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Revamping the Restriction Strategy by
, 2007
"... This study continues the antirealist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the antirealist’s knowability principle ‘ϕ, therefore ✸Kϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the r ..."
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This study continues the antirealist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the antirealist’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 socalled Rule of Factiveness ‘✸Kϕ therefore ϕ’. The proposed restriction appears to block another Fitchstyle derivation that uses the KKthesis in order to get around the Cartesian restriction on applications of the knowability principle. ∗ To appear in Joseph Salerno, ed., All Truths are Known: New Essays on the Knowability Paradox, Oxford University Press. This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder.
(to appear in J. Salerno, ed., New Essays on the Knowability Paradox, Oxford: Oxford University Press) Tennant’s Troubles
"... First, some reminiscences. In the years 197380, 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 antirealism. It end ..."
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First, some reminiscences. In the years 197380, 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 antirealism. It endorsed something like the replacement of truthconditional semantics by verificationconditional 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 antirealism 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
FITCH’S PARADOX AND THE PROBLEM OF SHARED CONTENT
"... According to the “paradox of knowability”, the moderate thesis that (necessarily) all truths are knowable – ‘∀p (p ⊃ ◊Kp) ’ – implies the seemingly preposterous claim that all truths are actually known – ‘∀p (p ⊃ Kp) ’ –, i.e. that we are omniscient. If Fitch’s argument were successful, it would am ..."
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According to the “paradox of knowability”, the moderate thesis that (necessarily) all truths are knowable – ‘∀p (p ⊃ ◊Kp) ’ – implies the seemingly preposterous claim that all truths are actually known – ‘∀p (p ⊃ Kp) ’ –, i.e. that we are omniscient. If Fitch’s argument were successful, it would amount to a knockdown rebuttal of antirealism by reductio. In the paper I defend the nowadays rather neglected strategy of intuitionistic revisionism. Employing only intuitionistically acceptable rules of inference, the conclusion of the argument is, firstly, not ‘∀p (p ⊃ Kp)’, but ‘∀p (p ⊃ ¬¬Kp)’. Secondly, even if there were an intuitionistically acceptable proof of ‘∀p (p ⊃ Kp)’, i.e. an argument based on a different set of premises, the conclusion would have to be interpreted in accordance with Heyting semantics, and read in this way, the apparently preposterous conclusion would be true on conceptual grounds and acceptable even from a realist point of view. Fitch’s argument, understood as an immanent critique of verificationism, fails because in a debate dealing with the justification of deduction there can be no interpreted formal language on which realists and antirealists could agree. Thus, the underlying problem is that a satisfactory solution to the “problem of shared content ” is not available. I conclude with some remarks on the proposals by J. Salerno and N. Tennant to reconstruct certain arguments in the debate on antirealism by establishing aporias. In 1963 Frederic Fitch presented an argument that is now commonly regarded as one of the most promising arguments against antirealism or “verificationism”. If, for sake of simplicity, we take Fitch’s highly plausible ‘theorem 1 ’ (1963: 138) (T1) ¬◊K(p & ¬Kp) for granted, one can show in a straightforward manner that the distinctive thesis of semantic antirealism, namely the principle of knowability (PK) ∀p (p ⊃ ◊Kp) leads to a rather implausible result. As the principle of knowability is supposed to be universally valid, we may infer:
Knowability and a Modal Closure Principle
, 2005
"... Does a factive conception of knowability figure in ordinary use? There is some reason to think so. ‘Knowable ’ and related terms such as ‘discoverable’, ‘observable’, and ‘verifiable ’ all seem to operate factively in ordinary discourse. Consider the following example, a dialog between colleagues A ..."
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Does a factive conception of knowability figure in ordinary use? There is some reason to think so. ‘Knowable ’ and related terms such as ‘discoverable’, ‘observable’, and ‘verifiable ’ all seem to operate factively in ordinary discourse. Consider the following example, a dialog between colleagues A and B: A: We could be discovered. B: Discovered doing what? A: Someone might discover that we're having an affair. B: But we are not having an affair! A: I didn’t say that we were. A’s remarks sound contradictory. In this context the factivity of ‘someone might discover that ’ explains this fact. So there is some reason to believe that knowability and related modalities are factive in ordinary use. For factive treatments of knowability in the context of epistemic theories of truth, compare Tennant (2000: 829) and Wright (2001: 5960, n. 17). The factivity of knowability will not be questioned herein, since the paper is concerned to examine some problems that arise in connection with factive knowability. In particular, the paper examines a clutch of issues concerning principles of modal epistemic logic and the knowability of truth. It begins with a puzzle, a closure paradox of knowability, that threatens to show that a factive interpretation of knowability entails the invalidity of a modest modal closure principle. The negative argument is that the puzzle in its original form does not tell against the joint validity of closure and factivity. That is because the puzzle rests on contingent assumptions whose compossibility is doubtful. The positive argument is that there is a formulation of the puzzle that does prescribe 1 revision of normal modal principles. The result may be taken as a new and improved paradox of knowability and as data for the future analysis of factive concepts of possible knowledge. 2 The puzzle was introduced by Sven Rosenkranz (2004) and threatens to show that a factive interpretation of knowability entails the invalidity of a familiar closure principle. The closure principle says that possibility is closed over necessary implication: ◊ϕ, �(ϕ → ψ)  – ◊ψ. The discussion focuses on instances of the principle where our propositional variables take epistemic propositions of the form 'Kϕ', to be read 'somebody at some time knows that ϕ'. Accordingly the focus is on instances of the closure principle (CL) that have the following form:
Knowability from a Logical Point of View
, 2010
"... The wellknown ChurchFitch paradox shows that the verificationist knowability principle all truths are knowable, yields an unacceptable omniscience property. Our semantic analysis establishes that the knowability principle fails because it misses the stability assumption ‘the proposition in questio ..."
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The wellknown ChurchFitch paradox shows that the verificationist knowability principle all truths are knowable, yields an unacceptable omniscience property. Our semantic analysis establishes that the knowability principle fails because it misses the stability assumption ‘the proposition in question does not change from true to false in the process of discovery, ’ hidden in the verificationist approach. Once stability is made explicit, the resulting stable knowability principle accurately represents verificationist knowability, does not yield the omniscience property, and can be offered as a resolution of the knowability paradox. Two more principles are considered: total knowability stating that it is possible to know whether a proposition holds or not, and monotonic knowability stemming from the intrinsically intuitionistic reading of knowability. The study of these four principles yields a “knowability diamond ” describing their logical strength. These results are obtained within a logical framework which opens the door to the systematic study of knowability from a logical point of view. 1