This book uses mathematical models of language to explain why there are certain gaps in language: things that we might expect to be able to say but can't. For instance, why can we say I ran for five minutes but not *I ran to the store for five minutes? Why is five pounds of books acceptable, but *five pounds of book not acceptable? What prevents us from saying *sixty degrees of water to express the temperature of the water in a (...) swimming pool when sixty inches of water can express its depth? And why can we not say *all the ants in my kitchen are numerous? The constraints on these constructions involve concepts that are generally studied separately: aspect, plural and mass reference, measurement, and distributivity. In this book, Lucas Champollion provides a unified perspective on these domains, connects them formally within the framework of algebraic semantics and mereology, and uses this connection to transfer insights across unrelated bodies of literature and formulate a single constraint that explains each of the judgments above. (shrink)

Brasoveanu (Linguist Philos 34:93–168, 2011) argues that “different” exhibits what he calls association with distributivity: a distributive operator such as “each” creates a two-part context that propagates through the compositional semantics in a way that can be accessed by a subordinate “different”. We show that Brasoveanu’s analysis systematically undergenerates, failing to provide interpretations of sentences such as “Every1 boy claimed every girl read a different1 poem”, in which “different” can associate with a non-local distributive operator. We provide a generalized (...) version of association with distributivity, implemented using de Groote’s (in: Proceedings of semantics and linguistic theory XVI, 2006) continuation-based dynamic semantics. We compare our analysis with the one in Brasoveanu (2011), drawing conclusions about computational tractability, scope of indefinites, and whether it is possible or even desirable to arrive at a unified analysis of internal and external readings of “different”. (shrink)

This article proposes a new logical framework for NL quantification. The framework is based on Generalized Quantifiers, Skolem-like functional dependencies, and Maximality of the involved sets of entities. Among the readings available for NL sentences, those where two or more sets of entities are independent of one another are particularly challenging. In the literature, examples of those readings are known as Collective and Cumulative readings. This article briefly analyzes previous approaches to Cumulativity and Collectivity, and indicates (Schwarzschild in Pluralities. Kluwer, (...) Dordrecht, 1996) as the best proposal so far to deal with these readings. Then, it incorporates its insights in the logical framework defined in Robaldo (J Philos Log 39(1):23–58, 2009a), leading to a scalable logical account for NL quantification. (shrink)

This paper argues that lexical and operator-based analyses of distributivity are not in conflict, but are both necessary components of any theory of distributivity that aims to account for all the relevant data. I use several contrasts between plural definites and group NPs to show that we need an operator-based analysis of distributivity; this kind of distributivity is available with plural definites but not with group subjects, which can be explained under the common assumption that group (...) NPs denote atoms rather than sums and hence do not allow quantification over their individual parts. At the same time, we need a lexical theory of distributivity to account for the various distributive interpretations that we do find with groups; a formalisation of such a theory is outlined in the final section of this paper. (shrink)

The following four chapters take up the features of quantities of matter that distinguish them from individuals by their mereological structure. This structure is reflected in the character of certain predicates expressing the property of being a particular kind of substance, say water, and exhibiting one or more phases, such as being liquid and being gas. Such predicates seem to ascribe an amorphous character to their subjects, which authors have tried to more precisely capture in terms of a distributive and (...) a cumulative condition, formulated in terms of the mereological part relation and sum operation, respectively. These conditions are presented in this chapter and generalised to apply to relational predicates. This generalisation is obviously needed for relational predicates like “is the same substance as” and “is warmer than”. But substance and phase predicates, by contrast with the monadic predicates “is a time” and “is a spatial region”, are also taken to be relational, standing between a quantity and a time. The relational interpretation follows the chemist’s understanding of the permanence of matter throughout chemical change, which has been standard at least since the time of Lavoisier. Aristotle and the Stoics can be reasonably interpreted, it seems, to have thought that the distributive condition holds of substance and phase predicates, as discussed in Chap. 5. Whether the conditions can be upheld from a modern perspective is discussed in Chap. 7.Substance and phase predicates fall under the linguistic category of mass predicates in accordance with grammatical criteria for their use with articles in singular inflexion and qualification as much/little rather than many/few. This category is more extensive and includes terms whose normal translations into some foreign languages don’t comply with the grammatical criteria for mass predicates whilst lacking terms whose normal translations into some foreign languages do comply with the grammatical criteria. But whilst linguistically-inspired characterisations of mass predicates in general don’t necessarily say anything about the nature of substances and phases, Quine presents a general reason for rejecting the distributive condition on all mass predicates which he specifically illustrates with substance predicates based on his understanding of what science tells us about the atomic nature of matter. Quine’s case is discussed in Chap. 4, but as we will see in Chap. 7, it isn’t the last word on the import of the microscopic view of matter presented by modern science for the distributive condition. Taking advantage of what has been said about the occupies relation in Chap. 2, versions of the general distributive and cumulative conditions restricted to spatial parts are formulated. This raises the issue of cooccupancy, and some of what has been about this and the very little argument against the feasibility of the idea is reviewed in the final section of this chapter. (shrink)

This article describes a novel pattern of interpretations associated with universal determiners like ‘each’ and ‘every’. It is demonstrated that these canonically distributive quantifiers can give rise to surprising collective readings when they quantify into sub-clausal constituents, especially other Determiner Phrases. For instance, ‘two cards from each player’ can be understood to pick out a single assorted deck of cards, one whose contents co-vary with the players. Yet this deck as a whole may be said to participate in a range (...) of collective activities (being shuffled together, being traded en masse, not fitting into a standard pack, etc.). Such examples are shown to differ from more familiar _cumulative_ readings of the same quantifiers. A compositional analysis is offered that generalizes Krifka’s ( 2001 ) method of quantification into speech acts in order to accommodate quantification into a larger class of non-truth-denoting semantic objects, including in these cases, entities. (shrink)

We prove that a enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no $m$-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.

We study the gameGfin related to weak distributivity of a given Boolean algebraB. Consider the following implication: ifBis weakly -distributive then player I does not have a winning strategy in the gameGfin. We show that this implication is true for properBbut it is not true in general.

Binominal each is known to exhibit selectional requirements on the noun phrase that immediately precedes it. The goal of this paper is to reduce these selectional requirements to a single requirement of monotonic growth of measurement in relation to the ‘size’ of distributivity. More concretely, it is argued that binominal each imposes a constraint on the functional dependencies arising from distributive quantification, requiring that the measurement of its host grows monotonically with the number of values being distributively quantified. To (...) make constraints on dependencies formally explicit, I devise a version of dynamic plural logic with features from van den Berg (1996) and Brasoveanu (2008, 2013) to semantically represent dependencies arising from evaluating distributive quantification. The use of a dynamic logic, coupled with a delayed evaluation mechanism in terms of higher order meaning (Cresti, 1995; Swart, 2000; Charlow, 2021, allows the constraint to act as an output context constraint on distributive quantification, which mirrors the use of output constraints pioneered by Farkas (1997, 2002b) and and further developed in Brasoveanu (2013), Henderson (2014) and Kuhn (2017). (shrink)

In this paper we propose a unified semantics for singular and plural superlative expressions that makes use of the ‘**’ (“double star”) distributivity operator (an operator whose role is to pluralize 2-place predicates). The analysis aims to solve two problems: (a) the distributivity problem (the fact that a superlative expression doesn’t distribute over the atomic parts of the plural individual it is predicated of); and (b) the cut-off problem (the fact that a plural superlative expression cannot simultaneously be (...) predicated of two distinct yet overlapping plural individuals). We argue that any solution to these problems that posits two distinct superlative morphemes, one corresponding to a superlative operator for singular individuals and one corresponding to a superlative operator for plural individuals, is challenged by the lack of cross-linguistic morphological evidence. We provide a unified analysis, and account for the differences between plural and singular superlative expressions by appealing to pragmatic principles. (shrink)

In this note, we study the complementedness and the distributivity of upper semilattices of Kleene degrees assuming V = L. K denotes the upper semilattice of all Kleene degrees. We prove that if V = L, then some sub upper semilattices of K are non-complemented and some are non-distributive.

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ1. This answers a question of Halbeisen.

In this article, we shall show the generalized notions of distributivity of Boolean algebras have essential relations with several axioms and properties of set theory, say the Axiom of Choice, the Axiom of Dependence Choice, the Prime Ideal Theorems, Martin's axioms, Lebesgue measurability and so on.

This paper begins with a discussion ofcumulativity (e.g., ‘P(a) & P(b) implies P(a+b)’), formalized using a verb phrase operator. Next, the meanings of distributivity markers such aseach and non-distributivity indicators such astogether are considered. An existing analysis ofeach in terms of quantification over parts of a plurality is adopted. However,together is problematic, for it involves a cancellation or negation of the quantification associated witheach. (The four boys together owned exactly three cars could not be true if each of (...) the boys owned three cars, though the same sentence withouttogether could be.) A refinement of this idea of negated quantification over parts is proposed as an analysis for non-distributivity operators. It is then worked out in the system described in Cooper (1983), in which positive and negative extensions are assigned. Presuppositions connected with plural reference are considered at this point as well. Finally, the cumulativity operator is argued to be quantificational and therefore sensitive to contextual domain selection. This context sensitivity is claimed to be the source of distributive readings that appear in the absence of modifiers likeeach and non-distributive or collective readings that arise withouttogether. (shrink)

We introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive law $a \vee \bigwedge_{i \in I} b_i = \bigwedge_{i \in I} (a \vee b_i)$ in the lattice of chapters, in which the chapters a and b i are compact. (Counterexamples in which a is not (...) compact have been found previously.). (shrink)

This paper presents a diagnostic for identifying distributive constructions and shows that it applies to pseudopartitives and for -adverbials. On this basis, a unified account is proposed for the parallels between the constructions involved. This account explains why for -adverbials reject telic predicates (*run to the store for five hours), why pseudopartitives reject count nouns (*five pounds of book ), and why both reject certain measure functions like temperature and speed (*30 of water, *drive for 5 mph). These restrictions all (...) follow from a general constraint on distributive constructions. Related concepts such as the D operator (Link, 1987), the subinterval property (Bennett and Partee, 1972), and divisive reference (Cheng, 1973) can be understood as formalizing special cases of this constraint. (shrink)

We examine 'weak-distributivity' as a rewriting rule $??$ defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proof-nets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for $??$: starting from a set of simple generators (proof-nets which are a n-ary ⊗ of &-ized axioms), any mono-conclusion MLL proof-net can be reached by $??$ rewriting (up to ⊗ and (...) & associativity and commutativity). (shrink)

We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with h > N₁ is consistent. We work with a class of axiom A forcings with countable conditions such that q ≥ n p is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for (...) forcings with creatures that are translated into trees. Both lead to new models of the club principle. (shrink)

In this talk, I first systematize the analogies and complete the picture in some corners. Then, I formally relate the three constructions, and I derive their properties from a single operator. Previous analyses were not designed to account for all three constructions at the same time. Accordingly, I not only assess how well they generalize beyond their intended purpose, but I also evaluate them in their own right. Even so, this analysis improves on previous accounts in several ways. Although I (...) will not focus on this in the talk, the analysis of distributive quantification is compatible with more general analyses of cumulative and collective quantification such as Landman (2000). (shrink)

We construct a class K of infinite convex geometries by taking unions of chains of finite convex geometries, and prove that every member of K is both lower semi-modular and join semi-distributive. These properties, while automatic in the finite case, can fail for infinite convex geometries in general. We also show that not all infinite convex geometries arise via this construction, so that K is a proper subclass of the class of all infinite convex geometries.

The phenomenon of distributed knowledge is well-known in epistemic logic. In this paper, a similar phenomenon in ethics, somewhat neglected so far, is investigated, namely distributed morality. The article explains the nature of distributed morality, as a feature of moral agency, and explores the implications of its occurrence in advanced information societies. In the course of the analysis, the concept of infraethics is introduced, in order to refer to the ensemble of moral enablers, which, although morally neutral per se, can (...) significantly facilitate or hinder both positive and negative moral behaviours. (shrink)

Synthesizing the domains of investigation highlighted in current research in distributed cognition and related fields, this paper offers an initial taxonomy of the overlapping types of resources which typically contribute to distributed or extended cognitive systems. It then outlines a number of key dimensions on which to analyse both the resulting integrated systems and the components which coalesce into more or less tightly coupled interaction over the course of their formation and renegotiation.

According to the hypotheses of distributed and extended cognition, remembering does not always occur entirely inside the brain but is often distributed across heterogeneous systems combining neural, bodily, social, and technological resources. These ideas have been intensely debated in philosophy, but the philosophical debate has often remained at some distance from relevant empirical research, while empirical memory research, in particular, has been somewhat slow to incorporate distributed/extended ideas. This situation, however, appears to be changing, as we witness an increasing level (...) of interaction between the philosophy and the empirical research. In this editorial, we provide a high-level historical overview of the development of the debates around the hypotheses of distributed and extended cognition, as well as relevant theory and empirical research on memory, considering both the role of memory in theoretical debates around distributed/extended ideas and strands of memory research that resonate with those ideas; we emphasize recent trends towards increased interaction, including new empirical paradigms for investigating distributed memory systems. We then provide an overview of the special issue itself, drawing out a number of general implications from the contributions, and conclude by sketching promising directions for future research on distributed memory. (shrink)

There are various philosophical approaches and theories describing the intimate relation people have to artifacts. In this paper, I explore the relation between two such theories, namely distributed cognition and distributed morality theory. I point out a number of similarities and differences in these views regarding the ontological status they attribute to artifacts and the larger systems they are part of. Having evaluated and compared these views, I continue by focussing on the way cognitive artifacts are used in moral practice. (...) I specifically conceptualise how such artifacts (a) scaffold and extend moral reasoning and decision-making processes, (b) have a certain moral status which is contingent on their cognitive status, and (c) whether responsibility can be attributed to distributed systems. This paper is primarily written for those interested in the intersection of cognitive and moral theory as it relates to artifacts, but also for those independently interested in philosophical debates in extended and distributed cognition and ethics of (cognitive) technology. (shrink)