Nicholas J. J. Smith

In the context of classical (crisp, precise) sets, there is a familiar connection between the notions of counting, ordering and cardinality. When it comes to vague collections, the connection has not been kept in central focus: there have been numerous proposals regarding the cardinality of vague collections, but these proposals have tended to be discussed in isolation from issues of counting and ordering. My main concern in this paper is to draw focus back onto the connection between these notions. I propose a natural generalisation to the vague case of the familiar process of counting precise collections. I then discuss the relationships between this process of counting and various notions of ordering and cardinality for vague sets. Some existing views concerning the cardinality of vague collections fit better than others with my proposal about how to count the members of such a collection. In particular, the idea that we should approach cardinality via certain formulas of a logical language -- which has been prominent in the recent literature -- is less attractive than other existing proposals.

In an earlier paper I argued that time travellers cannot change the past: alleged models of changing the past either fall into contradiction or else involve avoiding, not changing, the past. Goddu has responded to my argument, maintaining that his hypertime model involves time travellers changing (not avoiding) the past. In the present paper I first discuss what would be required to substantiate the claim that a given model involves changing rather than avoiding the past. I then consider Goddu's hypertime model and an earlier model due to Meiland. I argue that neither author does what would be required to substantiate the claim that the model involves changing (not avoiding) the past. I go on to give reasons for the stronger claim that no-one can present a coherent model and also substantiate the claim that it involves changing (not avoiding) the past.

In this position paper we present a logical framework for modelling reasoning with graded predicates. We distinguish several types of graded predicates and discuss their ubiquity in rational interaction and the logical challenges they pose. We present mathematical fuzzy logic as a set of logical tools that can be used to model reasoning with graded predicates, and discuss a philosophical account of vagueness that makes use of these tools. This approach is then generalized to other kinds of graded predicates. Finally, we propose a general research program towards a logic-based account of reasoning with graded predicates.

The degrees of belief of rational agents should be guided by the evidence available to them. This paper takes as a starting point the view—argued elsewhere—that the formal model best able to capture this idea is one that represents degrees of belief using Dempster–Shafer belief functions. However degrees of belief should not only respect evidence: they also guide decision and action. Whatever formal model of degrees of belief we adopt, we need a decision theory that works with it: that takes as input degrees of belief so represented. The task of this paper is to develop such a decision theory for the belief function model of degrees of belief. This is not the first paper to attempt that task, but compared to the existing literature it takes a more abstract route to its destination, via a consideration of the very idea of rational decision making in light of one’s beliefs and desires. After presenting the new decision theory and comparing it to existing views, the paper goes on to consider diachronic decision situations.

In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox.

The received model of degrees of belief represents them as probabilities. Over the last half century, many philosophers have been convinced that this model fails because it cannot make room for the idea that an agent’s degrees of belief should respect the available evidence. In its place they have advocated a model that represents degrees of belief using imprecise probabilities (sets of probability functions). This paper presents a model of degrees of belief based on Dempster–Shafer belief functions and then presents arguments for belief functions over imprecise probabilities as a model of evidence-respecting degrees of belief. The arguments cover three kinds of issue: theoretical virtues (simplicity, interpretability and flexibility); motivations; and problem cases (dilation and belief inertia).

In this paper I present a new theory of propositions, according to which propositions are abstract mathematical objects: well-formed formulas together with models. I distinguish the theory from a number of existing views and explain some of its advantages  chief amongst which are the following. On this view, propositions are unified and intrinsically truth-bearing. They are mind- and language-independent and they are governed by logic. The theory of propositions is ontologically innocent. It makes room for an appropriate interface with formal semantics and it does not enforce an overly fine or overly coarse level of granularity.

A common objection to theories of vagueness based on fuzzy logics centres on the idea that assigning a single numerical degree of truth -- a real number between 0 and 1 -- to each vague statement is excessively precise. A common objection to Bayesian epistemology centres on the idea that assigning a single numerical degree of belief -- a real number between 0 and 1 -- to each proposition is excessively precise. In this paper I explore possible parallels between these objections. In particular I argue that the only good objection along these lines to fuzzy theories of vagueness does not translate into a good objection to Bayesian epistemology. An important part of my argument consists in drawing a distinction between two different notions of degree of belief, which I call dispositional degree of belief and epistemic degree of belief.

One of Tarski’s stated aims was to give an explication of the classical conception of truth—truth as ‘saying it how it is’. Many subsequent commentators have felt that he achieved this aim. Tarski’s core idea of defining truth via satisfaction has now found its way into standard logic textbooks. This paper looks at such textbook definitions of truth in a model for standard first-order languages and argues that they fail from the point of view of explication of the classical notion of truth. The paper furthermore argues that a subtly different definition—also to be found in classic textbooks but much less prevalent than the kind of definition that proceeds via satisfaction—succeeds from this point of view.

In formal modelling, it is essential that models be supplied with an interpretative story: there must be a clear and coherent account of how the formal model relates to the phenomena it is supposed to model. The traditional representation of degrees of belief as mathematical probabilities comes with a clear and simple interpretative story. This paper argues that the model of degrees of belief as imprecise probabilities (sets of probabilities) lacks a workable interpretation. The standard interpretative story given in the literature is shown to lead to unacceptable results -- and, it is argued, there is no way for imprecise probabilists to restrict, replace or finesse this story so as either to avoid or to be able to live with the consequences. Thus the imprecise probabilist lacks a viable account of how a set of probability functions models a belief state: a story about which properties of the model correspond to facts about the thing modelled and how exactly we are to extract information about an agent's belief state from a set of probabilities.

The storage hypothesis—as described by Norby—is a descriptive thesis (for it yields systematic predictions of human behaviour across a wide range of situations) that has as a core commitment that degrees of belief are stable, persistent states. It is not clear to me that such a view is widely held in philosophy. If the storage hypothesis is not widely held, then arguments against it become less interesting. But is Norby’s argument against the view compelling in any case? I shall argue that it is not.

This paper defends the idea that there might be vagueness or indeterminacy in the world itself---as opposed to merely in our representations of the world---against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague *properties and relations*; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague *objects*; we argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the *world as a whole* might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects.

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a good model: I shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth -- supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth-functional (at least in certain circumstances) -- and supervaluationism does not do so (in those circumstances).

This dissertation is concerned with the problem of giving a correct account of the semantics of vague predicates such as '...is tall', '...is bald' and '...is near...'. ;In Chapter 1 I present a definition of vagueness that aims to capture, in a useful form, all our fundamental intuitions about the vagueness of predicates such as those mentioned above; such a definition is lacking in the literature. I also present an abstract characterisation of the Sorites paradox: one that is independent of the particular forms in which the paradox may be presented, and brings to light the essence of the paradox. ;In Chapter 2 I examine existing theories of vagueness. In light of the definition of vagueness defended in Chapter 1, I argue that we need a semantics for vague language that countenances degrees of truth. I distinguish two sorts of degree theory. One sort---which includes the degree form of supervaluation semantics---sees vagueness as an essentially semantic matter. The other sort---which includes accounts based on fuzzy set theory---accounts for vagueness in language in terms of vagueness in the world. I argue that we need a degree theory of the latter, worldly sort. ;In Chapter 3 I examine the fuzzy theory in detail. I show that many of the objections to the fuzzy view that have been raised in the literature do not carry weight. In particular, I defend the coherence of the idea of degrees of truth. However, I isolate a problem for the fuzzy view---the problem of higher-order vagueness---that is serious enough to render the view unacceptable. ;In Chapter 4 I present a new theory of vagueness: one that is intended to share the advantages of the fuzzy view, while avoiding its disadvantages. In particular, this theory accommodates the phenomenon of higher-order vagueness. The theory involves degrees of truth, but they are not the same as the degrees of truth involved in the fuzzy theory. Although the theory involves a non-classical semantics for vague language, this semantics validates classical logic

This paper defends the idea that there might be vagueness or indeterminacy in the world itself---as opposed to merely in our representations of the world---against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague *properties and relations*; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague *objects*; we argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the *world as a whole* might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects.