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Knowability Noir: 1945–1963

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Abstract

This chapter analyzes Chapters 1 and 2. It argues that Fitch's intent was to pinpoint a disruptive set of logical properties that lend themselves to the trivialization of conditional analyses. Or, at the very least, Fitch included the central theorems to demonstrate a sort of conditional fallacy that threatens, although not irredeemably, against his own analysis of value. If this is right, then Fitch does not take the knowability proofs to be paradoxical, but instead takes them to be a lesson about how intensional operators interact, surprisingly, to thwart the efforts of conditional analyses. Fitch's demonstration of the knowability proofs may be understood as a logical lesson in how to avoid the so-called 'conditional fallacy' in philosophical analysis.

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... The truth of a sentence is its agreement with (or correspondence to) reality. 17 (ii) In section I.4 the same intention is made precise by requiring that the definition satisfies the material adequacy condition. Hence, the material adequacy condition 'does justice' to the intuitive notion of truth as correspondence. ...
... Of course p 2 is a procedure such that its execution yields, after a finite time, either a proof of K α or a proof of K β; we can therefore take p 2 as f (π ). 17 Reference [21], pp. 342-3. ...
... In other words, the schema (17) ∼α → ¬α; ...
Chapter
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An intuitionistic solution to the Paradox of Knowability is given. It consists (i) in accepting αKα\alpha \mathbin {\rightarrow }{{\mathrm{\mathrm {K}}}}\alpha , the ordinary formalization of the principle of Radical Anti-Realism (RAR) that “Every truth is known”, since, intuitionistically understood, it means that proofs are epistemically transparent; and (ii) in accepting (RAR) itself, on the basis of the fact that knowledge is an intuitionistic internal truth notion. Some neo-verificationist approaches are criticized. Finally the problem of how to frame a rational discussion between Classicism and Intuitionism is briefly discussed.
... 5 Cf. Church (2009), Salerno (2009a) 6 Note that these theorems include second-order quantification over propositions. In order to not confuse an instance of a particular proposition with a bound second-order variable, I will use 'ρ 1 , ρ 2 , . . ...
... Cf.Routley (1981) 9 Cf.Salerno (2009a) for a comprehensive list of historical figures which might be considered committed to this position. To my knowledge, Dummett is the only figure to actually assent to this attribution. ...
Article
In 1963 Frederich Fitch published a paper titled "A Logical Analysis of Some Value Concepts". 1 In this paper he introduced six theorems, of which little was discussed. The significance of these theorems, particularly, Theorem 4, credited to an anonymous referee–now known to be Alonzo Church 2 –, and Theorem 5, a generalization on Theorem 4, was not noticed until thirteen years after the original publication. 3 Hart and McGinn (1976) pointed out that Theorem 5 collapses a weak epistemic principle commonly attributed to verificationists or anti-realists into a position of naive idealism: all truths are known by some person at some time. It is the purpose of this paper to examine theorems 4 and 5, as it has been noted 4 that these theorems establish necessary limits on one's epistemic ability independently of any considerations related to the com-mitments of the anti-realist, and to then show that the only way in which this collapse can be generated is through singular thought about abstract objects. The argument will proceed in two steps: in 1, I outline the modal and epistemic principles required in order to generate Theorems 4 and 5, and then show how these theorems, in conjunction with the weak verificationist principle, generate the modal-epistemic collapse; in 2, I argue that the only possible reading of the fitch proposition which generates the collapse requires singular thought about propositions, and I then examine one way in which singular thought about a proposition might make sense.
... But Fitch attributes the paradox argument to an anonymous referee of an earlier paper by him that was not published. According toSalerno (2009b), that referee was Church. That is why some authors prefer to call the paradox the 'Church-Fitch paradox' . ...
Article
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.
... 2 This paradox was first published by Frederic Fitch (1963), but originates with Alonzo Church, who relayed it to Fitch as an anonymous referee. Church's reports have since been published in Church (2009); see Salerno (2009) for more on the history of the paradox. Church's role in creating this paradox is important to point out both for reasons of historical accuracy and, even more crucially, to justify the title of this paper. ...
Article
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We can find in the passages that set out the Master Argument a precursor to the paradox of knowability. That paradox shows that if all truths are knowable, all truths are known. Similarly, Berkeley might be read as proposing that if all sensible objects are (distinctly) conceivable, then all sensible objects are conceived.
... For the sake of simplicity, we focus on KK throughout, even though everything we say already applies to its weaker, capped version. 5 Fitch credits the reasoning to an anonymous reviewer, who was later discovered by Joe Salerno (and by one of the present authors) to be Alonzo Church (for details, see Salerno, 2009). 6 As a referee has pointed out, the situation here is analogous to a 2-day version of the paradox, in which the surprise exam is announced, say on Wednesday, to take place between Thursday and Friday. ...
Article
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The Surprise Exam Paradox is well‐known: a teacher announces that there will be a surprise exam the following week; the students argue by an intuitively sound reasoning that this is impossible; and yet they can be surprised by the teacher. We suggest that a solution can be found scattered in the literature, in part anticipated by Wright and Sudbury, informally developed by Sorensen, and more recently discussed, and dismissed, by Williamson. In a nutshell, the solution consists in realising that the teacher's announcement is a blindspot that can only be known if the week is at least 2 days long. Along the way, we criticise Williamson's own treatment of the paradox. In Williamson's view, the Surprise is similar to the Paradox of the Glimpse and, because of their similarities, both these paradoxes ought to receive a uniform treatment—one that involves locating an illicit application of the KK Principle. We argue that there's no deep analogy between the Surprise and the Glimpse and that, even if there were, the Surprise reasoning reaches a paradoxical conclusion before the KK Principle is used. Rather, in both the Surprise and the Glimpse, the blame should be put on other epistemic principles—respectively, a knowledge retention and a margin for error principle.
... Because what we have, what these arguments tell us, eventually, is the insidious and creative power of language, viz. the power of logic. I focus here on the knowability paradox, also known as Fitch-Church proof [22] showing that it may offer some interesting insights about onto-theology, and more generally about the peculiar connection between language and existence, logic and ontology, involved in theological thought. ...
Article
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In virtue of Fitch-Church proof, also known as the knowability paradox, we are able to prove that if everything is knowable, then everything is known. I present two ‘onto-theological’ versions of the proof, one concerning collective omniscience and another concerning omnificence. I claim these arguments suggest new ways of exploring the intersection between logical and ontological givens that is a grounding theme of religious thought. What is more, they are good examples of what I call semi-paradoxes: apparently sound arguments whose conclusion is not properly unacceptable, but simply arguable.
... When publishing it, Fitch acknowledged the essential contribution of an anonymous referee. Indeed, it was Fitch's footnote that led scholars to inquire curiously about the origin of the idea (Routley, [1981(Routley, [ ] 2010Künne, 2003, p. 425n), and ultimately to find it (Salerno, 2009b). ...
Article
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.
Chapter
In the last twenty years, knowledge-centered approaches have become increasingly popular in analytic epistemology. Rather than trying to account for knowledge in other terms, these approaches take knowledge as the starting point for the elucidation of other epistemic notions (such as belief, justification, and rationality). Knowledge-centered approaches have been so influential that it now looks as if epistemology is undergoing a factive turn. However, relatively little has been done to explore how knowledge-centered views fare in new fields inside and beyond epistemology strictly understood. This volume aims to remedy this situation by putting together contributions that investigate the significance of knowledge in debates where its roles have been less explored. The goal is to see how far knowledge-centered views can go by exploring new prospects and identifying new trends of research for the knowledge-first program. Extending knowledge-centered approaches in this way promises not only to deliver novel insights into these neglected fields but also to revisit more traditional debates from a fresh perspective. As a whole, the volume develops and evaluates the knowledge-first program in original and fertile ways.
Article
A minimal constraint on normative reasons seems to be that if some fact is a reason for an agent to ϕ (act, believe, or feel), the agent could come to know that fact. This constraint is threatened by a well known type of counterexamples. Self-effacing reasons are facts that intuitively constitute reasons for an agent to ϕ, but that if they were to become known, they would cease to be reasons for that agent. The challenge posed by self-effacing reasons bears important structural similarities with a range of epistemic paradoxes, most notably the Knowability Paradox. In this article, we investigate the similarities and differences between the two arguments. Moreover, we assess whether some of the approaches to the Knowability Paradox could help solve the challenge posed by self-effacing reasons. We argue that at least two popular approaches to the paradox can be turned into promising strategies for addressing the self-effacing reasons problem.
Chapter
Historically, most of the best-known modal logics had axiomatic characterizations long before either tableau systems or semantical approaches were available. While early modal axiom systems were somewhat circuitous by today’s standards, a natural and elegant system for S4\mathbf {S4} was given in Gödel (1933), and this has become the paradigm for axiomatizing modal logics ever since. It is how we do things here.
Chapter
In this chapter the significance of the Paradox of Knowability is discussed with respect to the question of how to conceive truth within an anti-realist conceptual framework. In Sect. 9.1 the Paradox is introduced; Sect. 9.2 articulates the intuitionistic equation of truth with knowledge, first by putting into evidence (Sects. 9.2.1–9.2.3) the conditions at which the equation is acceptable: transparency of knowledge and ‘disquotational property’ of truth; then by showing (Sects. 9.2.4 and 9.2.5) how the charge of inconsistency can be resisted. In Sect. 9.3 the neo-verificationist approaches to the Paradox are discussed, and it is shown how the Paradox hits the neo-verificationist idea of the necessity of a notion of truth irreducible to proof possession. In Sect. 9.4 the Dummettian problem is discussed of how a debate between alternative logics can be rationally shaped.KeywordsIntuitionismKnowability paradoxAnti-realistic theory of meaningTruth notionsInternal truthBHK-ExplanationNeo-VerificationismPhilosophy of logic
Book
Die Autorinnen und Autoren präsentieren in diesem Buch Argumente, die die Unmöglichkeit des Reduktionismus aus philosophischer, naturwissenschaftlicher bzw. mathematisch-logischer Perspektive zu begründen suchen. Der Reduktionismus behauptet, dass Eigenschaften auch von komplexen Systemen (bis hin zu Lebensvorgängen und menschlichem Bewusstsein) vollständig auf ihre Bestandteile zurückgeführt werden können. Diese Position ist einflussreich, aber umstritten. Im Jahr 2019 hat der Kurt Gödel Freundeskreis einen Essaywettbewerb veranstaltet, um schlagende Argumente gegen den Reduktionismus zu finden. Unter den internationalen Teilnehmern waren neben weltweit führenden Forschern auch Wissenschaftlerinnen und Wissenschaftler, die noch am Beginn ihrer Kariere stehen. Dieser Band versammelt die Beiträge der Preisträger und weitere ausgewählte Aufsätze. Aus dem Inhalt: · Kausalität als antireduktionistisches Hausmittel – Martin Breul · Reduktionismus im Diskurs – Hanna Hueske · Monads, Types, and Branching Time – Kurt Gödel’s approach towards a theory of the soul – Tim Lethen · The limits of reductionism: thought, life, and reality – Jesse M. Mulder · True or Rational? A Problem for a Mind-Body Reductionist – Michał Pawłowski · Why reductionism does not work – George F. R. Ellis · Physik ohne Reduktion – Rico Gutschmidt · Is there an Axiom for everything? – Jean-Yves Béziau · Unerklärliche Wahrheiten – Marco Hausmann · Gödel, mathematischer Realismus und Antireduktionismus – Reinhard Kahle Die Herausgeber Oliver Passon ist Privatdozent an der Bergischen Universität Wuppertal und lehrt Physik und ihre Didaktik. Zu seinen Hauptarbeits- und Interessensgebieten gehört die Didaktik, Geschichte und Philosophie der modernen Physik. Christoph Benzmüller ist Professor für KI/Informatik, Logik und Mathematik an der Freien Universität Berlin. Er war der erste UNA Europa Gastprofessor und er kooperiert derzeit mit einem Berliner Startup Unternehmen.
Chapter
Manche Menschen haben hohe Erwartungen an die Wissenschaft. Manche Menschen erwarten von der Wissenschaft, dass sie die Welt vollständig beschreiben wird. Die Wissenschaft wird diese Erwartung aber sicher nie erfüllen – zumindest dann nicht, wenn sie versucht, alle Wahrheiten über die Welt einzeln aufzuzählen. Aber auch der Versuch aus einer endlichen Anzahl von Annahmen eine vollständige Beschreibung zu reduzieren ist zum Scheitern verurteilt.
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Chapter
In this chapter I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism.
Chapter
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Article
Full-text available
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.
Article
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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.
Article
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This article argues against fallibilist evidentialism on the basis of considerations about knowability. KeywordsEpistemology–Fallibilism–Knowability–Unknowability–Evidentialism–Evidence–Justification
Article
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.
Article
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A logical argument known as Fitch’s Paradox of Knowability, starting from the assumption that every truth is knowable, leads to the consequence that every truth is also actually known. Then, given the ordinary fact that some true propositions are not actually known, it concludes, by modus tollens, that there are unknowable truths. The main literature on the topic has been focusing on the threat the argument poses to the so called semantic anti-realist theories, which aim to epistemically characterize the notion of truth; according to those theories, every true proposition must be knowable. But the paradox seems to be a problem also for epistemology and philosophy of science: the conclusion of the paradox – the claim that there are unknowable truths – seems to seriously narrow our epistemic possibilities and to constitute a limit for knowledge. This fact contrasts with certain views in philosophy of science according to which every scientific truth is in principle knowable and, at least at an ideal level, a perfected, “all-embracing”, omniscient science is possible. The main strategies proposed in order to avoid the paradoxical conclusion, given their effectiveness, are able to address only semantic problems, not epistemological ones. However, recently Bernard Linsky (2008) proposed a solution to the paradox that seems to be effective also for the epistemological problems. In particular, he suggested a possible way to block the argument employing a type-distinction of knowledge. In the present paper, firstly, we introduce the paradox and the threat it represents for a certain views in epistemology and philosophy of science; secondly, we show Linsky’s solution; thirdly, we argue that this solution, in order to be effective, needs a certain kind of justification, and we suggest a way of justifying it in the scientific field; fourthly, we show that the effectiveness of our proposal depends on the degree of reductionism adopted in science: it is available only if we do not adopt a complete reductionism.
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