Kp 2, FACT ¬K(p ∧ ¬Kp) 1-3, ¬-I ¬K(p ∧ ¬Kp) 4, Necessitation ¬♦K(p ∧ ¬Kp) 5, ¬φ → ¬♦φ

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For our formal regimentation of the proof, we adopt quantification in sentence position instead of schemata. Nothing in what follows hinges on this choice. 35 Alternatively, we may frame this two principle as inference rules as follows: (FACT)

Kφ ; φ

(DIST)

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K(φ ∧ ψ) . Kφ ∧ Kψ

We then proceed to show that, given (6), WVER collapses on SVER:36 (7) (8) (9) (10) (11) (12) (13) (14) (15)

4.2

∀p(p → ♦Kp) Assumption for →-I (p ∧ ¬Kp) → ♦K(p ∧ ¬Kp) 7, ∀ E p ∧ ¬Kp Assumption for ¬-I ♦K(p ∧ ¬Kp) 8, 9, →-E ♦K(p ∧ ¬Kp) ∧ ¬♦K(p ∧ ¬Kp) 10, 6, ∧-I ¬(p ∧ ¬Kp) 9-10, ¬-I p → Kp 12, PC (classical) ∀p(p → Kp) 13, ∀-I ∀(p → ♦Kp) → ∀p(p → Kp) 7-14, →-I

Weak Verificationism Under Threat

From WVER and the seemingly innocuous assumption that some truths are forever unknown, a contradiction follows. For we cannot know, for any given truth, that it is forever unknown, on pain on contradiction. If contradictions are not allowed in our system, something must be discharged. The antirealist will discharge the second assumption, thereby committing herself—by an exclusively classical step—to SVER. Yet, if our matchbox example is correct, SVER is plainly false. Therefore,WVER should be regarded as false as well. As Colin McGinn and William Hart put it: In presence of obvious truths, [SVER] is deducible from [WVER]. [But] [SVER] is obviously false and is an objectionably strong thesis of idealism [. . . ]. Therefore [WVER] is false: there are truths which absolutely cannot be known. (Hart and McGinn, 1976, p. 139) 36

PC abbreviates ’Propositional Calculus’.

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The proof appears to be valid in classical modal logics as strong as K for any factive operator closed under ∧-E.37 The minimal modal principles it requires—essentially, the Rule of Necessitation—cannot be reasonably questioned (although we shall encounter one unreasonable exception in Chapter 4). Likewise, it would be terribly hard to deny that knowledge is factive and distributes under conjunction.38 Finally, although the derivation of SVER requires that the Law of the Excluded Middle (LEM) φ ∨ ¬φ holds good, neither the argument from (1) to (6) nor Fitch’s original Theorem 5 require any exclusively calssical step. However, a proof to the effect that there are unknowable truths should be enough cause of concern for the antirealists who claims that all truths are knowable. Should we therefore accept the Church-Fitch result?

5

What’s Paradoxical?

Before we turn to this question, it is worth pausing briefly on a separate issue—i.e. the problem of the nature of the Paradox of Knowability. Is the Paradox a mere counterexample to WVER? Or does it reveal a more profound, and perhaps more troubling, puzzle? If the second question were to receive an affirmative answer, the problem raised by Church-Fitch would afflict realists and anti-realists alike. This latter view has been recently defended by Jon Kvanvig in a number of writings.39 We shall argue, following 37

Where K is the logic obtained by adding to classical logic Necessitation and the axiom: (K) (φ → ψ) → (φ → ψ). 38 Althougth we shall encounter some attempts of weakening both FACT and DIST in Chapter II. 39 See Kvanvig (2006) and Kvanvig (forthcoming).

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Carrie Jenkins, that it is mistaken.40,41

5.1

Orthodoxy

Orthodoxy has it that the Church-Fitch proof essentially constitutes a problem for WVER, but nothing more. We find its clearest statement in Tim Williamson’s words: The argument is sometimes called ‘The Paradox of Unknowability’, although why it should be regarded as a paradox is quite unclear. The conclusion that there are unknowable truths is an affront to various philosophical theories, but not to common sense. If proponents (and opponents) of those theories long overlooked a simple counterexample, that is an embarrassment, not a paradox. (Williamson, 2000, p. 271) On any plausible definition of the term ‘paradox’, an argument counts as paradoxical only if (i) it starts from premises that are true as a matter of common sense, (ii) the inferential rules it relies upon are intuitively correct and (iii) it leads to conlusions that are not true as a matter of common sense. If all of this is minimally faithful to the meaning of the world ‘paradox’, then Williamson is indoubtably correct: the argument from WVER to SVER is not paradoxical, given that WVER is certainly not a matter of common sense. The orthodox view advocated by Williamson has recently came under attack, however.

5.2

The Mystery of the Disappearing Diamond

According to Jon Kvanvig, orthodoxy seriously overlooks the very nature of the Church-Fitch proof.42 The real nature of the proof, Kvanvig alleges, is 40

See Jenkins (2006), Jenkins (forthcoming). Many thanks to Cesare Cozzo, Carrie Jenkins and Tim Williamson for pressing me on this point. Many thanks to Jon Kvanvig for helpful correspondence. 42 See Kvanvig (2006) and Kvanvig (forthcoming). 41

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that of a modal paradox. As Kvanvig (quite mistakenly) puts it: The proof undermines the truism that there is a logical distinction between the actual and the possible. (Kvanvig, 2006, p. 54) Here we shall examine Kvanvig’s arguments and some of the problems they face. 5.2.1

The Lost Distinction

Is the proof of WVER → SVER a paradoxical result? Arguably not. As Williamson correctly points out, the truth of WVER is not really a matter of common sense. Still, the proof is quite surprising. Why should a plausible thesis collapse into a blatantly false one? According to Kvanvig, the metaphysical puzzle is just the emerging part of the iceberg. Since truth entails possibility, Kvanvig points out, the proof licenses the following equivalence: (LD) ∀p(p → Kp) ↔ ∀p(p → ♦Kp). LD, Kvanvig argues, obliterates the logical distinction “between actual and possible knowledge in the domain of truth” (Kvanvig, 2006, p. 2). But this, Kvanvig suggests, is a paradoxical result. In his words: though Fitch’s proof may provide an argument against verificationism and other philosophical positions that maintain some variant of the knowability claim, the paradoxicality of the proof is found elsewhere. (Kvanvig, 2006, p. 54) Since actuality and possibility are logical notions, Kvanvig concludes, the proof is a paradox for everyone—not only for the anti-realist committed to WVER.

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5.2.2

Jenkins’ Objections

Kvanvig’s view has been recently opposed by Carrie Jenkins.43 The proof, Jenkins argues, does not reveal that there is “no distinction between knowable and known truth” (Kvanvig, 2006, p. 205). For if this were the case, Jenkins observes, the following equivalence: (LD∗ ) ∀p(p → p) ↔ ∀p(p → ♦p) would be paradoxical too. But this is clearly unacceptable. Furthermore, Jenkins points out, LD does not show that there is no distinction between WVER and SVER. For even if the former is assumed, she writes, all the proof forces us to accept is that the class of known truths is identical to the class of knowable truths (because both are identical to the class of all truths). Provided one acknowledges a distinction between a property and its extension, this does nothing towards establishing that the property of being a knowable truth has collapsed into the property of being a known truth. It is normally acknowledged that even necessarily coextensive properties may be distinct. (Jenkins, 2006, p. 1142) LD can be surprising, but not paradoxical. In Jenkins’ view, the proof just tells us that an intuitively plausible claim—WVER—is in fact false.44 The Church-Fitch proof strikes us as paradoxical, Jenkins argues, only because, when we consider WVER, we do not think of p ∧ ¬Kp as one of its possible values. However, she concludes, the surprise elapses once we check the proof, step by step, and we feel compelled by its validity. 43

See Jenkins (2006) and Jenkins (forthcoming). Similarly, according to Jenkins, Russell showed that the intuitively correct na¨ıve comprehension principle is in fact incorrect. Strictly speaking, she suggests, neither proof is a paradox. 44

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5.2.3

Mystery Yet Again?

To this, it might be argued that LD is nevertheless surprising. For let us suppose that WVER is necessarily false. Then, by LD (left-to-right), it follows that SVER is necessarily false too. What seems to be puzzling, however, is that—in contrast with the case of WVER and SVER—necessarily false claims are not, in general, interderivable in the way WVER and SVER are (consider, for instance, ‘water = H3 O’ and ‘68 + 57 = 5’).45 Indeed, a proof of (16) (Water = H3 O) ↔ (68 + 57 = 5) that rested only on first order logic, plus minimal assumptions of epistemic and modal logic, would be paradoxical. So why LD should not be paradoxical too? 5.2.4

Response to Kvanvig

Albeit prima facie appealing, we suggest that a similar kind of thought should be resisted. For the Church-Fitch proof only shows that WVER and SVER are logically equivalent—not that they express the same proposition. The surprising element of the proof, therefore, can perfectly be located, to use Fregean terms, at the level of sense. More to the point, the analogy between LD and (16) does not seem to be that helpful. In the former case, we have a surprising equivalence, but— following Jenkins—our surprise can be easily accounted for in psychological terms. However, contrary to what (the online) Kvanvig suggests, in the latter case things might well proceed in an analogous way. We can imagine that a hypothetical proof of (16) might well offer us an explanation of the otherwise surprising equivalence. The proof would convince us that the equivalence holds, and the surprise would disappear at some point. To appeal to the 45

As Kvanvig pointed out to me in correspondence and on his weblog Certain Doubts in April 2006.

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counterfactual paradoxicality of a proof of (16), therefore, appears to have the form of a petitio principii.

Conclusion In spite of our faithfulness to orthodoxy, we shall nevertheless follow tradition and continue to call the Church-Fitch proof a paradox. The argument, we shall argue, proves that there are unknowable truth. This result is stronger than the one established by G¨odel’s First Incompleteness Theorem. Whereas G¨odel’s theorem only applies to the notion of provability within formal systems of a certain strength, the Church-Fitch proof bears on the limits of our knowledge in general. Anti-realists have thus good reasons to be worried by the Church-Fitch result. In what follows, we shall examine a number of antirealist attempts to address the worry and block the slippery slope between EC and idealism.

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Contents 1 Realism and Potential Unknowability 1.1 Modesty and Presumption . . . . . . . . . . . . . . . . . . . . 1.2 Dummett’s Manifestation Challenge . . . . . . . . . . . . . . . 2 Truth, Justification and Idealized Assertibility 2.1 All Truths are Justifiably Believed? . . . . . . . . . . 2.1.1 Stability . . . . . . . . . . . . . . . . . . . . . 2.1.2 Matchboxes and Forever Unjustified Truths . 2.2 All Truths are Justifiably Believable? . . . . . . . . . 2.2.1 Kelp and Pritchard’s Proposal . . . . . . . . . 2.2.2 Objections to Kelp and Pritchard . . . . . . . 2.3 Idealized Assertibility . . . . . . . . . . . . . . . . . . 2.3.1 Wright on Truth and Warranted Assertibility 2.3.2 Peircian Biconditionals . . . . . . . . . . . . . 2.3.3 Putnam80 ’s Biconditionals . . . . . . . . . . . 2.3.4 Superassertibility . . . . . . . . . . . . . . . . 2.3.5 Potential Concerns . . . . . . . . . . . . . . . 2.3.6 Four Constraints . . . . . . . . . . . . . . . . 3 Truth and Knowability 3.1 Strong Verificationisms . . . . . . . . . 3.2 The Knowability Principle . . . . . . . 3.2.1 Standard Weak Verificationism 3.2.2 Edgingtonian Knowability . . .

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4 The Paradox of Knowability 20 4.1 The Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Weak Verificationism Under Threat . . . . . . . . . . . . . . . 22

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5 What’s Paradoxical? 5.1 Orthodoxy . . . . . . . . . . . . . . . . . . 5.2 The Mystery of the Disappearing Diamond 5.2.1 The Lost Distinction . . . . . . . . 5.2.2 Jenkins’ Objections . . . . . . . . . 5.2.3 Mystery Yet Again? . . . . . . . . 5.2.4 Response to Kvanvig . . . . . . . .

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References Blome-Tillmann, M.: 2007, The folly of trying to define knowledge, Analysis 67(3), 214–19. Brogaard, B.: forthcoming, Temporalism and Eternalism. An Essay in the Metaphysics of Propositions, Oxford: Oxford University Press. Cogburn, J.: 2004a, The philosophical basis of what? the anti-realist case for dialethism, in G. Priest and J. Beall (eds), The Law of Non-Contradiction, Oxford: Oxford University Press. Cozzo, C.: 1994, What can we learn from the paradox of knowability?, Topoi 13, 71–78. Dummett, M.: 1976, What is a theory of meaning ii, in G. Evans and J. McDowell (eds), Truth and Meaning, Oxford: Oxford University Press. Dummett, M.: 1978, Truth and Other Enigmas, Duckworth, London. Dummett, M.: 1993a, Wittgenstein on necessity: Some reflections, The Seas of Language, Harvard (Mass.): Harvard University Press. Edgington, D.: 1985, The paradox of knowability, Mind 94, 557–68. Fitch, F.: 1963, A logical analysis of some value concepts, Journal of Philosophical Logic 28, 135–42. Frege, G.: 1956, The thought: A logical inquiry, Mind 65(265), 289–311. Hart, W. and McGinn, C.: 1976, Knowledge and necessity, Journal of Philosophical Logic 5, 205–208. Jenkins, C.: 2006, Review of Jonathan Kvanvig, The Paradox of Knowability, Mind 115, 1141–47.

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Jenkins, C.: forthcoming, The mystery of the disappearing diamond, in J. Salerno (ed.), New Essays on the Knowability Paradox, Oxford: Oxford University Press. Kaplan, D.: 1989, Demonstratives, in J. Almog, J. Perry and H. Wettstein (eds), Themes from Kaplan, Oxford: Oxford University Press. Kelp, C. and Pritchard, D.: forthcoming, Two deflationary approaches to Fitch-style reasoning, in J. Salerno (ed.), New Essays on the Knowability Paradox, Oxford: Oxford University Press. Kripke, S.: 1979, A puzzle about belief, in A. Margalit (ed.), Meaning and Use, Dordrecht: Reidel, pp. 239–83. K¨ unne, W.: 2003, Conceptions of Truth, Oxford University Press, Oxford. Kvanvig, J.: 2006, The Knowability Paradox, Oxford University Press, Oxford. Kvanvig, J.: forthcoming, Restriction strategies on knowability: some lessons in false hope, in J. Salerno (ed.), New Essays on the Knowability Paradox, Oxford: Oxford University Press. Peacocke, C.: 1999, being Known, Oxford: Clarendon Press. Plantinga, P.: 1982, How to be an anti-realist, Proceedings and Addresses of the American Philosophical Association, Vol. 56. Prawitz, D.: 1998, Comments on Michael Dummett’s paper ‘truth from the constructive standpoint’, Theoria 64. Putnam, H.: 1981, Realism, Truth and History, Cambridge: Cambridge University Press. Putnam, J.: 1990, Realism with a Human Face, Cambridge, MA: Harvard University Press. 32

Rorty, R.: 1984, Life at the end of inquiry, London Review of Books 2, 6–7. Skorupski, J.: 1988, Review: Realism, meaning and truth, The Philosophical Quarterly 38(153), 500–25. Tennant, N.: 1997, The Taming of the True, Oxford University Press, Oxford. Williamson, T.: 2000, Knowledge and its Limits, Oxford University Press, Oxford. Wright, C.: 1992, Truth and Objectivity, Harvard University Press, Cambridge, (Mass.). Wright, C.: 1993, Realism, Meaning and Truth (2nd ed.), Harvard University Press, Harvard (Mass.). Wright, C.: 2000, Truth as sort of epistemic: Putnam’s peregrinations, The Journal of Philosophy 97(6), 335–64. Wright, C.: 2006, Intuititionism, realism, relativism and rhubarb, in P. Greenough and M. Linch (eds), Truth and Realism, Oxford: Oxford University Press.

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