After a quick overview of the field of study known as “Lexical Semantics”, where we advocate the need of accessing additional information besides syntax and Montague-style semantics at the lexical level in order to complete the full analysis of an utterance, we summarize the current formulations of a well-known theory of that field. We then propose and justify our own model of the Generative Lexicon Theory, based upon a variation of classical compositional semantics, and outline its formalization. (...) Additionally, we discuss the theoretical place of informational, knowledge-related data supposed to exist within the lexicon as well as within discourse and other linguistic constructs. The formalization of the structure of natural language utterances around a surface form (phenogrammatics), a deep structure (tectogrammatics) and the meaning thereof as a logical form (semantics) has developed from the original theories of Curry and Montague to form coherent, type-driven models. Most of these new theories rely upon variations of the compositional analysis of the sentence: from pheno to tectogrammatics, and then to semantics. Our contribution to this work aims at giving such a model a means to overcome the problems posed by polysemous lexical units during the semantical analysis of the tectogrammatical form. Building upon an assumed “deep structure”, we formalize parts of Pustejovsky’s Generative Lexicon Theory, linguistically motivated in Pustejovsky (The generative lexicon, MIT Press, Cambridge, MA, 1995), in a pre-processing of the semantics of the sentence. The mechanisms of Lexical Semantics we propose are an additional layer of classical Montague compositional semantics, and, as such, integrate smoothly within such an analysis; we proceed by converting the lexical data to modifiers of the logical form. This treatment of Lexical Semantics furthermore induces us to think that some sort of non-evident background knowledge of the common use of words is necessary to perform a correct semantic analysis of an utterance. This “commonsense metaphysics” would therefore not be strictly confined to pragmatics, as is often assumed. (shrink)

It is the aim of INDICES to document recent explorations in the various fields of philosophical logic and formal linguistics and their applications in other disciplines. The main emphasis of this series is on self-contained monographs covering particular areas of recent research and surveys of methods, problems, and results in all fields of inquiry where recourse to logical analysis and logical methods has been fruitful. INDICES will contain monographs dealing with the central areas of philosophical logic (extensional and intensional systems, (...) indexical logics, non-classical logics, philosophy of logic, etc.) as well as studies in which these systems are applied to specific issues in philosophy, in the formal semantics of natural languages, the foundations of linguistic theory, in computational linguistics, and in theoretical computer science. Constructive type theory was first presented in 1970, by the Swedish logician Per Martin-Lof. It has become one of the main approaches used in the foundations of mathematics and computer science. But it has remained relatively unknown among linguists and philosophers, although it provides a considerable extension of the concepts and techniques of logic. The book first gives an introduction to type theory from the point of view of linguistics and the philosophy of language. Type theory is then applied in the areas of quantification, anaphora, temporal reference, and the structure of text and discourse. By virtue of the type-theoretical concepts of proof object and context, various phenomena of dependence and progression in language can be discussed in precise terms, and several well-known problems can be solved. A categorial grammar is presented togenerate formally a fragment of English, together with an example of a computer implementation. (shrink)

The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a dedicated natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, (...) Dummett’s manifestation argument does not obtain, and assertions are always warranted, having grounds of assertion. The proof system is shown to satisfy Dummett’s harmony property, justifying the ND rules as meaning conferring. The semantics is suitable for incorporation into computational linguistics grammars, formulated in type-logical grammar. (shrink)

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as (...) a crucial advantage over natural deduction, where substitution is built into the general framework. (shrink)

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Löf) type theory [8, 9] by stating rules for (...) the formation, introduction, elimination and equality of atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs. (shrink)

The paper briefly surveys the sentential proof-theoretic semantics for fragment of English. Then, appealing to a version of Frege’s context-principle (specified to fit type-logical grammar), a method is presented for deriving proof-theoretic meanings for sub-sentential phrases, down to lexical units (words). The sentential meaning is decomposed according to the function-argument structure as determined by the type-logical grammar. In doing so, the paper presents a novel proof-theoretic interpretation of simple type, replacing Montague’s model-theoretic (...) type interpretation (in arbitrary Henkin models). The domains of derivations are collections of derivations in the associated “dedicated” natural-deduction proof-system, and functions therein (with no appeal to models, truth-values and elements of a domain). The compositionality of the semantics is analyzed. (shrink)

It has been argued that reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the same conclusion would have to be identified. In this paper it will be shown that the question, which reductions we accept in our system, is not only important if we see them as generating a theory of proof identity but is also (...) decisive for the more general question whether a proof has meaningful content. There are certain reductions which would not only force us to identify proofs of different arbitrary formulas but which would render derivations in a system allowing them meaningless. To exclude such cases, a minimal criterion is proposed which reductions have to fulfill to be acceptable. (shrink)

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Fregecontributions” to sentential meanings as determined by the function-argument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality (...) of the resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal. (shrink)

In this paper, we shall present a generalization of phrase structure grammar, in which all functional categories have type restrictions, that is, their argument types are specific domains. In ordinary phrase structure grammar, there is just one universal domain of individuals. The grammar does not make a distinction between verbs and adjectives in terms of domains of applicability. Consequently, it fails to distinguish between sentences like every line intersects every line, which is well typed, and every line intersects every (...) point, which is ill typed.Our generalization relates to ordinary phrase structure grammar in the same way as the higherlevel constructive type theory of Martin-Löf relates to the simple type theory of Church . Simple type theory has been used in linguistics and related with phrase structure grammar, especially in the tradition based on the work of Montague .Our definition of the grammar will be more formal than Montague's in the sense that we shall use a formal metalanguage, that of constructive type theory, for defining both the object language and its interpretation. The grammar, both syntax and semantics, is thus readily implementable in the type-theoretical proof system ALF . Inside type theory, a distinction can be made between the object language and the model, in other words, between syntactic and semantic types. It will turn out that the object language cannot be defined independently of the model, as in ordinary Tarski semantics. This is a direct consequence of introducing the typing restrictions.The grammar presented here can be seen as a formal linguistic elaboration of the work presented in Ranta 1991 and 1994. The organization of generative grammar as categorial grammar followed by sugaring should now look more familiar to the linguist, as the grammatical formalism on which sugaring operates is no longer full type theory but a set of phrase structure trees. (shrink)

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type (...) is definable in the Curry-Howard theory. (shrink)

We formulate a Curry-typed logic with fine-grained intensionality within Turner’s typed predicate logic. This allows for an elegant presentation of a theory that corresponds to Fox and Lappin’s property theory with curry typing, but without the need for a federation of languages. We then consider how the fine-grained intensionality of this theory can be given an operational interpretation. This interpretation suggests itself as expressions in the theory can be viewed as terms in the untyped lambda-calculus, which provides a model of (...) computation. (shrink)

I defend the view that linguistic meaning is a relation borne by an expression to a type of speech act, and that this relation holds in virtue of our overlapping communicative dispositions, and not in virtue of linguistic conventions. I argue that this theory gives the right account of the semantics–pragmatics interface and the best-available semantics for non-declarative clauses, and show that it allows for the construction of a rigorous compositional semantic theory with greater explanatory power than (...) both truth-conditional and dynamic semantics. (shrink)

We present an approach to anaphora and ellipsis resolution in which pronouns and elided structures are interpreted by the dynamic identification in discourse of type constraints on their semantic representations. The content of these conditions is recovered in context from an antecedent expression. The constraints define separation types in Property Theory with Curry Typing, an expressive first-order logic with Curry typing that we have proposed as a formal framework for natural language semantics.

This thesis provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. After an illustration of the basic principles of proof-theoretic semantics, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reject on them in the light of proof-theoretic semantic principles. The main original contributions of the present thesis are: (a) a revised version of the display-style calculus D.EAK, (...) which we argue to be more adequate from the proof-theoretic semantic viewpoint; the main feature of this revision is that a smoother proof (so-called Belnap-style) of cut-elimination holds for it, which is problematic for the original version of D.EAK. (b) The intro duction of a novel, multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The presence of types endows the language of the Dynamic Calculus with additional expressivity, and makes it possible to design rules with an even smoother behavior. We argue that this calculus paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly for proof systems beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination. (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)

This thesis examines procedures for ascribing ambiguity to particular sentences and words of a language. My discussion focuses on theories advanced by Keith Donnellan, Saul Kripke and David Kaplan regarding the alleged referential/attributive ambiguity of definite descriptions, and on arguments offered by Paul Ziff, David Wiggins and Jerrold Katz concerning the ambiguity of the word 'good'. I distinguish two kinds of semantic ambiguity, which I call "strong" and "weak", and develop a theoretical model of the former. Using this model, I (...) devise and evaluate a number of theoretically related practical tests designed to detect the presence of ambiguity in concrete cases. I argue against direct appeals to our intuitions of meaning, to Occam's Razor, to grammatical analyses and to foreign languages in settling disputed ascriptions of ambiguity. I favor procedures which indirectly test our intuitions of meaning by isolating the linguistic effects of mixing and shifting senses. My tests rely on the assumption that the effects which accompany the use of truly ambiguous expressions are distinguishable from the mere pragmatic uncertainty which permeates general language use. Semantically ambiguous terms are viewed as awkward linguistic devices whose use, for example, will occasionally produce humor, as in the case of a pun, or deceptively bad reasoning. The account I develop rests heavily on the traditional theory of meaning , and to the extent that the standards I erect furnish better evidence for an ascription of semantic ambiguity, to that extent they ratify the classical analysis of meaning. (shrink)

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we (...) apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

The constructive reformulation of the semantic theory suggests two basic principles to be assumed: first, the distinction between proper knowledge, expressed in judgemental form, and the assertion conditions for such knowledge; second, ...

Portable Grammar Format (PGF) is a core language for type-theoretical grammars. It is the target language to which grammars written in the high-level formalism Grammatical Framework (GF) are compiled. Low-level and simple, PGF is easy to reason about, so that its language-theoretic properties can be established. It is also easy to write interpreters that perform parsing and generation with PGF grammars, and compilers converting PGF to other formats. This paper gives a concise description of PGF, covering syntax, (...) class='Hi'>semantics, and parser generation. It also discusses the technique of embedded grammars, where language processing tasks defined by PGF grammars are integrated in larger systems. (shrink)

In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as predicates of entities as in Montague’s semantics. This brings about important advantages in linguistic interpretations but also leads to a limitation of expressive power because there are fewer operations on types as compared with those on predicates. The theory of coercive subtyping adequately extends the modern type theories and, as shown in this paper, plays a very useful role (...) in making type theories more expressive for formal semantics. It not only gives a satisfactory solution to the basic problem of ‘multiple categorisation’ caused by interpreting common nouns as types, but provides a powerful formal framework to model interesting linguistic phenomena such as copredication, whose formal treatment has been found difficult in a Montagovian setting. In particular, we show how to formally introduce dot-types in a type theory with coercive subtyping and study some type-theoretic constructs that provide useful representational tools for reference transfers and multiple word meanings in formal lexical semantics. (shrink)

The article deals with an approach to the analysis of propositional attitudes based on the type-theoretical semantics proposed by A. Ranta and originating from the type theory of P. Martin-Löf. Type-theoretical semantics contains the notion of context and tools of extracting information from it in an explicit form. This allows us to correctly formalize the dependence on contexts typical of propositional attitudes. In the article the context is presented as a dependent sum type (Record (...) type in the proof assistant Coq). Ranta’s approach is refined and applied to the analysis of Quine’s phrase “Ralph believes that someone is a spy”. Three variants of formalization for this phrase are described which differ in the content of contextual knowledge and the way the truth values of the phrase are derived. Contexts are connected through the function of conversion, making it possible to relate truth values. As a result, it is shown that the instruments for working with contexts provided by type-theoretical semantics allow us to avoid the problem of opacity described by Quine. Provided formalization along with proofs is coded in Coq and made freely available. (shrink)

As emphasized by Alonzo Church and David Kaplan (Church 1974, Kaplan 1975), the philosophies of language of Frege and Russell incorporate quite different methods of semantic analysis with different basic concepts and different ontologies. Accordingly we distinguish between a Fregean and a Russellian tradition in intensional semantics. The purpose of this paper is to pursue the Russellian alternative and to provide a language of intensional logic with a model-theoretic semantics. We also discuss the so-called Russell-Myhill paradox that (...) threatens simple Russellian type theory if propositions satisfies very strict principles of individuation. One way of avoiding the paradox is to adopt a ramified rather than a simple theory of types. (shrink)

Dynamic and proof-conditional approaches to discourse (exemplified by Discourse Representation Theory and Type-Theoretical Grammar, respectively) are related through translations and transitions labeled by first-order formulas with anaphoric twists. Type-theoretic contexts are defined relative to a signature and instantiated modeltheoretically, subject to change.

Transparent intensional logic (TIL) is a well-explored type-theoretical framework for semantics of natural language. However, its treatment of polymorphic functions, which are essential for the analysis of various natural language phenomena, is still underdeveloped. In this paper, we address this issue and propose an extension of TIL that introduces polymorphism via type variables ranging over types and generalized variables ranging over constructions and types. Furthermore, we offer an analysis of sentences involving non-specific notional attitudes of the general (...) form ‘_A_ considers (believes, desires, wants, seeks,...) something’. (shrink)

We develop a vector space semantics for verb phrase ellipsis with anaphora using type-driven compositional distributional semantics based on the Lambek calculus with limited contraction of Jäger. Distributional semantics has a lot to say about the statistical collocation based meanings of content words, but provides little guidance on how to treat function words. Formal semantics on the other hand, has powerful mechanisms for dealing with relative pronouns, coordinators, and the like. Type-driven compositional distributional (...) class='Hi'>semantics brings these two models together. We review previous compositional distributional models of relative pronouns, coordination and a restricted account of ellipsis in the DisCoCat framework of Coecke et al. :1079–1100, 2013). We show how DisCoCat cannot deal with general forms of ellipsis, which rely on copying of information, and develop a novel way of connecting typelogical grammar to distributional semantics by assigning vector interpretable lambda terms to derivations of LCC in the style of Muskens and Sadrzadeh Logical aspects of computational linguistics, Springer, Berlin, 2016). What follows is an account of ellipsis in which word meanings can be copied: the meaning of a sentence is now a program with non-linear access to individual word embeddings. We present the theoretical setting, work out examples, and demonstrate our results with a state of the art distributional model on an extended verb disambiguation dataset. (shrink)

A BSTRACT. We present Property Theory with Curry Typing, an intensional first-order logic for natural language semantics. PTCT permits fine-grained specifications of meaning. It also supports polymorphic types and separation types.1 We develop an intensional number theory within PTCT in order to represent proportional generalized quantifiers like most. We use the type system and our treatment of generalized quantifiers in natural language to construct a typetheoretic approach to pronominal anaphora that avoids some of the difficulties that undermine previous (...) type-theoretic analyses of this phenomenon. (shrink)

We present Property Theory with Curry Typing (PTCT), an intensional first-order logic for natural language semantics. PTCT permits fine-grained specifications of meaning. It also supports polymorphic types and separation types. We develop an intensional number theory within PTCT in order to represent proportional generalized quantifiers like “most.” We use the type system and our treatment of generalized quantifiers in natural language to construct a type-theoretic approach to pronominal anaphora that avoids some of the difficulties that undermine (...) previous type-theoretic analyses of this phenomenon. (shrink)

We propose a lexical account of event nouns, in particular of deverbal nominalisations, whose meaning is related to the event expressed by their base verb. The literature on nominalisations often assumes that the semantics of the base verb completely defines the structure of action nominals. We argue that the information in the base verb is not sufficient to completely determine the semantics of action nominals. We exhibit some data from different languages, especially from Romance language, which show that (...) nominalisations focus on some aspects of the verb semantics. The selected aspects, however, seem to be idiosyncratic and do not automatically result from the internal structure of the verb nor from its interaction with the morphological suffix. We therefore propose a partially lexicalist approach view of deverbal nouns. It is made precise and computable by using the Montagovian generative lexicon, a type theoretical framework introduced by Bassac, Mery and Retoré in this journal in 2010. This extension of Montague semantics with a richer type system easily incorporates lexical phenomena like the semantics of action nominals in particular deverbals, including their polysemy and felicitous copredications. (shrink)

In this paper I discuss a certain kind of 'type confusion' which involves use of expressions of the wrong grammatical category, as in the string 'runs eats'. It is (nearly) universally accepted that such strings are meaningless. My purpose in this paper is to question this widespread assumption (or as I call it, 'the last dogma'). I discuss a range of putative reasons for accepting the last dogma: in §II, semantic and metaphysical reasons; in §III, logical reasons; and in (...) §IV, syntactic reasons. I argue that none of these reasons is conclusive, and that consequently we should be willing to question this last dogma of type confusions. (shrink)

This paper introduces some of the main components of a novel type theoretical semantics for quantifi- cation with plural noun phrases. This theory, unlike previous ones, sticks to the standard generalized quantifier treatment of singular noun phrases and uses only one lifting operator per semantic category (predicate, quantifier and determiner) for quantification with plurals. Following Bennett (1974), plural individuals are treated as functions of type ¢¡ . Plural nouns and other plural predicates accordingly denote £ ¢¡¥¤¦¡ functions. (...) Such predicates do not match the standard £ ¢¡¥¤ £§£ ¢¡¨¤©¡¥¤ type of determiners. Following Partee and Rooth (1983), type mismatches are resolved using type shifting operators. These operators derive collectivity with plurals, keeping the analysis of singular noun phrases, where no type mismatch arises, as in Barwise and Cooper (1981). A single type shifting operator for determiners combines into one reading the existential shift and the counting (neutral) shift of Scha (1981) and Van der Does (1993). This operator combines the conservativity principle of generalized quantifier theory with Szabolcsi’s (1997) existential quantification over witness sets. The unified lift prevents unmotivated ambiguity as well as the monotonicity ill of existential lifts pointed out by Van Benthem.. (shrink)

This paper studies the properties of eight semantic consequence relations defined from a Tarski-logic L and a preference relation ≺. They are equivalent to Shoham’s so-called preferential entailment for smooth model structures, but avoid certain problems of the latter in non-smooth configurations. Each of the logics can be characterized in terms of what we call multi-selection semantics. After discussing this type of semantics, we focus on some concrete proposals from the literature, checking a number of meta-theoretic (...) properties and elaborating on their intuitive motivation. As it turns out, many of their meta-properties only hold in case ≺ is transitive. To tackle this problem, we propose slight modifications of each of the systems, showing the resulting logics to behave better at the intuitive level and in metatheoretic terms, for arbitrary ≺. (shrink)

The Montagovian hypothesis of direct model-theoretic interpretation of syntactic surface structures is supported by an account of the semantics of binding that makes no use of variables, syntactic indices, or assignment functions & shows that the interpretation of a large portion of so-called variable-binding phenomena can dispense with the level of logical form without incurring equivalent complexity elsewhere in the system. Variable-free semantics hypothesizes local interpretation of each surface constituent; binding is formalized as a type-shifting operation (...) on expressions that denote functions, & sentences containing a free pronoun are analyzed as a function from individuals to propositions having a meaning of type (e,t). Standard weak crossover effects & binding patterns in sentences with multiple pronouns are shown to submit to straightforward type-theoretic treatments that do not rely on indexation. The variable-free semantics smoothly implements full surface compositionality & requires less machinery than standard accounts to handle functional questions, their answers, sloppy inferences, & across-the-board binding. 73 References. J. Hitchcock. (shrink)

Types are fundamental for conceptual modeling and knowledge representation, being an essential construct in all major modeling languages in these fields. Despite that, from an ontological and cognitive point of view, there has been a lack of theoretical support for precisely defining a consensual view on types. As a consequence, there has been a lack of precise methodological support for users when choosing the best way to model general terms representing types that appear in a domain, and for building sound (...) taxonomic structures involving them. For over a decade now, a community of researchers has contributed to the development of the Unified Foundational Ontology (UFO) - aimed at providing foundations for all major conceptual modeling constructs. At the core of this enterprise, there has been a theory of types specially designed to address these issues. This theory is ontologically well- founded, psychologically informed, and formally characterized. These results have led to the development of a Conceptual Modelling language dubbed OntoUML, reflecting the ontological micro-theories comprising UFO. Over the years, UFO and OntoUML have been successfully employed on conceptual model design in a variety of domains including academic, industrial, and governmental settings. These experiences exposed improvement opportunities for both the OntoUML language and its underlying theory, UFO. In this paper, we revise the theory of types in UFO in response to empirical evidence. The new version of this theory shows that many of OntoUML’s meta-types (e.g. kind, role, phase, mixin) should be considered not as restricted to substantial types but instead should be applied to model endurant types in general, including relator types, quality types, and mode types. We also contribute with a formal characterization of this fragment of the theory, which is then used to advance a new metamodel for OntoUML (termed OntoUML 2). To demonstrate that the benefits of this approach are extended beyond OntoUML, the proposed formal theory is then employed to support the definition of UFO-based lightweight Semantic Web ontologies with ontological constraint checking in OWL. Additionally, we report on empirical evidence from the literature, mainly from cognitive psychology but also from linguistics, supporting some of the key claims made by this theory. Finally, we propose a computational support for this updated metamodel. (shrink)

What makes our utterances mean what they do? In this work I formulate and justify a structural constraint on possible answers to this key question in the philosophy of language, and I show that accepting this constraint leads naturally to the adoption of an algebraic formalization of truth-theoretic semantics. I develop such a formalization, and show that applying algebraic methodology to the theory of meaning yields important insights into the nature of language. ;The constraint I propose is, roughly, (...) this: the meaning of an utterance of a natural language sentence is derived from the place of that sentence within a system that includes other sentences, and is structured in virtue of primitive semantic facts about complete sentences. This constraint calls for a formal semantics program where the meaning of natural language sentences is represented by an appropriate assignment of the elements of an algebraic structure to sentences, an assignment that represents facts concerning complete sentences. I call the above constraint 'The Algebraic Thesis', and the corresponding formal semantics program 'Algebraic Semantics.' ;I introduce AT by carving it out of the holistic views of language of Quine and Davidson, and I show that it is independent of other important features of these philosophers' views. I then argue for AT, taking into account recent work by Brandom, Dummett, Fodor and Lepore, Gaifman and Perry. In developing Algebraic Semantics I present structures of two basic types: Boolean algebras and cylindric algebras; these structures are construed in the context of AT in a novel way: they are treated as applying to classes of sentences as numbers apply to physical objects in measurement. Applying algebraic methodology to formal semantics and to the theory of meaning, I develop these results: ; An account of the semantics of necessity and possibility that avoids both Quine's dismissal of these notions and their reduction by David Lewis to quantification over possible worlds; ; The application of the model-theoretic notion of expansion to capture the process of our learning progressively more of the meaning of expressions in our first languages. (shrink)

The orthodox view of proper names, Millianism, provides a very simple and elegant explanation of the semantic contribution of referential uses of names–names that occur as bare singulars and as the argument of a predicate. However, one problem for Millianism is that it cannot explain the semantic contribution of predicative uses of names. In recent years, an alternative view, so-called the-predicativism, has become increasingly popular. According to the-predicativists, names are uniformly count nouns. This straightforwardly explains why names can be used (...) predicatively, but is prima facie less congenial to an analysis of referential uses. To address this issue, the-predicativists argue that referential names are in fact complex determiner phrases consisting of a covert definite determiner and a count noun—and so, a referential name is a definite description. In this paper, I will argue that despite the appearance of increased theoretical complexity, the view that names are ambiguous between predicative and referential types is in fact superior to the unitary the-predicativist view. However, I will also argue that to see why this ambiguity view is better, we need to give up the standard Millian analysis. Consequently, I will first propose an alternative analysis of referential names that retains the virtues of Millianism, but provides an important explanatory connection to the predicative uses. Once this analysis of names is adopted, the explanation for why names are systematically ambiguous between referential and predicative types is both simple and elegant. Second, I will argue that the-predicativism has the appearance of being simpler than an ambiguity view, but is in fact unable to account for certain key properties of referential names without making ad hoc stipulations. (shrink)

This paper gives a simple method for providing categorial brands of feature-based unification grammars with a model-theoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see Gabbay (1990)) in order to combine the two components of a feature-based grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Rounds-style feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the basic (...) syntactic types, by formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np (num: sg & pers: 3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left- and right-residuals of these string sets, does not match the intuitive, unification-based proof theory. Unification implements a global bookkeeping with respect to a proof whereas the direct fibring method restricts its view to the atoms of the grammar logic. The solution is to interpret the (embedded) feature terms as global feature constraints while maintaining the same kind of fibred structures. For this adjusted semantics, the anticipated proof system is sound and complete. (shrink)

We propose a lexical account of event nouns, in particular of deverbal nominalisations, whose meaning is related to the event expressed by their base verb. The literature on nominalisations often assumes that the semantics of the base verb completely defines the structure of action nominals. We argue that the information in the base verb is not sufficient to completely determine the semantics of action nominals. We exhibit some data from different languages, especially from Romance language, which show that (...) nominalisations focus on some aspects of the verb semantics. The selected aspects, however, seem to be idiosyncratic and do not automatically result from the internal structure of the verb nor from its interaction with the morphological suffix. We therefore propose a partially lexicalist approach view of deverbal nouns. It is made precise and computable by using the Montagovian generative lexicon, a type theoretical framework introduced by Bassac, Mery and Retoré in this journal in 2010. This extension of Montague semantics with a richer type system easily incorporates lexical phenomena like the semantics of action nominals in particular deverbals, including their polysemy and felicitous copredications. (shrink)

Far from being of mere historical interest, concept horse-style expressibility problems arise for versions of type-theoretic semantics in the tradition of Montague. Grappling with expressibility problems yields lessons about the philosophical interpretation and empirical limits of such type-theories.

This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects of (...) classical type-free truth. In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- . We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory. (shrink)

In virtue of what does a sign have meaning? This is the question raised by Wittgenstein's rule-following considerations. Semantic dispositionalism is a (type of) theory that purports to answer this question. The present paper argues that semantic dispositionalism faces a heretofore unnoticed problem, one that ultimately comes down to its reliance on unanalyzed notions of repeated types of signs. In the context of responding to the rule-following paradox—and offering a putative solution to it—this amounts to simply assuming a solution (...) to the problem in one domain and using it to solve the same problem in another. Given, moreover, the level at which the rule-following paradox undercuts dispositionalism—the level of the notion of a sign's repetition—the objections made to the view also rule out causal/informational theories of meaning as well as communitarian/assertion-theoretic ones as potential solutions to the rule-following paradox. (shrink)

In a lengthy review article, C. Anthony Anderson criticizes the approach to property theory developed in Quality and Concept (1982). That approach is first-order, type-free, and broadly Russellian. Anderson favors Alonzo Church’s higher-order, type-theoretic, broadly Fregean approach. His worries concern the way in which the theory of intensional entities is developed. It is shown that the worries can be handled within the approach developed in the book but they remain serious obstacles for the Church approach. The discussion (...) focuses on: (1) the fine-grained/coarse-grained distinction, (2) proper names and definite descriptions, (3) the paradox of analysis and Mates’ puzzle, and (4) the logical, semantical, and intentional paradoxes. (shrink)

Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to (...) the literature for details. Each of the three steps considered below is based on more recent logical research than the preceding one. The first step from sets to types is essentially the familiar idea of set theoretic semantics for a syntactic system, i.e. giving a model; we take a brief glance at this step from the current point of view, mainly just to fix ideas and notation. The second step from types to categories is known to categorical logicians as the construction of a “syntactic category”; we give some specifics for the benefit of the reader who is not familiar with it. The third step from categories to sets is based on quite recent work, but captures in a precise way an intuition from the early days of foundational studies. With these pieces in place, we can then draw some conclusions regarding the differences between the three schemes, and their relative merits. In particular, it is possible to state more precisely why the methods of category theory are more appropriate to philosophical structuralism. (shrink)

This paper argues that questions have an important role to to play in logic, both semantically and proof-theoretically. Semantically, we show that by generalizing the classical notion of entailment to questions, we can capture not only the standard relation of logical consequence, which holds between pieces of information, but also the relation of logical dependency, which holds between information types. Proof-theoretically, we show that questions may be used in inferences as placeholders for arbitrary information of a given type; by (...) manipulating such placeholders, we may construct formal proofs of dependencies. Finally, we show that such proofs have a specific kind of constructive content: they do not just witness the existence of a certain dependency, but actually encode a method for transforming information of the types described by the assumptions into information of the type described by the conclusion. (shrink)

Distributional semantic models (DSMs) are a primary method for distilling semantic information from corpora. However, a key question remains: What types of semantic relations among words do DSMs detect? Prior work typically has addressed this question using limited human data that are restricted to semantic similarity and/or general semantic relatedness. We tested eight DSMs that are popular in current cognitive and psycholinguistic research (positive pointwise mutual information; global vectors; and three variations each of Skip-gram and continuous bag of words (CBOW) (...) using word, context, and mean embeddings) on a theoretically motivated, rich set of semantic relations involving words from multiple syntactic classes and spanning the abstract–concrete continuum (19 sets of ratings). We found that, overall, the DSMs are best at capturing overall semantic similarity and also can capture verb–noun thematic role relations and noun–noun event-based relations that play important roles in sentence comprehension. Interestingly, Skip-gram and CBOW performed the best in terms of capturing similarity, whereas GloVe dominated the thematic role and event-based relations. We discuss the theoretical and practical implications of our results, make recommendations for users of these models, and demonstrate significant differences in model performance on event-based relations. (shrink)

Without violating the spirit of Game-Theoretical semantics, its results can be re-worked in Martin-Löf''s Constructive Type Theory by interpreting games as types of Myself''s winning strategies. The philosophical ideas behind Game-Theoretical Semantics in fact highly recommend restricting strategies to effective ones, which is the only controversial step in our interpretation. What is gained, then, is a direct connection between linguistic semantics and computer programming.

Information-theoretic approaches to formal logic analyze the "common intuitive" concepts of implication, consequence, and validity in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; one given proposition is a consequence of a second if the latter contains all of the information contained by the former; an argument is valid if the conclusion contains no information beyond that of the premise-set. (...) This paper locates information-theoretic approaches historically, philosophically, and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyze validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information-theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. (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)