Knowability Paradox Research Articles

Introduction to knowability and beyond

Beyond the limits of human knowledge lies nothing at all—no understandable proposition whose truth may outrun our idealized epistemic capacities. So says the semantic anti-realist, who advocates an epistemic theory of truth—one for which, as a matter of metaphysical necessity, anything true is knowable in principle. The interesting idealization remains elusive but is meant to lie somewhere between the triviality of equating truth with God’s knowledge (uninformative realism) and the naivety of equating it with what humans actually know (naive idealism). The former offers no explanatory gain, and the latter fails to appreciate the objectivity and discoverability of truth. The middle way aims for something like this: barring vagueness and ambiguity, we could in principle know the truth value of any fully understood proposition, given only finite improvements to our epistemic capacities, resources or environment. Call this “moderate anti-realism”. Many forms of non-realism are in the neighborhood of the thesis, including some versions of ethical expressivism, Michael Dummett’s mathematical intuitionism, Putnamian internal realism, Peircean pragmatism, logical positivism, Kantian transcendental realism, and Berkeleyian idealism. One great problem for our understanding of the middle way is the knowability paradox—also known as “Fitch’s paradox” because Frederic Fitch first published the result in 1963, and the “Church–Fitch paradox” because Church was the anonymous referee who conveyed it to Fitch in 1945.1 It consists of a proof that appears

The incarnation and the knowability paradox

The best defense of the doctrine of the Incarnation implies that traditional Christianity has a special stake in the knowability paradox, a stake not shared by other theistic perspectives or by non-traditional accounts of the Incarnation. Perhaps, this stake is not even shared by antirealism, the view most obviously threatened by the paradox. I argue for these points, concluding that these results put traditional Christianity at a disadvantage compared to other viewpoints, and I close with some comments about the extent of the burden incurred.

The Paradox of Idealization

A well-known proof by Alonzo Church, first published in 1963 by Frederic Fitch, purports to show that all truths are knowable only if all truths are known. This is the Paradox of Knowability. If we take it, quite plausibly, that we are not omniscient, the proof appears to undermine metaphysical doctrines committed to the knowability of truth, such as semantic antirealism. Since its rediscovery by Hart and McGinn (1976), many solutions to the paradox have been offered. In this article, we present a new proof to the effect that not all truths are knowable, which rests on different assumptions from those of the original argument published by Fitch. We highlight the general form of the knowability paradoxes, and argue that anti-realists who favour either an hierarchical or an intuitionistic approach to the Paradox of Knowability are confronted with a dilemma: they must either give up anti-realism or opt for a highly controversial interpretation of the principle that every truth is knowable.

From the Knowability Paradox to the existence of proofs

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Lof using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if $${\varphi}$$ is true, then there exists a proof of $${\varphi}$$ . Building on the work of Artemov (Bull Symb Log 7(1): 1–36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.

Knowability and bivalence: intuitionistic solutions to the Paradox of Knowability

In this paper, I focus on some intuitionistic solutions to the Paradox of Knowability. I first consider the relatively little discussed idea that, on an intuitionistic interpretation of the conditional, there is no paradox to start with. I show that this proposal only works if proofs are thought of as tokens, and suggest that anti-realists themselves have good reasons for thinking of proofs as types. In then turn to more standard intuitionistic treatments, as proposed by Timothy Williamson and, most recently, Michael Dummett. Intuitionists can either point out the intuitionistc invalidity of the inference from the claim that all truths are knowable to the insane conclusion that all truths are known, or they can outright demur from asserting the existence of forever-unknown truths, perhaps questioning—as Dummett now suggests—the applicability of the Principle of Bivalence to a certain class of empirical statements. I argue that if intuitionists reject strict finitism—the view that all truths are knowable by beings just like us—the prospects for either proposal look bleak.

Faint-hearted Anti-realism and Knowability

It is often claimed that anti-realists are compelled to reject the inference of the knowability paradox, that there are no unknown truths. I call those anti-realists who feel so compelled ‘faint-hearted’, and argue in turn that anti-realists should affirm this inference, if it is to be consistent. A major part of my strategy in defending anti-realism is to formulate an anti-realist definition of truth according to which a statement is true only if it is verified by someone, at some time. I also liberalize what is meant by a verification to allow for indirect forms of verification. From this vantage point, I examine a key objection to anti-realism, that it is committed to the necessary existence of minds, and reject a response to this problem set forth by Michael Hand. In turn I provide a more successful anti-realist response to the necessary minds problem that incorporates what I call an ‘agential’ view of verification. I conclude by considering what intellectual cost there is to being an anti-realist in the sense I am advocating.

Truth and the Enigma of Knowability

Since its disc overy by Fitch, the paradox of knowability has been a thorn in the anti-realist's side. Recently both Dummett and Tennant have sought to relieve the anti-realist by restricting the applicability of the knowability principle – the principle that all truths are knowable – which has been viewed as both a cardinal doctrine of anti-realism and the assumption for reductio of Fitch's argument. In this paper it is argued that the paradox of knowability is a peculiarly acute manifestation of a syndrome affecting anti-realism, against which Dummett's and Tennant's manoeuvres are not finally efficacious. The anti-realist can only cope with the syndrome by being much clearer about her notion of knowability. In fact, she'll have to offer an account which relativises the notion of knowability both to the world at which knowability is assessed and to the content of the proposition to which it is applied. This is not, however, merely an ad hoc manoeuvre to counter the problematic syndrome; rather it is just what we should expect from the anti-realist's intuitive use of the notion. A preliminary investigation indicates that there is no way of providing a general, systematic explanation of such a notion of knowability and thus an inherent restriction on the principle of knowability – but one differing from those offered by either Dummett or Tennant – is developed.

Anti-realism and Epistemic Accessibility

I argue that Fitch’s ‘paradox of knowability’ presents no special problem for the epistemic anti-realist who believes that reality is epistemically accessible to us. For the claim which is the target of the argument (If p then it is possible to know p) is not a commitment of anti-realism. The epistemic anti-realist’s commitment is (or should be) to the recognizability of the states of affairs which render true propositions true, not to the knowability of the propositions themselves. A formal apparatus for discussing the recognizability of states of affairs is offered, and other prima facie similar approaches to the paradox argument are reviewed.

Review: The Knowability Paradox

Review: The Knowability Paradox

Verificationists Versus Realists: The Battle Over Knowability

Verificationism is the doctrine stating that all truths are knowable. Fitch’s knowability paradox, however, demonstrates that the verificationist claim (all truths are knowable) leads to “epistemic collapse”, i.e., everything which is true is (actually) known. The aim of this article is to investigate whether or not verificationism can be saved from the effects of Fitch’s paradox. First, I will examine different strategies used to resolve Fitch’s paradox, such as Edgington’s and Kvanvig’s modal strategy, Dummett’s and Tennant’s restriction strategy, Beall’s paraconsistent strategy, and Williamson’s intuitionistic strategy. After considering these strategies I will propose a solution that remains within the scope of classical logic. This solution is based on the introduction of a truth operator. Though this solution avoids the shortcomings of the non-standard (intuitionistic) solution, it has its own problems. Truth, on this approach, is not closed under the rule of conjunction-introduction. I will conclude that verificationism is defensible, though only at a rather great expense.

The Knowability Paradox

The Knowability Paradox

The Knower Paradox in the Light of Provability Interpretations of Modal Logic

The Knower Paradox in the Light of Provability Interpretations of Modal Logic

The Knower Paradox in the light of provability interpretations of modal logic

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368--387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338-- 355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287--304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy.

The Paradox of the Knower without Epistemic Closure?

Does the Paradox of the Knower, as introduced by Kaplan and Montague (1960), provide us with compelling reason to doubt that, even when a fair amount of idealization is tolerated, knowledge is closed under entailment? Not so, says Cross (2001). Charles Cross has recently offered a variant of the Paradox of the Knower that dispenses with Kaplan and Montague's assumption that knowledge is closed under entailment. Cross's Paradox of the Knowledge-Plus Knower is not directly concerned with knowledge but rather with what he calls 'knowledgeplus', which is, roughly speaking, derivability from some known premiss. However, Cross (2001) suggests that the Paradox of the KnowledgePlus Knower renders implausible the view that Kaplan and Montague's requirement of epistemic closure is to be blamed for the Paradox of the Knower. The purpose of this note is to suggest that Cross's Paradox of the Knowledge-Plus Knower involves an assumption that is more problematic than Kaplan and Montague's original assumption of epistemic closure. I would like to argue, in addition, that the interplay between knowledge-plus and provability undermines the use of the Paradox of the Knowledge-Plus Knower against the diagnosis of the Paradox of the Knower that blames the contradiction on Kaplan and Montague's assumption that knowledge is closed under entailment. In the end, however, I would like to suggest, the moral of the Knower Paradox remains uncertain. The Paradox of the Knower can be developed in any reasonable theory capable of proving a certain amount of arithmetic. In what follows, I will assume the axioms of Peano Arithmetic as formalized in L(K), the language of arithmetic augmented with a monadic primitive predicate K. Thus I allow for occurrences of the predicate K in the induction axioms of PA(K). Though the full power of PA(K) is not needed to develop the Paradox of the Knower, PA(K) is a simple theory capable of proving

Analogues of Knowability

An interesting recent reply to the Paradox of Knowability is Neil Tennant's proposal: to restrict the anti-realist's knowability thesis to truths the knowing of which is logically consistent. However, this proposal is egregiously ad hoc unless motivated by something other than the wish to save anti-realism from embarrassment. We examine Tennant's argument that his restriction is motivated by parallel considerations in cases that are neutral with respect to debates about realism. We conclude that the cases are not neutral, nor the considerations parallel. The failure of Tennant's argument provides an opportunity to reflect on, among other things, the nature of Moore's paradox, and the role of idealization in doxastic logic.

Knowability and Epistemic Truth

The so-called knowability paradox results from Fitch's argument that if there are any unknown truths, then there are unknowable truths. This threatens recent versions of semantical antirealism, the central thesis of which is that truth is epistemic. When this is taken to mean that all truths are knowable, antirealism is thus committed to the conclusion that no truths are unknown. The correct antirealistic response to the paradox should be to deny that the fundamental thesis of the epistemic nature of truth entails the knowability of all truths. Correctly understood, the antirealistic conditions on a proposition's truth do not require that the proposition possess a verification-procedure which, when executed under the given conditions, issues in an agent's recognition of truth, but merely that there be a verification-procedure which, under these conditions, takes the value true. The knowability paradox and the related idealism problem (that antirealism seems, but is not, committed to the necessary existence of an epistemic agent) draw attention to the fact that certain propositions, those that are about verification-procedures themselves, may under certain conditions take the value true despite their unperformability under these circumstances. Thus these propositions' procedures can only be performed when the propositions are false, and they gain the appearance of antirealistic impossibility (e.g., that there is an unknown truth). This differs from the unperformability that antirealists object to, pertaining merely to matters of execution rather than to the logical structure of the procedures themselves. The force of antirealism's notion of epistemic truth is piecemeal, rather than consisting in a blanket characterization of truth as knowable.

Diamonds are a Philosopher's Best Friends

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.

Clues to the paradoxes of knowability: reply to Dummett and Tennant

In 'Victor's Error' Dummett (2001) considers the semantic anti-realist's conception of truth as knowability. He ponders Fitch's paradox of knowability,1 which threatens any such conception. Dummett maintains that the anti-realist's error is to offer a blanket characterization of truth, expressed by the following knowability principle: any statement A is true if and only if it is possible to know A. Formally,

The Paradox of the Knower without Epistemic Closure

I am pleased to announce the following result: a version of the Knower Paradox remains even when Kaplan and Montague's epistemic closure assumption is dropped. In this essay I establish this result and show that it vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. Finally, I assess the use of the Knower Paradox and the variant presented here in arguments against syntactical treatments of knowledge.

Fitch's proof, verificationism, and the knower paradox

Summary I have argued that without an adequate solution to the knower paradox Fitch's Proof is- or at least ought to be-ineffective against verificationism. Of course, in order to follow my suggestion verificationists must maintain that there is currently no adequate solution to the knower paradox, and that the paradox continues to provide prima facie evidence of inconsistent knowledge. By my lights, any glimpse at the literature on paradoxes offers strong support for the first thesis, and any honest, non-dogmatic reflection on the knower paradox provides strong support for the second. Whether verificationists want to go the route I've suggested is not for me todecide. As in the previous section my aim has been that of defending the mere viability of verificationism in the face of what many, many philosophers have taken to be its death-knell, namely Fitch's Proof. But, as the final objection makes clear, showing that verificationism can live in the face of Fitch's Proof is one thing; showing that it should live is another project. If nothing else, I hope that this papershows that verificationists still have a project to pursue; Fitch's Proof, contrary to popular opinion, need not bury verificationism.13 This paper arose in response to Colin Cheyne's ‘Fitch's Proof and Beyond’ (typescript), which was read at the 1998 AAP(NZ) conference at the University of Waikato, Hamilton, NZ. The paper has undergone significant revision in the face of comments from two anonymous referees, comments for which I am very grateful. Thanks to John Bishop, Colin Cheyne, Jay Garfield, Fred Kroon, and especially Graham Priest and Greg Restall for very helpful discussion.