1.1 Contradictoriness and inconsistency, consistency and non-contradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory

Heinrich Wansing Dresden University of Technology 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 o#ered 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.

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, Tr(A) iff ‡K(A) To remedy the error, Dummett’s proposes the following inductive characterization of truth: (i) Tr(A) iff ‡K(A), if A is a basic statement; (ii) Tr(A and B) iff Tr(A) & Tr(B); (iii) Tr(A or B) iff Tr(A) v Tr(B); (iv) Tr(if A, then B) iff (Tr(A) Æ Tr(B)); (v) Tr(it is not the case that A) iff ¬Tr(A), where the logical constant on the right-hand side of each biconditional clause is understood as subject to the laws of intuitionistic logic. 2 The only other principle in play in Dummett’s discussion is

ing of justifi cation is, in my terms, that (1) the epistemologist has to choose between (a) mind-internalist foundationalism (e.g., evidentialism), (b) mind-externalist foundationalism (e.g., process reliabilism), and (c) mind-internalist coherentism (e.g., simple coherentism), and (2) objections such as the Alternative-Systems Objection and the Isolation Objection dictate against choosing (c) and, therefore, in favor of choosing either (a) or (b). I have challenged this orthodox line on two fronts. I have argued, fi rst, that the epistemologist has a fourth option: (d) mindexternalist coherentism. I have argued, second, that mind-externalist coherentism is immune to both the Alternative-Systems Objection and the Isolation Objection—so that even if these objections are decisive against mind-internalist coherentism, it in no way follows that the epistemologist has to choose between mind-internalist foundationalism and

Antirealists have hitherto offered at best sketches of a theory of truth. This paper presents an antirealist theory of truth in some formal detail. It is shown that the theory is able to deal satisfactorily with some problems that are standardly taken to beset antirealism. According to antirealists, there is an intimate connection between truth and human cognitive capacities which holds of conceptual necessity. While antirealists differ about the exact nature of the connection, the point of conceptual necessity is undisputed; it distinguishes the antirealist conception of truth from a realist one accompanied by some methodological view to the effect that, by natural selection or just by good fortune perhaps, our epistemic powers happen to be so attuned to the world we inhabit that there exist no truths which are beyond our ken in principle. So far antirealists have proposed constraints to be met by antirealist theories of truth, and even a sporadic “informal elucidation ” of antirealist truth (Putnam [1981:56]), but an antirealist theory of truth, comparable, if only just remotely, in formal precision to Tarski’s [1956] theory

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. Even in 1970s Oxford, intuitionistic anti-realism was a strictly minority view, but many others regarded it as a live theoretical option in a way that now seems very distant. As the extreme verificationist commitments of the view have combined with accumulating decades of failure to reply convincingly to criticisms of the arguments in its favour or to carry out the programme of generalizing intuitionistic semantics for 1 mathematics to empirical discourse, even in toy examples, the impression has been

Abstract: Fitch showed that not every true proposition can be known in due time; in other words, that not every proposition is knowable. Moore showed that certain propositions cannot be consistently believed. A more recent dynamic phrasing of Moore-sentences is that not all propositions are known after their announcement, i.e., not every proposition is successful. Fitch’s and Moore’s results are related, as they equally apply to standard notions of knowledge and belief (S 5 and KD45, respectively). If we interpret ‘successful ’ as ‘known after its announcement ’ and ‘knowable ’ as ‘known after some announcement’, successful implies knowable. Knowable does not imply successful: there is a proposition j that is not known after its announcement but there is another announcement after which j is known. We show that all propositions are knowable in the more general sense that for each proposition, it can become known or its negation can become known. We can get to know whether it is true: �(Kj ⁄ K¬j). This result comes at a price. We cannot get to know whether the proposition was true. This restricts the philosophical relevance of interpreting ‘knowable ’ as ‘known after an announcement’. Keywords: modal logic, knowability, Fitch’s paradox, dynamic epistemics, public announcements 1. Successful – the Historical Record