Instability and Contraction

Journal of Philosophical Logic - In other works, I’ve proposed a solution to the semantic paradoxes which, at the technical level, basically relies on failure of contraction. I’ve also...     

K ⊈ E

In a series of very influential works, Tim Williamson has advanced and defended a much discussed theory of evidence containing, among other claims, the thesis that, if one knows P, P is part of one's evidence (K ? E). I argue that K ? E is false, and indeed that it is so for a reason that Williamson himself essentially provides in arguing against the thesis that, if one has a justified true belief in P, P is part of one's evidence: together with a very plausible principle governing the acquisition of knowledge by non‐deductive inference based on evidence, K ? E leads, in a sorites‐like fashion, to what would seem a series of unacceptably bootstrapping expansions of one's evidence. I then develop some considerations about the functions of and conditions for evidence which are suggested by the argument against K ? E. I close by discussing the relationship of the argument with anti‐closure arguments of the style exemplified by the preface paradox: I contend that, if closure is assumed, it is extremely plausible to expect that the diagnosis of what goes wrong in the preface‐paradox‐style argument cannot be used to block my own argument.     

SQUEEZING AND STRETCHING: HOW VAGUENESS CAN OUTRUN BORDERLINENESS

The paper develops a critical dialectic with respect to the nowadays dominant approach in the theory of vagueness, an approach whose main tenet is that it is in the nature of the vagueness of an expression to present borderline cases of application, conceived of as enjoying some kind of distinctive normative status. Borderlineness is used to explain the basic phenomena of vagueness, such as, for example, our ignorance of the location of cut-offs in a soritical series. Every particular theory of vagueness exemplifying the approach makes use, in the vague object language, of a definiteness operator which, however substantially interpreted, unavoidably inherits the vagueness of the expressions on which it operates ('higher-order vagueness'). It is first argued that finite soritical series force a surprising collapse result concerning a particular set of expressions involving the definiteness operator. It is then shown that, under two highly plausible assumptions about higher-order vagueness (the existence of 'absolutely definitely' positive and negative cases and the 'radical' character of higher-order vagueness itself), the collapse result implies the inadequacy of the dominant approach as a theory of vagueness, as its main tenet can be, at best, not absolutely definitely true.     

Context and consequence. An intercontextual substructural logic

Some apparently valid arguments crucially rely on context change. To take a kind of example first discussed by Frege, ‘Tomorrow, it’ll be sunny’ taken on a day seems to entail ‘Today, it’s sunny’ taken on the next day, but the first sentence taken on a day sadly does not seem to entail the second sentence taken on the second next day. Mid-argument context change has not been accounted for by the tradition that has extensively studied the distinctive logical properties of context-dependent languages, for that tradition has focussed on arguments whose premises and conclusions are taken at the same context. I first argue for the desiderability of having a logic that accounts for mid-argument context change and I explain how one can informally understand such context change in a standard framework in which the relation of logical consequence holds among sentences. I then propose a family of simple temporal “intercontextual” logics that adequately model the validity of certain arguments in which the context changes. In particular, such logics validate the apparently valid argument in the Fregean example. The logics lack many traditional structural properties (reflexivity, contraction, commutativity etc.) as a consequence of the logical significance acquired by the sequence structure of premises and conclusions. The logics are however strong enough to capture in the form of logical truths all the valid arguments of both classical logic and Kaplan-style “intracontextual” logic. Finally, I extend the framework by introducing new operations into the object language, such as intercontextual conjunction, disjunction and implication, which, contrary to intracontextual conjunction, disjunction and implication, perfectly match the metalinguistic, intercontextual notions of premise combination, conclusion combination and logical consequence by representing their respective two operands as taken at different contexts.     

Naive Modus Ponens

The paper is concerned with a logical difficulty which Lionel Shapiro’s deflationist theory of logical consequence (as well as the author’s favoured, non-deflationist theory) gives rise to. It is argued that Shapiro’s non-contractive approach to solving the difficulty, although correct in its broad outlines, is nevertheless extremely problematic in some of its specifics, in particular in its failure to validate certain intuitive rules and laws associated with the principle of modus ponens. An alternative non-contractive theory is offered which does not suffer from the same problem.     

Higher-Order Sorites Paradox

The naive theory of vagueness holds that the vagueness of an expression consists in its failure to draw a sharp boundary between positive and negative cases. The naive theory is contrasted with the nowadays dominant approach to vagueness, holding that the vagueness of an expression consists in its presenting borderline cases of application. The two approaches are briefly compared in their respective explanations of a paramount phenomenon of vagueness: our ignorance of any sharp boundary between positive and negative cases. These explanations clearly do not provide any ground for choosing the dominant approach against the naive theory. The decisive advantage of the former over the latter is rather supposed to consist in its immunity to any form of sorites paradox. But another paramount phenomenon of vagueness is higher-order vagueness: the expressions (such as ‘borderline’ and ‘definitely’) introduced in order to express in the object language the vagueness of the object language are themselves vague. Two highly plausible claims about higher-order vagueness are articulated and defended: the existence of “definitely ^ω ” positive and negative cases and the “radical” character of higher-order vagueness itself. Using very weak logical principles concerning vague expressions and the ‘definitely’-operator, it is then shown that, in the presence of higher-order vagueness as just described, the dominant approach is subject to higher-order sorites paradoxes analogous to the original ones besetting the naive theory, and therefore that, against the communis opinio, it does not fare substantially better with respect to immunity to any form of sorites paradox.     

Knowledge-how, true indexical belief, and action

Intellectualism is the doctrine that knowing how to do something consists in knowing that something is the case. Drawing on contemporary linguistic theories of indirect interrogatives, Jason Stanley and Timothy Williamson have recently revived intellectualism, proposing to interpret a sentence of the form ‘DP know how to VP’ as ascribing to DP knowledge of a certain way w of VPing that they could VP in w. In order to preserve knowledge-how’s connection to action and thus avoid an overgeneration problem, they add that this knowledge must be had under a “practical” mode of presentation of w. I argue that there can be non-knowledgeable true beliefs under a practical mode of presentation and that some such beliefs would nevertheless be sufficient to establish knowledge-how’s characteristic connection to action, and thus count as knowledge-how. If so, Stanley and Williamson’s account is faced with a serious undergeneration problem. Moreover, the structural features on which the argument relies make it likely to present a quite general challenge for intellectualist strategies.     

A Model of Tolerance

According to the naive theory of vagueness, the vagueness of an expression consists in the existence of both positive and negative cases of application of the expression and in the non-existence of a sharp cut-off point between them. The sorites paradox shows the naive theory to be inconsistent in most logics proposed for a vague language. The paper explores the prospects of saving the naive theory by revising the logic in a novel way, placing principled restrictions on the transitivity of the consequence relation. A lattice-theoretical framework for a whole family of (zeroth-order) “tolerant logics” is proposed and developed. Particular care is devoted to the relation between the salient features of the formal apparatus and the informal logical and semantic notions they are supposed to model. A suitable non-transitive counterpart to classical logic is defined. Some of its properties are studied, and it is eventually shown how an appropriate regimentation of the naive theory of vagueness is consistent in such a logic.