Elia Zardini

After setting up the framework of _logical inferentialism_ and explaining that view’s relevance for the question of a priori justification, this chapter presents and discusses ‘the _underdetermination challenge_’ to the view. The challenge consists in arguing that, contrary to logical inferentialism, some logical words are such that no set of formal, recursively enumerable rules of inference can pin down their meaning, in that, for any candidate such set, one can show that the set is adequate also with respect to a meaning different from the intended one. I substantiate the challenge by looking at four broad classes of logical words: _1ary intensional operators_, _conditionals_, _variable-binding operators_, and _congruence predicates_. I close by offering a tentative diagnosis of why logical inferentialism is subject to the underdetermination challenge.

Chapter 1 first presents a particular way of understanding the problematic character of a priori justification. It then develops the three main questions around which the book is structured. The first question concerns the significance of a priori justification: the chapter explains the difficulties in articulating the crucial notion of independence from experience. The second question concerns the sources of a priori justification: the chapter discusses the prospects for intuitions and understanding to play that role. The third question concerns the extent of a priori justification: the chapter examines a couple of contested cases (introspection and testimony). The chapter also explains the aims and origin of the book, and it closes with an exposition of the essential contents of each chapter.

Traditionally, a justification for believing something is a priori if and only if it is independent of experience. Throughout Western philosophy ever since Plato, some of the most divisive questions have been whether a priori justification exists, how it is possible, and how far it reaches. This book is structured around these three main questions. The first part concerns a recent modulation of the question of the existence of a priori justification in terms of its significance: given the unclarities and presuppositions attaching to standard explications of the notion of a priori justification, is it still a useful notion for epistemology? In a tight dialectic, the chapters in this part either attack or defend the theoretical importance of that notion. The second part concerns the sources of a priori justification: since a priori justification is not grounded in experience—our arguably best understood source of justification—in what can it then be grounded? The chapters in this part explore in related ways the possibility that a priori justification is grounded either in intuitions or in understanding. The third part concerns the extent of a priori justification: beyond core cases like mathematical proof, what methods can yield a priori justification? The chapters in this part variously investigate to what degree and why methods like introspection, testimony, and others have import for a priori justification. Overall, the book aims at showcasing as well as pushing further some of the latest contemporary trends in thinking about these questions.

The chapter proposes that, contrary intuitions notwithstanding, dogmatists accept a variety of Moorean arguments as legitimate ways to acquire a justification for their conclusions. The chapter counters an ensuing objection from Bayesian confirmation theory, arguing by way of example that acquisition of justification is compatible with the probability lowering that Bayesian confirmation theory predicts to take place in many Moorean arguments. Appealing to considerations concerning defeat, it then shows that a minimal strengthening of dogmatism is incompatible with the classical, additive theory of the very structure of probabilities, and it consequently investigates a more dogmatist-friendly non-classical, super-additive theory of both the structure and dynamics of probabilities, in which the probability of the conclusion of a Moorean argument rises no less than the probability of its premise, and in which perception can play this epistemic role without either presupposing or directly contributing justification for a principle linking appearances with reality.

This chapter concerns logics that restrict the transitivity property of consequence. The chapter canvasses a range of philosophically interesting non-transitive consequence relations, then discusses of two very influential objections to the philosophical use of non-transitive consequence relations. The chapter argues that one can interpret non-transitive logics as a hybrid, joining important aspects of the metaphysics characteristic of deductive consequence relations as opposed to defeasible ones (i.e. the guarantee of truth preservation) with important aspects of the normativity characteristic of defeasible consequence relations as opposed to deductive ones (i.e. the relevance of the premises' pedigree to the normative force of a valid argument). The chapter closes by considering the behaviour of closure notions (such as theories, worlds and situations) in a non-transitive framework.

The paper explores an argument offered by C. Wright against the “verdict-exclusion” view (according to which we can knowledgeably rule out that, in the presence of a borderline case of Fness, that case is either feasibly knowably F or feasibly knowably ∼F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim } F$$\end{document}). The view is shared by otherwise very different theories of vagueness, such as the “third-possibility” view (according to which an object’s being a borderline case of Fness is a state which is incompatible both with F’s applying to it and with ∼F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim } F$$\end{document}’s applying to it) and “pessimistic classical epistemicism” (according to which not only we are not in a position to rule out the possibility that, after all, F does apply to the case in question nor the possibility that, after all, ∼F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim } F$$\end{document} does apply to it, we also know that either F or ∼F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim } F$$\end{document} applies to it, and we just cannot know which). After presenting Wright’s original argument against the verdict-exclusion view and proposing a strengthening thereof, the paper considers and eventually rejects a restriction of the verdict-exclusion view, restriction which could be proposed in order to block the argument against it. The paper then identifies a first fundamental problem with the argument, concerning a slide from something’s being a mere consequence to its being a full-blooded characterization. Further, the paper distinguishes two different kinds of vagueness other than first-order vagueness (higher-order vagueness proper and what will be called ‘cross-order vagueness’), and shows that the idea behind the argument against the verdict-exclusion view can be used to build an argument against the definiteness of instances of cross-order borderlineness in the first place, thereby making it clear both that the new argument actually undermines a crucial assumption of Wright’s argument against the verdict-exclusion view and that, anyway, the issue has nothing to do in particular with the verdict-exclusion view. The paper then inquires what the verdict-exclusion view should make of such a result, and argues that both third-possibility theorists and pessimistic classical epistemicists face the novel explanatory challenge of giving a principled answer to the question as to why they are not, after all, committed to the definiteness of some instances of cross-order borderlineness. The paper concludes by generalizing the challenge to all extant theories of vagueness and considering possible avenues of reply.

In a series of works, Alan Weir has suggested that the paradoxes are due not so much to the operational principles of classical logic, but to some of its structural principles. Weir’s insightful suggestion has opened up the programme of developing a system that, while validating the principles of naive truth (or of naive vagueness), preserves the classical essence of the logical operations and remains coherent by denying instead, in a philosophically motivated way, some of the classical structural principles. This paper first presents Weir’s logical system and argues that it does not adequately preserve operational classicism. The paper then presents Weir’s nontransitive system of vagueness and argues that it does not adequately preserve naive vagueness. The paper finally presents Weir’s nontransitive system of truth and argues that it does not adequately preserve naive truth. As a contrast, the paper mentions how these shortcomings are overcome by my favoured nontransitive system of vagueness and by my favoured noncontractive system of truth.

At the core of JC Beall and Greg Restall’s brand of logical pluralism is Generalised Tarski’s Thesis, according to which an argument is valid iff, in every relevant case where every premise is true, so is the conclusion. I argue that the thesis implies that many philosophically interesting substructural logics are not legitimate relations of logical consequence. I then diagnose the clash as due to the fact that, in important ways, the thesis is not sensitive to intensional connections and to plurality in occurrences, values and models. Next, I extend the argument to the effect that the more general conception of logical consequence as guaranteed truth preservation clashes with substructurality. I conclude with a proposal as to how, for the substructural logics in question, we can still uphold a broadly semantic conception of logical consequence: given any such logic L, we can reinterpret truth-preservation conditionals with the notions of conjunction and implication available in L, and say that the fact that, in L, φ0, φ1, φ2 …, φi entail ψ is grounded in the fact that, in L, the conditional ‘If “φ0” is true and “φ1” is true and “φ2” is true … and “φi” is true, “ψ” is true’ is a logical truth. On this proposal, contrary to the contemporary Tarskian vulgate, it is logical consequence that is grounded in logical truth rather than vice versa.

For centuries, the Sorites Paradox has spurred philosophers to think and argue about the problem of vagueness. This volume offers a guide to the paradox which is both an accessible survey and an exposition of the state of the art, with a chapter-by-chapter presentation of all of the main solutions to the paradox and of all its main areas of influence. Each chapter offers a gentle introduction to its topic, gradually building up to a final discussion of some open problems. Students will find a comprehensive guide to the fundamentals of the paradox, together with lucid explanations of the challenges it continues to raise. Researchers will find exciting new ideas and debates on the paradox.

Traditionally, a justification for believing something is a priori is if, and only if, it is independent of experience. Throughout Western philosophy since Plato, some of the most divisive questions have been whether a priori justification exists, how it is possible, and how far it reaches. This book is structured around these three main questions. The first question has recently been modulated as to concern the significance of a priori justification: given the unclarities and presuppositions attached to standard explications of the notion of a priori justification, is it still a useful notion for epistemology? In a tight dialectic, the chapters in this part either attack or defend the theoretical importance of that notion. The second part concerns the sources of a priori justification: since a priori justification is not grounded in experience--our arguably best understood source of justification--in what can it be grounded? The chapters in this part explore the possibility that a priori justification is grounded either in intuition or in understanding. The third part concerns the extent of a priori justification: beyond core cases like mathematical proof, what methods can yield a priori justification? The chapters in this part investigate to what degree and why methods like introspection, testimony, and others have an import on a priori justification. Overall, the book showcases and furthers some of the latest contemporary trends in thinking about these questions.

In a series of works, Pablo Cobreros, Paul Égré, David Ripley and Robert van Rooij have proposed a nontransitive system (call it ‘_K__3__L__P_’) as a basis for a solution to the semantic paradoxes. I critically consider that proposal at three levels. At the level of the background logic, I present a conception of classical logic on which _K__3__L__P_ fails to vindicate classical logic not only in terms of structural principles, but also in terms of operational ones. At the level of the theory of truth, I raise a cluster of philosophical difficulties for a _K__3__L__P_-based system of naive truth, all variously related to the fact that such a system proves things that would seem already by themselves repugnant, even in the absence of transitivity. At the level of the theory of validity, I consider an extension of the _K__3__L__P_-based system of naive validity that is supposed to certify that validity in that system does not fall short of naive validity, argue that such an extension is untenable in that its nontriviality depends on the inadmissibility of a certain irresistible instance of transitivity (whence the advertised “final cut”) and conclude on this basis that the _K__3__L__P_-based system of naive validity cannot coherently be adopted either. At all these levels, a crucial role is played by certain metaentailments and by the extra strength they afford over the corresponding entailments: on the one hand, such strength derives from considerations that would seem just as compelling in a general nontransitive framework, but, on the other hand, such strength wreaks havoc in the particular setting of _K__3__L__P_.

The paper critically discusses two prominent arguments against closure principles for knowledge. The first one is the “argument from aggregation”, claiming that closure under conjunction has the consequence that, if one individually knows i premises, one also knows their i-fold conjunction—yet, every one of the premises might exhibit interesting positive epistemic properties while the i-fold conjunction might fail to do so. The second one is the “argument from concatenation”, claiming that closure under entailment has the consequence that, if one knows a premise, one also knows each of its remote consequences one arrives at—yet, again, the premise might exhibit interesting positive epistemic properties while some of its remote consequences might fail to do so. The paper firstly observes that the ways in which these two arguments try to establish that the relevant closure principle has the relevant problematic consequence are strikingly similar. They both crucially involve showing that, given the features of the case, the relevant closure principle acts in effect as a soritical principle, which is in turn assumed to lead validly to the relevant problematic consequence. There are however nontransitive logics of vagueness where soritical principles do not have any problematic consequence. Assuming that one of these logics is the correct logic of vagueness, the paper secondly observes that both arguments describe situations where knowledge is arguably vague in the relevant respects, so that a tolerant logic should be used in reasoning about it, with the effect that the relevant soritical principle no longer validly leads to the relevant problematic consequence. This shows an interesting respect in which the gap between validity and good inference that arguably arises in a transitive framework can be bridged in a tolerant one, thereby approximating better certain features of our epistemic lives as finite subjects. Moreover, even for those who do not subscribe to tolerant logics, the paper’s two observations jointly indicate that, for all the arguments from aggregation and concatenation show, the status of the relevant closure principles should be no worse than that assigned by one’s favoured theory of vagueness to soritical principles, which only rarely is plain falsity and can indeed get arbitrarily close to full truth.

This anthology of the very latest research on truth features the work of recognized luminaries in the field, put together following a rigorous refereeing process. Along with an introduction outlining the central issues in the field, it provides a unique and unrivaled view of contemporary work on the nature of truth, with papers selected from key conferences in 2011 such as Truth Be Told (Amsterdam), Truth at Work (Paris), Paradoxes of Truth and Denotation (Barcelona) and Axiomatic Theories of Truth (Oxford).

Milne [2005] argued that a sentence saying of itself that it does not have a truthmaker is true but does not have a truthmaker. López de Sa and Zardini [2006] worried that, by parity of reasoning, one should conclude that a sentence saying of itself that it is not both true and short is true but not short. Recently, Milne [2013] and Gołosz [2015] have replied to López de Sa and Zardini’s worry, arguing in different ways that the worry is illfounded. In this paper, I’ll address these replies and argue that they fail to dispel López de Sa and Zardini’s worry, bringing out in the process some broader points concerning the use of self-referential sentences in arguments in philosophy of logic.

According to the dominant approach in the theory of vagueness, the nature of the vagueness of an expression ‘F’ consists in its presenting borderline cases in an appropriately ordered series: objects which are neither definitely F nor definitely not F (where the notion of definiteness can be semantic, ontic, epistemic, psychological or primitive). In view of the various problems faced by theories of vagueness adopting the dominant approach, the thesis proposes to reconsider the naive theory of vagueness, according to which the nature of the vagueness of an expression consists in its not drawing boundaries between any neighbouring objects in an appropriately ordered series. It is argued that expressions and concepts which do present this feature play an essential role in our cognitive and practical life, allowing us to conceptualize---in a way which would otherwise be impossible---the typically coarse-grained distinctions we encounter in reality. Despite its strong initial plausibility and ability to explain many phenomena of vagueness, the naive theory is widely rejected because thought to be shown inconsistent by the sorites paradox. In reply, it is first argued that accounts of vagueness based on the dominant approach are themselves subject to higher-order sorites paradoxes. The paradox is then solved on behalf of the naive theory by rejecting the unrestricted transitivity of the consequence relation on a vague language; a family of logics apt for reasoning with vague expressions is proposed and studied (using models with partially ordered values). The characteristic philosophical and logical consequences of this novel solution are developed and defended in detail. In particular, it is shown how the analysis of what happens in the attempt of surveying a sorites series and deciding each case allows the naive theory to recover a "thin" notion of a borderline case

Fitch’s paradox of knowability is an apparently valid reasoning from the assumption (typical of semantic anti-realism) that every true proposition is knowable to the unacceptable conclusion that every true proposition is known. The paper develops a critical dialectic wrt one of the best motivated solutions to the paradox which have been proposed on behalf of semantic anti-realism—namely, the intuitionistic solution. The solution consists, on the one hand, in accepting the intuitionistically valid part of Fitch’s reasoning while, on the other hand, exploiting the characteristic weakness of intuitionistic logic in order to preserve the consistency of such acceptance with the denial of omniscience. It is first remarked how the solution still commits one to acceptance of modal claims which are unwarranted even by the lights of standard intuitionistic semantics. A novel form of the paradox is then introduced, which focuses on infallibility rather than omniscience and derives, from semantic anti-realism and a highly plausible constraint on knowledge, that every believed proposition is not untrue. Because of the logical form of this conclusion, an analogue of the intuitionistic solution for the novel form of the paradox would require drawing the characteristic intuitionistic distinctions wrt decidable propositions, which cannot be done. Semantic anti-realism still intuitionistically entails the unacceptable conclusion that every believed (decidable) proposition is true.