In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, (...) which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases. (shrink)

The presupposition triggered by an expression E is generally satisfied by information that comes before rather than after E in the sentence or discourse. In Heim’s classic theory (1983), this left-right asymmetry is encoded in the lexical semantics of dynamic connectives and operators. But several recent analyses offer a more nuanced approach, in which presupposition satisfaction has two separate components: a general principle (which varies from theory to theory) specifies under what conditions a presupposition triggered by an expression E is (...) satisfied; and an ‘incremental’ component specifies that the principle must be checked on the basis of information that comes before E. Several researchers take this incremental component to be a processing bias, which can be overcome at some cost. If so, it should be possible, though costly, to satisfy presuppositions ‘symmetrically’, i.e. by taking into account linguistic material that comes both before and after the presupposition trigger. We test this claim with experimental means. Using inferential (and to some extent acceptability) tasks involving the anaphoric trigger aussi (‘too’) in French, we argue that symmetric readings are indeed possible (albeit degraded) in environments involving the connectives if, or, and unless. (shrink)

In this paper it is shown how a partial semantics for presuppositions can be given which is empirically more satisfactory than its predecessors, and how this semantics can be integrated with a technically sound, compositional grammar in the Montagovian fashion. Additionally, it is argued that the classical objection to partial accounts of presupposition projection, namely that they lack “flexibility,” is based on a misconception. Partial logics can give rise to flexible predictions without postulating any ad hoc ambiguities. Finally, it is (...) shown how the partial foundation can be combined with a dynamic system of common-ground maintenance to account for accommodation. (shrink)

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

Karttunen's seminal 1973 article Presuppositions of compound sentences, lays the groundwork for the elegant and fruitful theory of this subject which he subsequently presented in (1974). In (1973, pp. 185–8), however, he fallaciously argued that the regularities he discovered concerning the behavior of and, or, and if ... then in English cannot be embodied in any three-valued logic giving a truth-functional interpretation to these connectives. The present paper refutes Karttunen's argument by exhibiting an interpretation with the desired properties, and shows (...) further how the full articulation of his 1974 system can be developed naturally within a truth-conditional semantics for English if bivalence is abandoned — contrary to what Karttunen expected. (shrink)

This paper discusses some formal properties of trivalent approaches to presupposition projection, and in particular of the middle Kleene system of Peters (1977) and Krahmer (1998). After exploring the relationship between trivalent truth-functional accounts and dynamic accounts in the tradition of Heim (1983), I show how the middle Kleene trivalent account can be formulated in a way which shows that it meets the explanatory challenge of Schlenker (2006, 2008a,b), and provide some results relating to the application of the middle Kleene (...) approach to generalised quantifiers. (shrink)

This paper deals with the truth-Conditions and the logic for vague languages. The use of supervaluations and of classical logic is defended; and other approaches are criticized. The truth-Conditions are extended to a language that contains a definitely-Operator and that is subject to higher order vagueness.

Supervaluational theories of vagueness have achieved considerable popularity in the past decades, as seen in eg [5], [12]. This popularity is only natural; supervaluations let us retain much of the power and simplicity of classical logic, while avoiding the commitment to strict bivalence that strikes many as implausible. Like many nonclassical logics, the supervaluationist system SP has a natural dual, the subvaluationist system SB, explored in eg [6], [28].1 As is usual for such dual systems, the classical features of SP (...) (typically viewed as benefits) appear in SB in ‘mirror-image’ form, and the nonclassical features of SP (typically viewed as costs) also appear in SB in ‘mirror-image’ form. Given this circumstance, it can be difficult to decide which of two dual systems is better suited for an approach to vagueness.2 The present paper starts from a consideration of these two approaches— the supervaluational and the subvaluational—and argues that neither of them is well-positioned to give a sensible logic for vague language. §2 presents the systems SP and SB and argues against their usefulness. Even if we suppose that the general picture of vague language they are often taken to embody is accurate, we ought not arrive at systems like SP and SB. Instead, such a picture should lead us to truth-functional systems like strong Kleene logic (K3) or its dual LP. §3 presents these systems, and argues that supervaluationist and subvaluationist understandings of language are better captured there; in particular, that a dialetheic approach to vagueness based on the logic LP is a more sensible approach. §4 goes on to consider the phenomenon of higher-order vagueness within an LP-based approach, and §5 closes with a consideration of the sorites argument itself. (shrink)

Presents a version of dynamic semantics for a language with presuppositions that predicts basic facts about presupposition projection in a non-stipulative way.

: In the 1980s, the analysis of presupposition projection contributed to a ‘dynamic turn’ in semantics: the classical notion of meanings as truth conditions was replaced with a dynamic notion of meanings as Context Change Potentials. We argue that this move was misguided, and we offer an alternative in which presupposition projection follows from the combination of a fully classical semantics and a new pragmatic principle, which we call Be Articulate. This principle requires that a meaning pp’ conceptualized as involving (...) a pre-condition p should be articulated as … … rather than as … pp’ …, unless the full conjunction is ruled out because the first or the second conjunct is semantically idle. In particular, … … is infelicitous - and hence … pp’ … is acceptable - if one can determine as soon as p and is uttered that no matter how the sentence ends these words could be eliminated without affecting its contextual meaning. An equivalence theorem guarantees that this condition suffices to derive Heim’s results in almost all cases. Extensions of the condition lead to several new predictions, in particular concerning some ‘symmetric readings’, as well as presupposition projection in quantified structures, which displays a complex interaction between the nature of the trigger and the monotonicity of the quantifier. (shrink)