The paper responds to Neil Tennant's recent discussion of Fitch's so-called paradox of knowability in the context of intuitionistic logic. Tennant's criticisms of the author's earlier work on this topic are shown to rest on a principle about the assertability of disjunctions with the absurd consequence that everything we could make true already is true. Tennant restricts the anti-realist principle that truth implies knowability in order to escape Fitch's argument, but a more complex variant of the argument is shown to (...) elicit from his restricted principle exactly the consequences which it was intended to avoid. (shrink)

There is an argument which seems to show that if all truths are knowable then all truths are known. It may be viewed as a "reductio ad absurdum" of certain forms of antirealism. However, The claim has been made elsewhere that the argument fails against antirealists who employ constructivist rather than classical logic. The paper defends and amplifies this claim against criticisms by crispin wright and others. Relations between knowability and time are discussed. Suggestions are also made about the proper (...) form of the constructivist semantics for the conditional, Involving a non-Platonistic version of the distinction between types and tokens of proofs. (shrink)

Perennial philosophers' hopes are unlikely victims of swift, natural deduction. Yet anti-realism has been thought one. Not hoping for anti-realism myself I here show it, lest it be underestimated, to survive the following argument, adapted from W. D.Hart pp. 156, 164-5; he credits first publication to Fitch).

This is a reply to Timothy Williamson ’s paper ‘Tennant’s Troubles’. It defends against Williamson ’s objections the anti-realist’s knowability principle based on the author’s ‘local’ restriction strategy involving Cartesian propositions, set out in The Taming of the True. Williamson ’s purported Fitchian reductio, involving the unknown number of books on his table, is analyzed in detail and shown to be fallacious. Williamson ’s attempt to cause problems for the anti-realist by means of a supposed rigid designator generates a contradiction (...) with arithmetic right away, upon instantiating the obviously relevant theorem that every natural number is provably odd or provably even. The paper also explains and formulates a globally restricted knowability principle, which likewise blocks the attempted reductio. (shrink)

The well-known argument of Frederick Fitch, purporting to show that verificationism (= Truth implies knowability) entails the absurd conclusion that all the truths are known, has been disarmed by Dorothy Edgington''s suggestion that the proper formulation of verificationism presupposes that we make use of anactuality operator along with the standardly invoked epistemic and modal operators. According to her interpretation of verificationism, the actual truth of a proposition implies that it could be known in some possible situation that the proposition holds (...) in theactual situation. Thus, suppose that our object language contains the operatorA — it is actually the case that ... — with the following truth condition: vA iff w0, wherew 0 stands for the designated world of the model — the actual world. Then we can formalize the verificationist claim as follows. (shrink)

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented (...) following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)

Truth’s universal knowability entails its discovery. This threatens antirealism, which is thought to require it. Fortunately, antirealism is not committed to it. Avoiding it requires adoption (and extension) of Dag Prawitz’s position in his long-term disagreement with Michael Dummett on the notion of provability involved in intuitionism’s identification of it with truth. Antirealism (intuitionism generalized) must accommodate a notion of lost-opportunity truth (a kind of recognition-transcendent truth), and even truth consisting in the presence of unperformable verifications. Dummett’s position cannot abide (...) this, while Prawitz’s can. Antirealism’s epistemic notion of truth derives from general features of its meaning theory, not from a universal knowability principle. (shrink)

Fitch’s argument purports to show that for any unknown truth, p , there is an unknowable truth, namely, that p is true and unknown; for a contradiction follows from the assumption that it is possible to know that p is true and unknown. In earlier work I argued that there is a sense in which it is possible to know that p is true and unknown, from a counterfactual perspective; that is, there can be possible, non-actual knowledge, of the actual (...) situation, that in that situation, p is true and unknown. Here I further elaborate that claim and respond to objections by Williamson, who argued that there cannot be non-trivial knowledge of this kind. I give conditions which suffice for such non-trivial counterfactual knowledge. (shrink)

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-Löf 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 φ is true, then there exists a proof of φ 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. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

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 F --> K♢F, 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. (shrink)

A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the (...) paradox. (shrink)

The paradox of knowability poses real difficulities to our understanding of truth. It does so by claiming that if we assume a truth is knowable, we can demonstrate that it is known. This demonstration threatens our understanding of truth in two quite different ways, only one of which has been recognized to this point in the literature on the paradox. Jonathan Kvanvig first unearths the ways in which the paradox is threatening, and then delineates an approach to the paradox that (...) solves both of the problems raised by the paradox for our understanding of truth. His book will be of interest throughout philosophy, but especially to logicians and epistemologists. (shrink)

This collection assembles Church's referee reports, Fitch's 1963 paper, and nineteen new papers on the knowability paradox.

Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed ...

This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore 3Kϕ’ 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 ‘3Kϕ therefore ϕ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around.. (shrink)

First, some reminiscences. In the years 1973-80, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of anti-realism. It endorsed something like the replacement of truth-conditional semantics by verification-conditional semantics and of classical logic by intuitionistic logic, and the principle that all truths are knowable. It did not endorse the principle that all truths are known. (...) Nor did it mention the now celebrated argument, first published by Frederic Fitch (1963), that if all truths are knowable then all truths are known. (shrink)