What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

We present a formal system for reasoning about inclusion and exclusion in natural language, following work by MacCartney and Manning. In particular, we show that an extension of the Monotonicity Calculus, augmented by six new type markings, is sufficient to derive novel inferences beyond monotonicity reasoning, and moreover gives rise to an interesting logic of its own. We prove soundness of the resulting calculus and discuss further logical and linguistic issues, including a new connection to the classes of weak, strong, (...) and superstrong negative polarity items. (shrink)

Ways of Scope Taking is concerned with syntactic, semantic and computational aspects of scope. Its starting point is the well-known but often neglected fact that different types of quantifiers interact differently with each other and other operators. The theoretical examination of significant bodies of data, both old and novel, leads to two central claims. (1) Scope is a by-product of a set of distinct Logical Form processes; each quantifier participates in those that suit its particular features. (2) Scope interaction is (...) further constrained by the semantics of the interacting operators. The arguments are developed using Minimalist syntax, Generalized Quantify theory, Discourse Representation Theory, and algebraic semantics. The contributors (Beghelli, Ben-Shalom, Doetjes, Farkas, Gutiérrez Rexach, Honcoop, Stabler, Stowell, Szabolcsi and Zwarts) make tightly related theoretical assumptions and focus on related empirical phenomena, which include the direct and inverse scope of quantifiers, distributivity, negation, modal and intensional contexts, weak islands, event-related readings, interrogatives, wh/quantifier interactions, and Hungarian syntax. An introduction to the formal semantics background is provided. Audience: Linguists, philosophers, computational and psycholinguists; advanced undergraduates, graduate students and researchers in these fields. (shrink)

Type-logical semantics studies linguistic meaning with the help of the theory of types. The latter originated with Russell as an answer to the paradoxes, but has the additional virtue that it is very close to ordinary language. In fact, type theory is so much more similar to language than predicate logic is, that adopting it as a vehicle of representation can overcome the mismatches between grammatical form and predicate logical form that were observed by Frege and Russell. The grammatical forms (...) of ordinary language sentences consequently may be taken to be much less misleading than logicians in the first half of the 20th century often thought them to be. This was realized by Richard Montague, who used the theory of types to translate fragments of ordinary language into a logical language. Semantics is commonly divided into lexical semantics, which studies the meaning of words, and compositional semantics, which studies the way in which complex phrases obtain a meaning from their constituents. The strength of type-logical semantics lies with the latter, but type-logical theories can be combined with many competing hypotheses about lexical meaning, provided these hypotheses are expressed using the language of type theory. (shrink)

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

1 Logic in philosophy The century that was Logic has played an important role in modern philosophy, especially, in alliances with philosophical schools such as the Vienna Circle, neopositivism, or formal language variants of analytical philosophy. The original impact was via the work of Frege, Russell, and other pioneers, backed up by the prestige of research into the foundations of mathematics, which was fast bringing to light those amazing insights that still impress us to-day. The Golden Age of the 1930s (...) deeply affected philosophy, and heartened the minority of philosophers with a formalanalytical bent. As Brand Blanshard writes in Reason and Analysis (1964) – I quote from memory here, to avoid the usual disappointment when re-reading an original text. (shrink)

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations (...) from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction. (shrink)

A method is described for inducing a type-logical grammar from a sample of bare sentence trees which are annotated by lambda terms, called term-labelled trees . Any type logic from a permitted class of multimodal logics may be specified for use with the procedure, which induces the lexicon of the grammar including the grammatical categories. A first stage of semantic bootstrapping is performed, which induces a general form lexicon from the sample of term-labelled trees using Fulop’s (J Log Lang Inf (...) 14(1):49–86, 2005) procedure. Next we present a two-stage procedure for performing distributional learning by unifying the lexical types that are initially discovered. The first structural unification algorithm in essence unifies the initial family of sets of types so that the resulting grammar will generate all term-labelled trees that follow the usage patterns evident from the learning sample. Further altering the lexical categories to generate a recursively extended language can be accomplished by a second unification. The combined unification algorithm is shown to yield a new type-logical lexicon that extends the learning sample to a possibly infinite (and possibly context-sensitive) language in a principled fashion. Finally, the complete learning strategy is analyzed from the perspective of algorithmic learning theory; the range of the procedure is shown to be a class of term-labelled tree languages which is finitely learnable from good examples (Lange et al in Algorithmic learning theory, Vol 872 of lecture notes in artificial intelligence, Springer, Berlin, pp 423–437), and so is identifiable in the limit as a corollary. (shrink)

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which finite sequences of lambda terms are the basic data structures, (...) pairs of terms syntax, semantics for example. These sequences then combine with the help of simple generalizations of the usual abstraction and application operations. This theory, which I call Lambda Grammars and which is closely related to the independently formulated theory of Abstract Categorial Grammars (de Groote, 2001; de Groote, 2002), in fact is an implementation of Curry’s ideas: the level of tectogrammar is encoded by the sequences of lambda-terms and their ways of combination, while the syntactic terms in those sequences constitute the phenogrammatical level. In de Groote’s formulation of the theory, tectogrammar is the level of abstract terms, while phenogrammar is the level of object terms. (shrink)

In this paper we discuss the treatment of variables in dynamic semantics. Referent systems are introduced as a flexible mechanism for working with variables. In a referent system we carefully distinguish the variables themselves both from the machinery by which we manipulate them - their names - and from the information that we store in them - their values. It is shown that the referent systems provide a natural basis for dynamic semantics. The semantics with referent systems is compared with (...) the familiar formalisms in dynamic semantics, DRT and DPL. (shrink)

The development of the dynamic semantics of natural languagehas put issues of variable control on the agenda of formal semantics. Inthis paper we regard variables as names for stacks of values and makeexplicit several control actions as push and pop actions on stacks. Weapply this idea both to static and dynamic languages and compare theirfinite variable hierarchies, i.e., the relation between the number ofvariable stacks that is available and the expressivity of the language.This can be compared in natural languages with (...) relating the number ofpronouns available to the expressivity of the language.The results are obtained using techniques from static and dynamic modeltheory: model theoretic games, transition systems and bisimulation. (shrink)

Labeled transition systems are key structures for modeling computation. In this paper, we show how they lend themselves to ordinary logical analysis (without any special new formalisms), by introducing their standard first-order theory. This perspective enables us to raise several basic model-theoretic questions of definability, axiomatization and preservation for various notions of process equivalence found in the computational literature, and answer them using well-known logical techniques (including the Compactness theorem, Saturation and Ehrenfeucht games). Moreover, we consider what happens to this (...) general theory when one restricts attention to special classes of transition systems (in particular, finite ones), as well as extended logical languages (in particular, infinitary first-order logic). We hope that this puts standard logical formalisms on the map as a serious option for a theory of computational processes. As a side benefit, our approach increases comparability with several other existing formalisms over labeled transition systems (such as Process Algebra or Modal Logic). We provide some pointers to this effect, too. (shrink)

The idea that natural language grammar and planned action are relatedsystems has been implicit in psychological theory for more than acentury. However, formal theories in the two domains have tendedto look very different. This article argues that both faculties sharethe formal character of applicative systems based on operationscorresponding to the same two combinatory operations, namely functional composition and type-raising. Viewing them in thisway suggests simpler and more cognitively plausible accounts of bothsystems, and suggests that the language faculty evolved in the (...) speciesand develops in children by a rather direct adaptation of a moreprimitive apparatus for planning purposive action in the world bycomposing affordances of objects or tools. Theknowledge representation that underlies such planning is alsoreflected in the natural language semantics of tense, mood, andaspect, which the paper begins by arguing provides the key tounderstanding both systems. (shrink)

Recently M. Szabolcs [12] has shown that many substructural logics including Lambek CalculusL are complete with respect to relativized Relational Semantics. The current paper proves that it is sufficient forL to consider a relativization to the relation x dividesy in some fixed semigroupG.

Discontinuity refers to the character of many natural language constructions wherein signs differ markedly in their prosodic and semantic forms. As such it presents interesting demands on monostratal computational formalisms which aspire to descriptive adequacy. Pied piping, in particular, is argued by Pollard (1988) to motivate phrase structure-style feature percolation. In the context of categorial grammar, Bach (1981, 1984), Moortgat (1988, 1990, 1991) and others have sought to provide categorial operators suited to discontinuity. These attempts encounter certain difficulties with respect (...) to model theory and/or proof theory, difficulties which the current proposals are intended to resolve.Lambek calculus is complete for interpretation byresiduation with respect to the adjunction operation of groupoid algebras (Buszkowski 1986). In Moortgat and Morrill (1991) it is shown how to give calculi for families of categorial operators, each defined by residuation with respect to an operation of prosodic adjunction (associative, non-associative, or with interactive axioms). The present paper treats discontinuity in this way, by residuation with respect to three adjunctions: + (associative), (.,.) (split-point marking), andW (wrapping) related by the equations 1+s 2+s 3=(s 1,s 3)Ws 2. We show how the resulting methods apply to discontinuous functors, quantifier scope and quantifier scope ambiguity, pied piping, and subject and object antecedent reflexivisation. (shrink)

This paper shows how proof nets can be used to formalize the notion of incomplete dependency used in psycholinguistic theories of the unacceptability of center embedded constructions. Such theories of human language processing can usually be restated in terms of geometrical constraints on proof nets. The paper ends with a discussion of the relationship between these constraints and incremental semantic interpretation.

This is a good book. Its main message is that a particular approach to natural language called type-logical grammar can, in-principle, be equipped with a learning theory. In this review, I first identify what type-logical grammar is, then outline what the learning theory is. Then I try to articulate why this message is important for the logical, linguistic and information-theoretic parts of cognitive science. Overall, I think the book’s main message is significant enough to warrant patience with its scientific limitations.

A two-stage procedure is described which induces type-logical grammar lexicons from sentences annotated with skeletal terms of the simply typed lambda calculus. First, a generalized formulae-as-types correspondence is exploited to obtain all the type-logical proofs of the sample sentences from their lambda terms. The resulting lexicons are then optimally unified. The first stage constitutes the semantic bootstrapping (Pinker, Language Learnability and Language Development, Harvard University Press, 1984), while the unification procedure of Buszkowski and Penn represents a first attempt at structure-dependent (...) distributional learning of the syntactic and semantic categories. This effort extends earlier induction procedures (Buszkowski and Penn, 1990, Studia Logica 49, 431–454; Kanazawa, 1998, CSLI Publications and the European Association for Logic, Language and Information) for classical categorial grammar to at first the non-associative Lambek calculus, and then to a large class of type logics enriched by modal operators and structural rules. (shrink)

In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations (...) from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction.This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with arbitrary structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work. (shrink)

This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and respects theirmonotonicity (...) properties. However, some previously unnoticed factsindicate that monotonicity of plural determiners is not always preservedwhen they apply to collective predicates. We show that the proposedoperator describes this behavior correctly, and characterize themonotonicity of the collective determiners it derives. It is proved thatdeterminer fitting always preserves monotonicity properties ofdeterminers in their second argument, but monotonicity in the firstargument of a determiner is preserved if and only if it is monotonic inthe same direction in the second argument. We argue that this asymmetryfollows from the conservativity of generalized quantifiers innatural language. (shrink)

Preservation and interpolation results are obtained for L ∞ω and sublogics $\mathscr{L} \subseteq L_{\infty\omega}$ such that equivalence in L can be characterized by suitable back-and-forth conditions on sets of partial isomorphisms.

Lambek categorial grammars is a class of formal grammars based on the Lambek calculus. Pentus proved in 1993 that they generate exactly the class of context-free languages without the empty word. In this paper, we study categorial grammars based on the Lambek calculus with the permutation rule LP. Of particular interest is the product-free fragment of LP called the Lambek-van Benthem calculus LBC. Buszkowski in his 1984 paper conjectured that grammars based on the Lambek-van Benthem calculus (LBC-grammars for short) generate (...) exactly permutation closures of context-free languages. In this paper, we disprove this conjecture by presenting a language generated by an LBC-grammar that is not a permutation closure of any context-free language. Firstly, we introduce an ad-hoc modification of vector addition systems called linearly-restricted branching vector addition systems with states and additional memory (LRBVASSAMs for short) and prove that the latter are equivalent to LBC-grammars. Then we construct an LRBVASSAM that generates a non-semilinear set and thus disprove Buszkowski’s conjecture. Since Buszkowski’s conjecture is false, not so much is known about the languages generated by LBC-grammars or by LP-grammars. The equivalence of LRBVASSAMs and LBC-grammars allows us to establish a number of their properties. We show that LP-grammars generate the same class of languages as LBC-grammars; that is, removing product from LP does not decrease expressive power of corresponding categorial grammars. We also prove that this class of languages is closed under union, intersection, concatenation, and Kleene plus. (shrink)

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

The formalisation of Natural Language arguments in a formal language close to it in syntax has been a central aim of Moss’s Natural Logic. I examine how the Quantified Argument Calculus (Quarc) can handle the inferences Moss has considered. I show that they can be incorporated in existing versions of Quarc or in straightforward extensions of it, all within sound and complete systems. Moreover, Quarc is closer in some respects to Natural Language than are Moss’s systems – for instance, is (...) does not use negative nouns. The process also sheds light on formal properties and presuppositions of some inferences it formalises. Directions for future work are outlined. (shrink)

Involutive Nonassociative Lambek Calculus is a nonassociative version of Noncommutative Multiplicative Linear Logic, but the multiplicative constants are not admitted. InNL adds two linear negations to Nonassociative Lambek Calculus ; it is a strongly conservative extension of NL Logical aspects of computational linguistics. LNCS, vol 10054. Springer, Berlin, pp 68–84, 2016). Here we also add unary modalities satisfying the residuation law and De Morgan laws. For the resulting logic InNLm, we define and study phase spaces. We use them to prove (...) the cut elimination theorem for a one-sided sequent system for InNLm, introduced here. Phase spaces are also employed in studying auxiliary systems InNLm, assuming the k-cyclic law for negation. The latter behave similarly as Classical Nonassociative Lambek Calculus, studied in de Groote and Lamarche :355–388, 2002) and Buszkowski. We reduce the provability in InNLm to that in InNLm. This yields the equivalence of type grammars based on InNLm with -free) context-free grammars and the PTIME complexity of InNLm. (shrink)

Many variants of categorial grammar assume an underlying logic which is associative and linear. In relation to left extraction, the former property is challenged by island domains, which involve nonassociativity, and the latter property is challenged by parasitic gaps, which involve nonlinearity. We present a version of type logical grammar including ‘structural inhibition’ for nonassociativity and ‘structural facilitation’ for nonlinearity and we give an account of relativisation including islands and parasitic gaps and their interaction.

La noción de mundos posibles, sostiene que nuestro mundo es un mundo entre otros, un subconjunto de todas las cosas que existen. Esto implica aceptar que existen mundos estructuralmente equivalentes con sus propios lenguajes [formales], que entre sí no tienen ningún estatuto privilegiado, p.e., el mundo y lenguaje del arte o el mundo y lenguaje de la física; no obstante, la idea de aceptar otros mundos equivalentes como mundos posibles epistémica y ontológicamente legítimos para acceder y expresar lo cognoscible del (...) mundo no goza de gran aceptación. En este sentido, de entre las múltiples dimensiones de la noción de mundos posibles anotada, su dimensión lógica, es una de las más extendidas, ésta se entiende como un posible estado de cosas o un posible curso de eventos. Empero, esta concepción es la más restrictiva de la noción de mundos posibles, no sólo porque dicha noción transmuta en una especie de «técnica predictiva» que nos permite analizar y visualizar posibles escenarios de un estado de cosas o un curso de eventos, sino porque fundamentalmente, la noción de mundos posibles pierde gran parte de su potencia conceptual. Hablar de proposiciones como conjuntos de mundos posibles, no es un error, pero es limitar innecesariamente el alcance y campo de acción de la noción de mundos posibles. Si entendemos dicha noción en la versión primaría de Lewis [«Nuestro mundo actual es sólo un mundo entre otros»], sería plausible recuperar, entre otras, las dimensiones ontológica y epistémica de la noción de mundos posibles, y por ende la posibilidad objetiva de indagar en formas alternativas y legitimas de acceder y expresar lo cognoscible del mundo; en este sentido, en este trabajo me propongo: (1) retomar la noción primaria de mundos posibles de Lewis, para analizar y explicitar brevemente sus limitaciones, virtudes y requerimientos; (2) examinar las condiciones de posibilidad de un mundo; y, (3) sugerir los lineamientos epistémicos que deberían considerarse y explicitarse cuando se habla de y desde un mundo posible a la hora de expresar un enunciado. (shrink)

Nonindexical Context-Dependence and the Interpretation as Abduction Approach Inclusive nonindexical context-dependence occurs when the preferred interpretation of an utterance implies its lexically-derived meaning. It is argued that the corresponding processes of free or lexically mandated enrichment can be modeled as abductive inference. A form of abduction is implemented in Simple Type Theory on the basis of a notion of plausibility, which is in turn regarded a preference relation over possible worlds. Since a preordering of doxastic alternatives taken for itself only (...) amounts to a relatively vacuous ad hoc model, it needs to be combined with a rational way of learning from new evidence. Lexicographic upgrade is implemented as an example of how an agent might revise his plausibility ordering in light of new evidence. Various examples are given how this apparatus may be used to model the contextual resolution of context-dependent or semantically incomplete utterances. The described form of abduction is limited and merely serves as a proof of concept, but the idea in general has good potential as one among many ways to build a bridge between semantics and pragmatics since inclusive context-dependence is ubiquitous. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

This paper argues that relativity of truth to a world plays no significant role in empirical semantic theory, even as it is done in the model-theoretic tradition relying on intensional type theory. Some philosophical views of content provide an important notion of truth at a world, but they do not constrain the empirical domain of semantic theory in a way that makes this notion empirically significant. As an application of this conclusion, this paper shows that a potential motivation for relativism (...) based on the relativity of truth to a world fails. (shrink)

We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new fine-structure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.

We present a labelled version of Lambek Calculus without unit, and we use it to prove a completeness theorem for Lambek Calculus with respect to some relational semantics.

We prove a representation theorem for residuated algebras: each residuated algebra is isomorphically embeddable into a powerset residuated algebra. As a consequence, we obtain a completeness theorem for the Generalized Lambek Calculus. We use a Labelled Deductive System which generalizes the one used by Buszkowski [4] and Pankrat'ev [17] in completeness theorems for the Lambek Calculus.

Abstract -/- Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that depend on a previously introduced interpreter and interpretation time in a two-dimensional semantic framework on the basis of simple type theory and categorial grammar. Against Stanley (2002) it is argued that a suitable version of QDR can deal with superlatives like tallest. However, it is shown that NR is needed to account for utterances when the speaker intends to convey different restrictions for (...) multiple uses of the same quantifying determiner. We argue that NR generally fares better with such examples but also observe that examples like Every sailor waves at every sailor might be pragmatically anomalous. An account of contextual domain restriction is proposed that (i) excludes these anomalous readings (but it is shown how they could be included), (ii) makes it possible to express different contextual domain restrictions as long-range dependencies on an interpreter and an interpretation time, and (iii) additionally models restrictions based on locative constructions as general mereological constraints introduced by shifting the index. (shrink)

We give a condensed survey of recent research on generalized quantifiers in logic, linguistics and computer science, under the following headings: Logical definability and expressive power, Polyadic quantifiers and linguistic definability, Weak semantics and axiomatizability, Computational semantics, Quantifiers in dynamic settings, Quantifiers and modal logic, Proof theory of generalized quantifiers.

The scalar approach to negative polarity item (NPI) licensing assumes that NPIs are allowable in contexts in which the introduction of the NPI leads to proposition strengthening (e.g., Kadmon & Landman 1993, Krifka 1995, Lahiri 1997, Chierchia 2006). A straightforward processing prediction from such a theory is that NPI’s facilitate inference verification from sets to subsets. Three experiments are reported that test this proposal. In each experiment, participants evaluated whether inferences from sets to subsets were valid. Crucially, we manipulated whether (...) the premises contained an NPI. In Experiment 1, participants completed a metalinguistic reasoning task, and Experiments 2 and 3 tested reading times using a self-paced reading task. Contrary to expectations, no facilitation was observed when the NPI was present in the premise compared to when it was absent. In fact, the NPI significantly slowed down reading times in the inference region. Our results therefore favor those scalar theories that predict that the NPI is costly to process (Chierchia 2006), or other, nonscalar theories (Giannakidou 1998, Ladusaw 1992, Postal 2005, Szabolcsi 2004) that likewise predict NPI processing cost but, unlike Chierchia (2006), expect the magnitude of the processing cost to vary with the actual pragmatics of the NPI. (shrink)

The process [by which any individual settles into new opinions] is always the same. The individual has a stock of old opinions already, but he meets a new experience that puts them to a strain…. The result is an inward trouble to which his mind till then had been a stranger, and from which he seeks to escape by modifying his previous mass of opinions. He saves as much of it as he can, for in this matter of belief we (...) are all extreme conservatives. So he tries to change first this opinion, and then that (for they resist change very variously), until at last some new idea comes up which he can graft upon the ancient stock with a minimum of disturbance of the latter, some idea that mediates between the stock and the new experience and runs them into one most felicitously and expediently.The new idea is then adopted as the true one. It preserves the older stock of truths with a minimum of modification, stretching them just enough to make them admit the novelty, but conceiving that in ways as familiar as the case leaves possible. (William James, Lectures on Pragmatism, 1907). (shrink)

This paper argues for the idea that in describing language we should follow Haskell Curry in distinguishing between the structure of an expression and its appearance or manifestation . It is explained how making this distinction obviates the need for directed types in type-theoretic grammars and a simple grammatical formalism is sketched in which representations at all levels are lambda terms. The lambda term representing the abstract structure of an expression is homomorphically translated to a lambda term representing its manifestation, (...) but also to a lambda term representing its semantics. (shrink)

Using a version of situation semantics, this article argues that bare and complex demonstratives are interpreted as individual concepts.

The paper is about the interpretation of opaque verbs like “seek”, “owe”, and “resemble” which allow for unspecific readings of their (indefinite) objects. It is shown that the following two observations create a problem for semantic analysis: (a) The opaque position is upward monotone: “John seeks a unicorn” implies “John seeks an animal”, given that “unicorn” is more specific than “animal”. (b) Indefinite objects of opaque verbs allow for higher-order, or “underspecific”, readings: “Jones is looking for something Smith is looking (...) for” can express that there is something unspecific that both Jones and Smith are looking for. Given (a) and (b), it would seem that the following inference is hard to escape, if the premisses are construed unspecifically and the conclusion is taken on its under- specific reading: Jones is looking for a sweater. Smith is looking for a pen. Smith is looking for something Jones is looking for. (shrink)