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. (shrink)

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. (shrink)

A well-known proof by Alonzo Church, first published in 1963 by Frederic Fitch, purports to show that all truths are knowable only if all truths are known. This is the Paradox of Knowability. If we take it, quite plausibly, that we are not omniscient, the proof appears to undermine metaphysical doctrines committed to the knowability of truth, such as semantic anti-realism. Since its rediscovery by Hart and McGinn (1976), many solutions to the paradox have been offered. In this article, we (...) present a new proof to the effect that not all truths are knowable, which rests on different assumptions from those of the original argument published by Fitch. We highlight the general form of the knowability paradoxes, and argue that anti-realists who favour either an hierarchical or an intuitionistic approach to the Paradox of Knowability are confronted with a dilemma: they must either give up anti-realism or opt for a highly controversial interpretation of the principle that every truth is knowable. (shrink)

It is often maintained that co-referential terms can be substituted for one another whilst preserving truth-value in extensional contexts, and preserving grammaticality in all contexts. Crispin Wright calls this claim ‘The Reference Principle’ . Since Wright defines extensional contexts as those in which truth-value is determined only by reference, it is the assertion about substitution salva congruitate that is significant. Wright argues that RP is the key to understanding how Frege came to hold, paradoxically, that the concept horse is not (...) a concept, and to providing a resolution to the paradox .RP and Wright's application of it to Frege's Paradox have recently been criticized by Oliver . 1 Oliver argues that the syntax of natural languages is too complicated for the failure of two terms to fit into the same forms of speech to indicate a difference in semantic significance: RP is, he claims, an empirical falsehood.I shall argue that, although Oliver's arguments show that RP is false as it stands, a revised version of the principle, employing a more robust notion of substitution, is nevertheless correct. I shall end by outlining the implications of this revision for the appeal to RP and for Wright's resolution of Frege's Paradox.1. Referring terms and referential positionOliver's case against RP appeals to an impressive array of counter-examples. For our purposes it will be useful to regard these as being of three sorts. The first sort can be dealt with relatively swiftly: examples of the second and third sorts are more challenging.The first sort of counter-example involves referring terms that feature as parts of complex referring expressions. Oliver's examples include ‘William’ in the complex expression ‘William the Conqueror’ . The substitution of expressions co-referential with ‘William’ into sentences employing the complex referring expression often results in ungrammatical …. (shrink)

Some attempts have been made to block the Knowability Paradox and other modal paradoxes by adopting a type-theoretic framework in which knowledge and necessity are regarded as typed predicates. The main problem with this approach is that when these notions are simultaneously treated as predicates, a new kind of paradox appears. I claim that avoiding this paradox either by weakening the Knowability Principle or by introducing types for both predicates is rather messy and unattractive. I also consider the prospect of (...) using the truth predicate to emulate other modal notions. It turns out that this idea works quite well. (shrink)

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 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. (shrink)

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that (...) some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1 Introduction2 Why It Might Matter3 Two Further Examples and Preliminaries4 An Exclusive Epistemic Virtue of Proof?5 Analyses of Knowledge6 The Inductive Basis of Deduction7 Physical to Mathematical Linkages8 Conclusion. (shrink)

Epistemic transparency tells us that, if an agent S knows a given proposition p , then S knows that she knows that p . This idea is usually encoded in the so-called KK principle of epistemic logic. The paper develops an argument in favor of a moderate version of KK , which I dub quasi-transparency , as a normative rather than a descriptive principle. In the second Section I put forward the suggestion that epistemic transparency is not a demand of (...) ideal rationality, but of ideal epistemic responsibility, and hence that ideally responsible agents verify transparency principles of some sort; I also contend that their satisfaction should not be tied to an internalist epistemology. The central argument in favor of transparency is then addressed in Sections 3 to 8, through the development of a formal system. I show that, in a well-behaved formal setting, a moderate version of transparency is imposed upon us as a result of a number of independent decisions on the structure of higher-order probabilities, as long as we request that our probability and knowledge attributions cohere with each other. Thus I give a rationale to build a model for a hierarchy of languages with different levels of knowledge and probability operators; we obtain an analogous to KK for successive knowledge operators without actually demanding transitivity. The formal argument reinforces the philosophical intuition that epistemic transparency is an important desideratum we should not be too ready to dismiss. (shrink)

In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated response (...) to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.Halbach presents an apparent problem for this argument. Let ‘ N’ and ‘ P’, respectively, denote the necessity and possibility predicates, ‘ K 1’ the knowledge predicate of first type, and let ‘γ’ be a K-free sentence such that γ ↔¬ P⌜ K 1⌜γ⌝⌝; we know such a ‘γ’ exists by a standard diagonalization argument. If we assume that γ is knowable at the next knowledge type, that is, at type 1, and that the possibility typing does not interfere with the knowledge typing, a contradiction quickly ensues. Formally: γ ↔¬ P⌜ …. (shrink)

Recently, it has been observed that the usual type-theoretic restrictions are not enough to block certain paradoxes involving two or more predicates. In particular, when we have a self-referential language containing modal predicates, new paradoxes might appear even if there are type restrictions for the principles governing those predicates. In this article we consider two type-theoretic solutions to multimodal paradoxes. The first one adds types for each of the modal predicates. We argue that there are a number of problems with (...) most versions of this approach. The second one, which we favour, represents modal notions by using the truth predicate together with the corresponding modal operator. This way of doing things is not only useful because it avoids multimodal paradoxes, but also because it preserves the expressive capacity of the language. As an example of the sort of theory we have in mind, we provide a type-theoretic axiomatization that combines truth with necessity and knowledge. (shrink)

The KK principle states that knowing entails knowing that one knows. This historically popular principle has fallen out of favour among many contemporary philosophers in light of putative counterexamples. Recently, some have defended more palatable versions of KK by weakening the principle. These revisions remain faithful to their predecessor in spirit while escaping crucial objections. This paper examines the prospects of such a strategy. It is argued that revisions of the original principle can be captured by a generalized knowledge iteration (...) principle, Weak-KK, which states that knowing entails the possibility of knowing that one knows. But Weak-KK is vulnerable to an unknowability result and therefore must be rejected. The arguments here suggest that retreating to weaker iteration principles is not an option for the KK enthusiast. (shrink)

The Knowability Paradox is a logical argument to the effect that, if there are truths not actually known, then there are unknowable truths. Recently, Alexander Paseau and Bernard Linsky have independently suggested a possible way to counter this argument by typing knowledge. In this article, we argue against their proposal that if one abstracts from other possible independent considerations supporting reasons for typing knowledge and considers the motivation for a type-theoretic approach with respect to the Knowability Paradox alone, there is (...) no substantive philosophical motivation to type knowledge, except that of solving the paradox. Every attempt to independently justify the typing of knowledge is doomed to failure. (shrink)

The Fitch paradox poses a serious challenge for anti-realism. This paper investigates the option for an anti-realist to answer the challenge by restricting the knowability principle. Based on a critical discussion of Dummett's and Tennant's suggestions for a restriction desiderata for a principled solution are developed. In the second part of the paper a different restriction is proposed. The proposal uses the notion of uniform formulas and diagnoses the problem arising in the case of Moore sentences in the different status (...) propositional letters receive. The new proposal is able to avoid some of the criticism on its predecessors. (shrink)

The thesis includes six essays, each corresponding to a chapter, which have the target of widening the discussion on the limits of knowability through the consideration of some general problematics and the discussion of specific topics. The work is composed of two parts, each of three chapters. In the first part, the discussion is focused on a perspective proper of the philosophy of language. In particular, I consider the discussion on the limits of knowability from the point of view of (...) the debate between semantic realism and antirealism. The second part of the thesis is focused on the discussion on the limits of knowability from a perspective more strictly epistemological and proper of the philosophy of science. (shrink)