Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines the various (...) options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis ( 1976 ). (shrink)

This paper is a tribute to Delia Graff Fara. It extends her work on failures of meta-rules for validity as truth-preservation under a supervaluationist identification of truth with supertruth. She showed that such failures occur even in languages without special vagueness-related operators, for standards of deductive reasoning as materially rather than purely logically good, depending on a context-dependent background. This paper extends her argument to: quantifier meta-rules like existential elimination; ambiguity; deliberately vague standard mathematical notation. Supervaluationist attempts to qualify the (...) meta-rules impose unreasonable cognitive demands on reasoning and underestimate her challenge. (shrink)

Might it be that world itself, independently of what we know about it or how we represent it, is metaphysically indeterminate? This article tackles in turn a series of questions: In what sorts of cases might we posit metaphysical indeterminacy? What is it for a given case of indefiniteness to be 'metaphysical'? How does the phenomenon relate to 'ontic vagueness', the existence of 'vague objects', 'de re indeterminacy' and the like? How might the logic work? Are there reasons for postulating (...) this distinctive sort of indefiniteness? Conversely, are there reasons for denying that there is indefiniteness of this sort? (shrink)

Revisionary theories of logic or truth require revisionary theories of mind. This essay outlines nonclassically based theories of rational belief, desire, and decision making, singling out the supervaluational family for special attention. To see these nonclassical theories of mind in action, this essay examines a debate between David Lewis and Derek Parfit over what matters in survival. Lewis argued that indeterminacy in personal identity allows caring about psychological connectedness and caring about personal identity to amount to the same thing. The (...) essay argues that Lewis's treatment of two of Parfit's puzzle cases—degreed survival and fission—presuppose different nonclassical treatments of belief and desire. (shrink)

According to the plurivaluationist, our vague discourse doesn’t have a single meaning. Instead, it has many meanings, each of which is precise—and it is this plurality of meanings that is the source of vagueness. I believe plurivaluationist positions are underdeveloped and for this reason unpopular. This paper attempts to correct this situation by offering a particular development of plurivaluationism that I call supersententialism. The supersententialist leverages lessons from another area of research—the Problem of the Many—in service of the plurivaluationist position. (...) The Problem reveals theoretical reasons to accept that there are many cats where we thought there was one; the supersententialist claims that we are in a similar situation with respect to languages, propositions and sentences. I argue that the parallel suggested by the supersententialist reveals unappreciated advantages and lines of defense for plurivaluationism. (shrink)

Indeterminacy is metaphysical (or worldly) if it has its source in the way the world is (rather than how it is represented or known). There are two questions we could ask about indeterminacy. First: does it exist? Second: is indeterminacy derivative? I focus on the second question. Specifically, I argue that (at least some) metaphysical indeterminacy can be derivative, where this roughly means that facts about indeterminacy are metaphysically grounded in facts about what is determinate.

There is consensus among computer scientists, logicians, and philosophers that good reasoning with qualitative beliefs must have the structural property of cumulative transitivity or, for short, cut. This consensus is typically explicitly argued for partially on the basis of practical and mathematical considerations. But the consensus is also implicit in the approach philosophers take to almost every puzzle about reasoning that involves multiple steps: philosophers typically assume that if each step in reasoning is acceptable considered on its own, the whole (...) chain of reasoning must also acceptable. In this paper I focus on whether there are good philosophical reasons for thinking that the consensus that good reasoning must satisfy cut is true. My central claim is that we should not accept the consensus—good reasoning might not satisfy cut. In particular, I consider four arguments for the consensus and explain why they are unpersuasive. (§2-5). I then show that the issue of whether good reasoning is cut turns on a substantive yet until now unnoticed question in epistemology (§6). (shrink)

Kilimanjaro is an example of what some philosophers would call a ‘vague object’: it is only roughly 5895 m tall, its weight is not precise and its boundaries are fuzzy because some particles are neither determinately part of it nor determinately not part of it. It has been suggested that this vagueness arises as a result of semantic indecision: it is because we didn’t make up our mind what the expression “Kilimanjaro” applies to that we can truthfully say such things (...) as “It is indeterminate whether this particle is part of Kilimanjaro”. After reviewing some of the limitations of this approach, I will propose an alternative account, based on a new semantic relation—multiple reference—capable of holding in a one-many pattern between a term and several objects in the domain. I will explain how multiple reference works, what differentiates it from plural reference and how it might be used to accommodate at least some aspects of our ordinary discourse about vague objects. (shrink)

Central to discussion of supervaluationist accounts of vagueness is the extent to which they require revisions of classical logic and if so, whether those revisions are objectionable. In an important recent Journal of Philosophy article, J.R.G. Williams presents a powerful challenge to the orthodox view that supervaluationism is objectionably revisionary. Williams argues both that supervaluationism is non-revisionary and that even if it were, those revisions would be unobjectionable. This note shows that his arguments for both claims fail.

Many classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists (...) can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic _NT_. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented. (shrink)

Vagueness, as discussed in the philosophical literature, is the phenomenon that paradigmatically rears its head in the sorites paradox, one prominent version of which is: One grain of sand does not make a heap. For any n, if n grains of sand do not make a heap, then n + 1 grains of sand do not make a heap. So, ten billion grains of sand do not make a heap. It is common ground that the different versions of the sorites (...) paradox arise because of vagueness in a key expression, in this case ‘heap’. One central concern in the literature on vagueness is to find a solution to the sorites paradox. Vagueness also gives rise to borderline cases. Since ‘heap’ is a vague expression, ‘heap’ also gives rise to borderline cases: there are some possible entities such that they are neither clearly heaps nor clearly non-heaps, but instead are borderline heaps. Borderline cases can seem intuitively to present counter-examples to bivalence and the law of excluded middle . If Harry is a borderline case of baldness, ‘Harry is bald’ can seem neither true nor false, whence it presents doubts concerning bivalence. Similarly, one can find the corresponding instance of LEM, ‘Either Harry is bald or Harry is not bald’, suspect. There are three main theories that involve departures from bivalence, and two of them involve rejection of LEM as well. Standard three-valued logic invokes the idea of a third truth-value or truth-value gaps; fuzzy logic invokes the idea of continuum-many truth-values intermediate between truth and falsity. Both these views preserve truth-functionality. Traditional supervaluationism involves rejection of bivalence, but adherence to LEM. The idea is that if an expression is vague there is a range of precisifications associated with it, where, roughly, precisifications are ways the expression could be …. (shrink)

We discuss arguments against the thesis that the world itself can be vague. The first section of the paper distinguishes dialectically effective from ineffective arguments against metaphysical vagueness. The second section constructs an argument against metaphysical vagueness that promises to be of the dialectically effective sort: an argument against objects with vague parts. Firstly, cases of vague parthood commit one to cases of vague identity. But we argue that Evans' famous argument against will not on its own enable one to (...) complete the reductio in the present context. We provide a metaphysical premise that would complete the reductio, but note that it seems deniable. We conclude by drawing general morals from our case study. (shrink)

Supervaluational accounts of vagueness have come under assault from Timothy Williamson for failing to provide either a sufficiently classical logic or a disquotational notion of truth, and from Crispin Wright and others for incorporating a notion of higher-order vagueness, via the determinacy operator, which leads to contradiction when combined with intuitively appealing ‘gap principles’. We argue that these criticisms of supervaluation theory depend on giving supertruth an unnecessarily central role in that theory as the sole notion of truth, rather than (...) as one mode of truth. Allowing for the co-existence of supertruth and local truth, we define a notion of local entailment in supervaluation theory, and show that the resulting logic is fully classical and allows for the truth of the gap principles. Finally, we argue that both supertruth and local truth are disquotational, when disquotational principles are properly understood. (shrink)

Recent analytic theology has seen a wave of excellent work on the fundamental problem of Christology, the question of how one and the same person can be human full stop and divine full stop. Along the way, new objections have been raised for a venerable family of Christological views, whose distinctive is the employment of qua-devices to dissolve the difficulties stemming from the dual nature doctrine of Chalcedon and its successors. My objective in this article is twofold. First, I propose (...) to lay out a hierarchy of principles that should guide the search for a Christological theory. I then use these principles to illuminate the best qua-theoretic approach to Christology. Finally, I argue that the best qua-theory is at worst on a par with major recent views. (shrink)

In this paper, I argue that the disjunction elimination rule presupposes the principle that a true disjunction contains at least one true disjunct. However, in some contexts such as supervaluationism or quantum logic, we have good reasons to reject this principle. Hence, disjunction elimination is restricted in at least one respect: it is not applicable to disjunctions for which this principle does not hold. The insight that disjunction elimination presupposes the principle that a true disjunction contains at least one true (...) disjunct is applied to two arguments which argue for this very principle. I show that these arguments are rule-circular since they rest on disjunction elimination. I claim that rule-circularity better explains why the arguments fail than the explanations provided by Rumfitt (2015), which, for instance, rely on controversial principles about truth. (shrink)

It is claimed to be a crucial advantage of supervaluationism over other theories of vagueness that it avoids any commitment to sharp boundaries. This thesis will challenge that claim and argue that almost all forms of supervaluationism are committed to infinitely sharp boundaries and that some of these boundaries are interesting enough to be problematic. I shall argue that only iterated supervaluationism can avoid any commitment to sharp boundaries, but on the other hand that is the model that Terrance Horgan (...) has recently argued is a form of transvaluationism and thus logically incoherent. I shall first argue that infinitely higher-order vagueness gives rise to an infinite number of boundaries. I will then argue that an infinite number of these boundaries are, in the case of the vague term ‘tall’, located over a finite range of heights. I will argue that because of this, these boundaries must be infinitely sharp. I shall argue that on every plausible non-iterated supervaluationist model, some such boundary will mark a sharp boundary between heights that would make someone ‘more tall than not tall’ and heights that would not. Finally I shall argue that this is the sort of interesting sharp boundary supervaluationism must not admit. (shrink)

I explore the thesis that the future is open, in the sense that future contingents are neither true nor false. The paper is divided into three sections. In the first, I survey how the thesis arises on a variety of contemporary views on the metaphysics of time. In the second, I explore the consequences for rational belief of the ‘Aristotelian’ view that indeterminacy is characterized by truth-value gaps. In the third, I outline one line of defence for the Aristotelian against (...) the puzzles this induces: treating opinion about future contingents as a matter of fictional belief rather than simple belief. (shrink)