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Knowability and a modal closure principle
... If we take the concept of knowability on its face, it is tempting to define it as the possibility to know. In this sense, a proposition p is knowable if and only if ◊Kp is true, where 'K' stands for 'someone knows at some time that' . 2 Then KP is represented as the schema (1) p → ◊Kp 3 . The Church-Fitch's argument shows that this representation generates a serious difficulty if we accept, in addition to (1), the following intuitively appealing principles: ...
... The fully explicit form of Kp is $x$t (x is an agent and t is a time & Kx, tp) with 'Kx, t' being a sentential operator and standing for 'x knows at t that' . 3 Adopting quantification over propositions, we can use the formula ∀p(p → ◊Kp) instead of the schema (1). I prefer (1) because it simplifies formal details. 4 In intuitionistic logic, (VII) does not entail (VIII), which blocks the counterintuitive conclusion. ...
... But it is not clear at all whether (VIII) and its generalization to the schema ~(p & ~K), as well as the intuitionistically equivalent p → ~~Kp, is epistemologically acceptable. For discussion on this question see Brogaard and Salerno (2006), Salerno (2009a: Part II), Rosenkranz (2004) and Murzi (2010). I leave this issue aside and stick to classical logic. ...
The most straightforward interpretation of the principle of knowability is that every true proposition may be known. This, taken together with some intuitively appealing ideas, raises a problem known as the Church–Fitch paradox. There is a wide variety of alternative interpretations of the principle of knowability that have been offered to avoid the paradox. Some of them are based on rigidification of certain aspects of what is knowable. I examine three proposals representing this strategy, those by Edgington, Rückert and Jenkins. Edgington defines what is knowable as a proposition prefixed by the actuality operator. Rückert and Jenkins maintain that what makes a proposition knowable is the possibility of knowing de re (Rückert) or recognizing (Jenkins) the state of affairs that renders the proposition actually true. In both cases, the link to the actual world (or situation) rigidifies what is knowable in some aspect or other. I argue that all three theories have strongly counterintuitive consequences, and I offer an interpretation of the principle of knowability that is both free from rigidity and immune to the Church–Fitch argument.
... The most obvious way to do this is by constructing proofs that are not in normal form. 2 Thus the most obvious prophylactic against such deductive chicanery is to insist that, when determining whether the pre-condition is met, the proof that is to result from the contemplated application of the restricted rule should be in normal form. With this said by way of foreshadowing, we shall defer detailed illustrative examples to their most natural points of entry below. ...
... 4 Personal communication. See also [2]. ...
... Compare Brogaard and Salerno's proof of the KK-knowability paradox in[1].6 This was observed by Brogaard and Salerno in[2]. ...
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 '', therefore 3K'' should be formulated as a consistency requirement not on the premise ' of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ' 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 '3K' therefore ''. 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.
... knowable falsehoods are possible). It has been argued that our intuitive concept of knowability is factive (Brogaard and Salerno 2006;Fuhrmann 2014) or, in other words, that only truths are knowable. Moreover, Kvanvig (1995) has pointed out that, given that antirealists want to define truth as knowability, they can only accept a factive notion of knowability. ...
... Second, to the extent that philosophers such as Brogaard and Salerno (2006) and Fuhrmann (2014) are right about the claim that our intuitive concept of knowability is factive, it is indeed worth exploring factive concepts of knowability that are made formally precise. Brogaard and Salerno (2006, p. 261) ask us to consider the following imaginary dialogue between two colleagues, A and B: ...
Antirealists who hold the knowability thesis, namely that all truths are knowable, have been put on the defensive by the Church–Fitch paradox of knowability. Rejecting the non-factivity of the concept of knowability used in that paradox, Edgington has adopted a factive notion of knowability, according to which only actual truths are knowable. She has used this new notion to reformulate the knowability thesis. The result has been argued to be immune against the Church–Fitch paradox, but it has encountered several other triviality objections. Schlöder in a forthcoming paper defends the general approach taken by Edgington, but amends it to save it in turn from the triviality objections. In this paper I will argue, first, that Schlöder’s justification for the factivity of his version of the concept of knowability is vulnerable to criticism, but I will also offer an improved justification that is in the same spirit as his. To the extent that some philosophers are right about our intuitive concept of knowability being a factive one, it is important to explore factive concepts of knowability that are made formally precise. I will subsequently argue that Schlöder’s version of the knowability thesis overgenerates knowledge or, in other words, it leads to attributions of knowledge where there is ignorance. This fits a general pattern for the research programme initiated by Edgington. This paper also contains preliminary investigations into the internal and logical structure of lines of inquiries, which raise interesting research questions.
... To address this point, we require a factive concept of knowability, so that in the relevant sense of 'knowable', only actual truths are knowable. For a factive concept of knowability, the following will hold: Factivity kφ → φ Brogaard & Salerno (2006) Nowadays there are alternative conceptualizations of knowability that restrict its range to (f )actual truths: having the counterfactual possibility to know that something is actually true (Edgington, 1985); actually having the capacity to know that something is actually true (Fara, 2010); having the potential to know (Fuhrmann, 2014); having the ability to know 3 There are other reasons to be careful about the factivity of possible knowledge. Heylen (2013, p. 96) notes the following consequence. ...
Many philosophical discussions hinge on the concept of knowability. For example, there is a blooming literature on the so-called paradox of knowability. How to understand this notion, however? In this paper, we examine several approaches to the notion: the naive approach to take knowability as the possibility to know, the counterfactual approach endorsed by Edgington (1985) and Schlöder (2019) , approaches based on the notion of a capacity or ability to know (Fara 2010, Humphreys 2011), and finally, approaches that make use of the resources of dynamic epistemic logic (van Benthem 2004, Holliday 2017).
... ) q 5, AR C right-left 11 Due to Brogaard and Salerno (2006), building on work in Williamson (1992), Brogaard and Salerno (2002), and Rosenkranz (2004). ...
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.
... (4K) also features in other Fitch-like derivations of trouble from the knowability principle (seeBrogaard and Salerno 2006 for an example). ...
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.
I present two arguments that aim to establish logical limits on what we can know. More specifically, I argue for two results concerning what we can know about questions that we cannot answer. I also discuss a line of thought, found in the writings of Pierce and of Rescher, in support of the idea that we cannot identify specific scientific questions that will never be answered.
This paper draws out and connects two neglected issues in Kant’s conception of a priori knowledge. Both concern topics that have been central to contemporary epistemology and to formal epistemology in particular: knowability and luminosity. Does Kant commit to some form of knowability principle according to which certain necessary truths are in principle knowable to beings like us? Does Kant commit to some form of luminosity principle according to which, if a subject knows a priori, then they can know that they know a priori? I defend affirmative answers to both of these questions. And by considering the special kind of modality involved in Kant’s conceptions of possible experience and the essential completability of metaphysics, I argue that the combination of knowability and luminosity principles leads Kant into difficulty.
In 1945, Alonzo Church issued a pair of referee reports in which he anonymously conveyed to Frederic Fitch a surprising proof showing that wherever there is (empirical) ignorance there is also logically unknowable truth. Fitch published this and a generalization of the result in 1963. Ever since, philosophers have been attempting to understand the significance and address the counter-intuitiveness of this, the so-called paradox of knowability. This book assembles Church's referee reports, Fitch's 1963 paper, and nineteen new papers on the knowability paradox. The book contains a general introduction to the paradox and the background literature, and is divided into seven sections that roughly mark the central points of debate. The sections include the history of the paradox, Michael Dummett's constructivism, issues of paraconsistency, developments of modal and temporal logics, Cartesian restricted theories of truth, modal and mathematical fictionalism, and reconsiderations about how, and whether, we ought to construe an anti-realist theory of truth.
I argue that the standard anti-realist argument from manifestability to intuitionistic logic is either unsound or invalid. Strong interpretations of the manifestability of understanding are falsified by the existence of blindspots for knowledge. Weaker interpretations are either too weak, or gerrymandered and ad hoc. Either way, they present no threat to classical logic.
In an attempt to improve upon Alexander Pruss’s work (The principle of sufficient reason: A reassessment, pp. 240–248, 2006), I (Weaver, Synthese 184(3):299–317, 2012) have argued that if all purely contingent events could be caused and something like a Lewisian analysis of causation is true (per, Lewis’s, Causation as influence, reprinted in: Collins, Hall and paul. Causation and counterfactuals, 2004), then all purely contingent events have causes. I dubbed the derivation of the universality of causation the “Lewisian argument”. The Lewisian argument assumed not a few controversial metaphysical theses, particularly essentialism, an incommunicable-property view of essences (per Plantinga’s, Actualism and possible worlds, reprinted in: Davidson (ed.) Essays in the metaphysics of modality, 2003), and the idea that counterfactual dependence is necessary for causation. There are, of course, substantial objections to such theses. While I think a fight against objections to the Lewisian argument can be won, I develop, in what follows, a much more intuitive argument for the universality of causation which takes as its inspiration a result from Frederic B. Fitch’s work (J Symb Logic 28(2):135–142, 1963) [with credit to who we now know was Alonzo church’s, Referee Reports on Fitch’s Definition of value, in: (Salerno (ed.), New essays on the knowability paradox, 2009)] that if all truths are such that they are knowable, then (counter-intuitively) all truths are known. The resulting Church–Fitch proof for the universality of causation is preferable to the Lewisian argument since it rests upon far weaker formal and metaphysical assumptions than those of the Lewisian argument.
In light of the paradox of knowability anti-realists ought to revise their wholesale equation of truth and knowability, lest they be committed to the absurd conclusion that there are no truths that will never be known. The task accordingly becomes to identify the problematic statements the knowability of whose truth would force that conclusion and to restrict the equation in appropriate ways to all but the problematic statements. This restriction strategy was first implemented by Tennant. However, recently Williamson and Brogaard and Salerno have argued that the restriction strategy, and in particular Tennant's implementation of it, fail to avert the paradoxical conclusion. Here I argue, first, that the arguments devised by Brogaard and Salerno are ineffective because they rely on an invalid closure principle and, second, that while Williamson's argument may suffice to undermine Tennant's specific proposal, it fails to discredit the restriction strategy as such. To this end, I give a better characterisation of the problematic cases, which is immune to Williamson's criticism, and then show how the restricted anti-realist thesis fares in light of the meaning-theoretical arguments anti-realists typically advance in support of their view.
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.
This paper presents a generalized form of Fitch’s paradox of knowability, with the aim of showing that the questions it raises
are not peculiar to the topics of knowledge, belief, or other epistemic notions. Drawing lessons from the generalization,
the paper offers a solution to Fitch’s paradox that exploits an understanding of modal talk about what could be known in terms
of capacities to know. It is argued that, in rare cases, one might have the capacity to know that p even if it is metaphysically impossible for anyone to know that p, and that recognizing this fact provides the resources to solve Fitch’s paradox.
Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even con-troversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from our list: a claim expressing some sort of common ground in the debate; a characteristically anti-realist thesis about truth and knowability; and a characteristically realist thesis about determinacy of truth value. Various patterns of reductio proof for these inconsistent triads are generated, so as to display their variety. The reductio proofs use only the inferential rules set out earlier. The philosophical utility of each aporia for the anti-realist is then assessed. This involves consideration of the acceptability of the premiss expressing common ground; the strength and plausibility of the anti-realist premiss; the strength of the realist premiss that is the ultimate target of the reductio; and the analytic status of the inferential rules applied within the proof.
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.
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.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.
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 assertibility 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.
This paper addresses an objection raised by Timothy Williamson to the ‘restriction strategy’ that I proposed, in The Taming of The True, in order to deal with the Fitch paradox. Williamson provides a newversion of a Fitch-style argument that purports to show that even the restricted principle of knowability suffers the same fate as the unrestricted one. I show here that the new argument is fallacious. The source of the fallacy is a misunderstanding of the condition used in stating the restricted knowability principle. I also rebut WilliamsonÃÂs criticism of my argument for the claim that any proposition of the form ‘it is known thatφ’ is decidable if φ is decidable
The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classified as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do , and value for . Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic. It will also be assumed that quantifiers over propositions have been included in extensions of these systems.
This paper addresses three problems: the problem of formulating a coherent relativism, the Sorites paradox and a seldom noticed difficulty in the best intuitionistic case for the revision of classical logic. A response to the latter is proposed which, generalised, contributes towards the solution of the other two. The key to this response is a generalised conception of indeterminacy as a specific kind of intellectual bafflement - Quandary. Intuitionistic revisions of classical logic are merited wherever a subject matter is conceived both as liable to generate Quandary and as subject to a broad form of evidential constraint. So motivated, the distinctions enshrined in intuitionistic logic provide both for a satisfying resolution of the Sorites paradox and a coherent outlet for relativistic views about, e.g., matters of taste and morals. An important corollary of the discussion is that an epistemic conception of vagueness can be prised apart from the strong metaphysical realism with which its principal supporters have associated it, and acknowledged to harbour an independent insight.