Fitch's Paradox in Fusions of Epistemic and Alethic Logics

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 knowability paradox, or Fitch's paradox, is thought to threaten semantical (Dummettian) antirealism. This chapter suggests that the lesson of the paradox concerns the theoretical location at which to impose the antirealist's ‘epistemic’ constraints on truth, i.e., on the ‘central notion’ of the antirealist's meaning theory. In particular, the knowability principle — that every truth is knowable — is not a successful way of capturing the antirealistic insight that truth is epistemically conditioned.

In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle all truths are knowable, when expressed as a bi-modal principle $${\diamondsuit}$$ , yields an unacceptable omniscience property all truths are known. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it relies, but rather to the knowability principle itself. Further, we show that, from a verifiability perspective, the knowability principle fails in the classical logic setting because it is missing the explicit incorporation of a hidden assumption of stability: ‘the proposition in question does not change from true to false in the process of discovery.’ Once stability is taken into account, the resulting stable knowability principle and its nuanced versions more accurately represent verification-based knowability and do not yield omniscience.

In this paper we undertake an analysis of the knowability paradox in the light of modal epistemic logics and of the phenomena of unsuccessful updates. The knowability paradox stems from the Church-Fitch observation that the plausible knowability principle, according to which all truths are knowable, yields the unacceptable conclusion that all truths are known. We show that the phenomenon of an unsuccessful update is the reason for the paradox arising. Based on this diagnosis, we propose a restriction on the knowability principle which resolves the paradox.

There is an argument (first presented by Fitch), which tries to show by formal means that the anti-realistic thesis that every truth might possibly be known, is equivalent to the unacceptable thesis that every truth actually is known (at some time in the past, present or future). First, the argument is presented and some proposals for the solution of Fitch's Paradox are briefly discussed. Then, by using Wehmeier's modal logic with subjunctive marker (S5*), it is shown how the derivation can be blocked if one respects adequately the distinction between the indicative and the subjunctive mood. Essentially, this proposal amounts to the one by Edgington which was formulated with the help of the actuality-operator. Finally it is shown how the criticisms by Williamson against Edgington can be answered by the formulation of a new conception of possible knowledge that α (thereby α being in the indicative mood and thus referring to the actual world). This conception is based on the concept of same de re knowledge in different possible worlds.

This chapter continues the anti-realist's quest for a principled way to avoid Fitch's paradox. It proposes that the Cartesian restriction on the anti-realist's knowability principle ‘φ, therefore ◇Kφ’ should be formulated as a consistency requirement not on the premise φ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of φ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘◇Kφ therefore φ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK-thesis in order to get around the Cartesian restriction on applications of the knowability principle.

It is rather discouraging that forty years have passed since Frederic Fitch first propounded his paradox of knowability without philosophers having achieved agreement on a solution. As a general rule, when modal phenomena prove puzzling, it is a good idea to look at the corresponding temporal phenomena. This chapter examines not the knowability principle that whatever is true can be known, but rather the discovery principle that whatever is true will be known.

This chapter further investigates and defends Dummett's newly favoured knowability principle, p→¬¬Kp. It discusses the ‘mapping objection’, which points out that Gödel's 1933 mapping of intuitionistic logic into S4 fails to preserve the original formulation of the knowability principle, and that this fact counts against the original formulation as an expression of intuitionistic anti-realism.

A novel solution to the knowability paradox is proposed based on Kant’s transcendental epistemology. The ‘paradox’ refers to a simple argument from the moderate claim that all truths are knowable to the extreme claim that all truths are known. It is significant because anti-realists have wanted to maintain knowability but reject omniscience. The core of the proposed solution is to concede realism about epistemic statements while maintaining anti-realism about non-epistemic statements. Transcendental epistemology supports such a view by providing for a sharp distinction between how we come to understand and apply epistemic versus non-epistemic concepts, the former through our capacity for a special kind of reflective self-knowledge Kant calls ‘transcendental apperception’. The proposal is a version of restriction strategy: it solves the paradox by restricting the anti-realist’s knowability principle. Restriction strategies have been a common response to the paradox but previous versions face serious difficulties: either they result in a knowability principle too weak to do the work anti-realists want it to, or they succumb to modified forms of the paradox, or they are ad hoc. It is argued that restricting knowability to non-epistemic statements by conceding realism about epistemic statements avoids all versions of the paradox, leaves enough for the anti-realist attack on classical logic, and, with the help of transcendental epistemology, is principled in a way that remains compatible with a thoroughly anti-realist outlook.

One diagnosis of Fitch’s paradox of knowability is that it hinges on the factivity of knowledge: that which is known is true. Yet the apparent role of factivity (in the paradox of knowability) and non-factive analogues in related paradoxes of justified belief can be shown to depend on familiar consistency and positive introspection principles. Rejecting arguments that the paradox hangs on an implausible consistency principle, this paper argues instead that the Fitch phenomenon is generated both in epistemic logic and logics of justification by the interaction of analogues of the knowability principle and positive introspection theses that are characteristic of, even if not entailed by, epistemic internalism.

The article shows that Fitch's Paradox of knowability can be resolved through the adoption of additional ontological obligation - the principle of referential conditionality of knowledge.

This chapter defends p→¬¬Kp as the best expression of semantic antirealism.

The paper examines the logic of the knowability paradox and a structural analogue, a new paradox of happiness. We develop a general understanding of what it is to be a Fitch paradox, and follow a natural thread in the literature that attempts to block or resolve Fitch paradoxes. We conclude that, in the case of the attitude of happiness, the new paradox remains even if one finds the knowability analogue non-threatening.

The Knower Paradox has had a brief but eventful history, and principles of epistemic closure (which say that a subject automatically knows any proposition she knows to be materially implied, or logically entailed, by a proposition she already knows) have been the subject of tremendous debate in epistemic logic and epistemology more generally, especially because the fate of standard arguments for and against skepticism seems to turn on the fate of closure. As far as I can tell, however, no one working in either area has emphasized the result I emphasize in this paper: the Knower Paradox just falsifies even the most widely accepted general principles of epistemic closure. After establishing that result, I discuss five of its more important consequences.

Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable us to derive a version of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability (as discussed for instance by Bas van Fraassen) need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source.