Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient for (...) identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part. (shrink)

Orthodoxy holds that there is a determinate fact of the matter about every arithmetical claim. Little argument has been supplied in favour of orthodoxy, and work of Field, Warren and Waxman, and others suggests that the presumption in its favour is unjustified. This paper supports orthodoxy by establishing the determinacy of arithmetic in a well-motivated modal plural logic. Recasting this result in higher-order logic reveals that even the nominalist who thinks that there are only finitely many things should think that (...) there is some sense in which arithmetic is true and determinate. (shrink)

Tato stať je zamýšlena jako úvod do spojité problematiky explikace pravdivosti a sémantic-kých paradoxů, které onu explikaci pronásledují. Tyto paradoxy jsou viděny jako prostředek ukázání, že nedostatečně promyšlená explikace našeho pojmu pravdivosti (významu pravdi-vostního predikátu) je explikací nevyhovující. Poté jsou předvedeny rysy obou klasických přístupů, Russellovy teorie typů a Tarského hierarchie metajazyků. Nakonec jsou podány zá-kladní ideje hlavních názorových škol, které přišly poté.

There is growing debate about what is the correct methodology for research in the ontology of artworks. In the first part of this essay, I introduce my view: I argue that semantic descriptivism is a semantic approach that has an impact on meta-ontological views and can be linked with a hermeneutic fictionalist proposal on the meta-ontology of artworks such as works of music. In the second part, I offer a synthetic presentation of the four main positive meta-ontological views that have (...) been defended in philosophical literature about artworks and of some criticisms that can be lodged against them: Amie Thomasson’s global descriptivism, Andrew Kania’s local descriptivism, Julian Dodd’s folk-theoretic modesty, and David Davies’ rational accountability view. In the conclusion, I show the advantages of my view. (shrink)

This paper revisits two fundamental issues in question semantics—what does a question mean, and how is this meaning compositionally derived? Drawing on observations with the distribution of wh-words in questions and free relatives as well as quantificational variability effects in question-embeddings, I argue that the nominal meanings of short answers must be derivable from question denotations, which therefore calls for a categorial approach to defining questions, including embedded questions. I provide a novel hybrid categorial approach to compose questions. This approach (...) overcomes the problems with traditional categorial approaches in defining bare wh-indefinites, composing multi-wh questions, and accounting for coordinations of questions. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

The article develops and justifies, on the basis of the epistemological argumentation theory, two central pieces of the theory of evaluative argumentation interpretation: 1. criteria for recognizing argument types and 2. rules for adding reasons to create ideal arguments. Ad 1: The criteria for identifying argument types are a selection of essential elements from the definitions of the respective argument types. Ad 2: After presenting the general principles for adding reasons (benevolence, authenticity, immanence, optimization), heuristics are proposed for finding missing (...) reasons, for deductive arguments, e.g., semantic tableaux are suggested. (shrink)

Here, I offer a rapid overview of the theory of metaphor, in order to situate the contributions to this volume in relation to one another and within the field more generally.

The thesis examines A.N. Whitehead and B. Russell’s Ramified Theory of Types. It consists of three parts. The first part is devoted to understanding the source of impredicativity implicit in the induction principle. The question I raise here is whether second-order explicit definitions are responsible for cases when impredicativity turns pathological. The second part considers the interplay between the vicious-circle principle and the no-class theory. The main goal is to give an explanation for the predicative restrictions entailed by the vicious-circle (...) principle. The explanation is that set-existence is parasitic upon prior predicative specifications. The justification for this claim is given by employing the method of hierarchy of languages. Supposing the natural number structure and the language of Peano Arithmetic as given, I describe the construction of a set-theoretic language equipped with substitutionally interpreted quantifiers ranging over arithmetically definable sets. The third part considers the proposition-theoretic version of Russell’s antinomy. A solution to this paradox is offered on the basis of the ramified hierarchy propositions. (shrink)

Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predica-tivity have since taken a life of their own, (...) but have also produced a deeper understanding of the notion of predicativity, therefore witnessing the " light logic throws on problems in the foundations of mathematics. " (Feferman 1998, p. vii) Predicativity has been at the center of a considerable part of Feferman's work: over the years he has explored alternative ways of explicating and analyzing this notion and has shown that predicative mathematics extends much further than expected within ordinary mathematics. The aim of this note is to outline the principal features of predicativity, from its original motivations at the start of the past century to its logical analysis in the 1950-60's. The hope is to convey why predicativity is a fascinating subject, which has attracted Feferman's attention over the years. (shrink)

The paper proposes a new semantics with dependent types for indefinites, encompassing both the data related to their exceptional scopal behavior and the data related to their anaphoric properties. The proposal builds on the formal system combining generalized quantifiers with dependent types in [Grudzińska & Zawadowski 2014] and [Grudzińska & Zawadowski 2016].

Lecture notes from Husserl's logic lectures published during the last 20 years offer a much better insight into his doctrine of the forms of meaning than does the fourth Logical Investigation or any other work published during Husserl's lifetime. This paper provides a detailed reconstruction, based on all the sources now available, of Husserl's system of logical grammar. After having explained the notion of meaning that Husserl assumes in his later logic lectures as well as the notion of form of (...) meaning as it features in ‘doctrine of the forms of meaning’, I present a system of rules that describes all the various forms of meaning that Husserl singles out in his lectures. (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 essay is divided into two parts. In the first part (§2), I introduce the idea of practical meaning by looking at a certain kind of procedural systems — the motor system — that play a central role in computational explanations of motor behavior. I argue that in order to give a satisfactory account of the content of the representations computed by motor systems (motor commands), we need to appeal to a distinctively practical kind of meaning. Defending the explanatory relevance (...) of semantic properties in a computationalist explanation of motor behavior, my argument concludes that practical meanings play a central role in an adequate psychological theory of motor skill. In the second part of this essay (§3), I generalize and clarify the notion of practical meaning, and I defend the intelligibility of practical meanings against an important objection. (shrink)

In this paper, I investigate type theories from several perspectives. First, I present and elaborate the philosophical and technical motivations for these theories. I then offer a formal analysis of various TTs, focusing on the cardinality of the set of types contained in each. I argue that these TTs can be divided into four formal categories, which are derived from the cardinality of the set of their basic elementary types and the finiteness of the lengths of their molecular types. The (...) paper provides proofs of the cardinality of the universe of types for each of the specified theories. (shrink)

This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too implausible to make these arguments troubling. (...) The second reason is conditional on a substitutional interpretation of quantification over types other than that of individuals. This reason stands up to investigation: a simple-type language will not sustain such an interpretation, but a ramified-type language will. And there is evidence that Russell was tacitly inclined towards such an interpretation. A strong construal of that interpretation opens a way to make sense of Russell’s simultaneous repudiation of propositions and his willingness to quantify over them. But that way runs into trouble with Russell’s commitment to the finitude of human understanding. (shrink)

This work explores the hypothesis that natural language is a tool for changing a language user's state of mind and, more specifically, the hypothesis that a sentence's meaning is constituted by its characteristic role in fulfilling this purpose. This view contrasts with the dominant approach to semantics due to Frege, Tarski and others' work on artificial languages: language is first and foremost a tool for representing the world. Adapted to natural language by Davidson, Lewis, Montague, et. al. this dominant approach (...) has crystalized as truth-conditional semantics: to know the meaning of a sentence is to know the conditions under which that sentence is true. Chapter 1 details the animating ideas of my alternative approach and shows that the representational function of language can be understood in terms of the more general function of changing representational mental states. Chapters 2-4 argue that the additional resources of this more general conception of meaning allow us to explain certain phenomena involving conditionals and grammatical mood that truth-conditional semantics does not. In the analysis of these specific phenomena and the articulation of the general approach on offer, it emerges that this approach combines insights and benefits from both use-theoretic and truth-theoretic work on meaning. (shrink)

It is argued that the source of complexity in language is twofold: repetition, and syntactic embedding. The former enables us to return again and again to the same subject across many sentences, and to maintain the coherence of an argument. The latter is governed by two forms of complexification: the functor-argument structure of all languages and the operator-bound-variable mechanism of familiar formal languages. The former is most transparently represented by categorial grammar, and an extension of this can adequately describe the (...) syntax of variable binders. Both developments have roots within the work of the Lvov-Warsaw School. (shrink)

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

The “dynamic turn” in semantic theory of natural language, which has been taking place roughly during the last decade, has resulted into seeing the meaning of a sentence as a “context-change-potential”, as a function which maps the set of possible contexts on itself. The development of theories of this kind has been stimulated especially by the effort to semantically cope with the anaphoric items of natural language . The most significant species of dynamic semantic theories are represented by Kamp´s “discourse (...) representations theory” and Groenendijk and Stokhof´s “dynamic logic”. The former sees sentences as means of building and rebuilding “discourse representations structures”; the latter, which tries to be an expressly logical objects to “discourse markers” into the same set. The paper presents basic principles of theories of both of these kinds and indicates how they can help us analyse natural language locutions involving anaphora. (shrink)

Ordinary Language Philosophy has largely fallen out of favour, and with it the belief in the primary importance of analyses of ordinary language for philosophical purposes. Still, in their various endeavours, philosophers not only from analytic but also from other backgrounds refer to the use and meaning of terms of interest in ordinary parlance. In doing so, they most commonly appeal to their own linguistic intuitions. Often, the appeal to individual intuitions is supplemented by reference to dictionaries. In recent times, (...) Internet search engine queries for expressions of interest have become quite popular. Apparently, philosophers attempt to surpass the limits of their own linguistic intuitions by appealing to experts or to factual uses of language. I argue that this attempt is commendable but that its execution is wanting. Instead of appealing to dictionaries or Internet queries, philosophers should employ computer-based linguistic corpora in order to confirm or falsify hypotheses about the factual use of language. This approach also has some advantages over methods employed by experimental philosophers. If the importance of ordinary language is stressed, the use of linguistic corpora is hardly avoidable. (shrink)

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic , logic (...) of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail. (shrink)

Distributive egalitarians believe that distributive justice is to be explained by the idea of distributive equality (DE) and that DE is of intrinsic value. The socio-relational critique argues that distributive egalitarianism does not account for the “true” value of equality, which rather lies in the idea of “equality as a substantive social value” (ESV). This paper examines the socio-relational critique and argues that it fails because – contrary to what the critique presupposes –, first, ESV is not conceptually distinct from (...) DE, and second, the idea of ESV cannot serve as a “foundation” or “root” of distributive egalitarianism. (shrink)

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

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)

So-called 'dynamic' semantic theories such as Kamp's discourse representation theory and Heim's file change semantics account for such phenomena as cross-sentential anaphora, donkey anaphora, and the novelty condition on indefinites, but compare unfavorably with Montague semantics in some important respects (clarity and simplicity of mathematical foundations, compositionality, handling of quantification and coordination). Preliminary efforts have been made by Muskens and by de Groote to revise and extend Montague semantics to cover dynamic phenomena. We present a new higher-order theory of discourse (...) semantics which improves on their accounts by incorporating a more articulated notion of context inspired by ideas due to David Lewis and to Craige Roberts. On our account, a context consists of a common ground of mutually accepted propositions together with a set of discourse referents preordered by relative salience. Employing a richer notion of contexts enables us to extend our coverage beyond pronominal anaphora to a wider range of presuppositional phenomena, such as the factivity of certain sentential-complement verbs, resolution of anaphora associated with arbitrarily complex definite descriptions, presupposition 'holes' such as negation, and the independence condition on the antecedents of conditionals. Formally, our theory is expressed within a higher-order logic with natural number type, separation-style subtyping, and dependent coproducts parameterized by the natural numbers. The system of semantic types builds on proposals due to Thomason and to Pollard in which the type of propositions (static meanings of sentential utterances) is taken as basic and worlds are constructed from propositions (rather than the other way around as in standard Montague semantics). (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)

This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...) paper is in three sections. The first is written in journalistic style:the story of the life of AlonzoChurch is told, including some of the many anecdotes I have collected from different sources. The secondpart is devoted to his work, but is far from being exhaustive. The last part is more original; in it I attempto show that Church’s great discovery was lambda calculus and that his remaining contributions weremainly inspired afterthoughts in the sense that most of his contributions as well as some of his pupils derivefrom that initial achievement. Included are Kleene’s Recursion Theory and the completeness proof ofHenkin. I have added an appendix in which is presented the typed lambda calculus and a proof of theundecidability of first-order logic. (shrink)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speaking) is (...) not inconsistent with a philosophy of types that does not think of propositional functions as mind- and language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary. (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)

This article presents an historical and conceptual overview on different approaches to logical abstraction. Two main trends concerning abstraction in the history of logic are highlighted, starting from the logical notions of concept and function. This analysis strictly relates to the philosophical discussion on the nature of abstract objects. I develop this issue further with respect to the procedure of abstraction involved by (typed) λ-systems, focusing on the crucial change about meaning and predicability. In particular, the analysis of the nature (...) of logical types in the context of Constructive Type Theory allows elucidation of the role of the previously introduced notions. Finally, the connection to the analysis of abstraction in computer science is drawn, and the methodological contribution provided by the notion of information is considered, showing its conceptual and technical relevance. Future research shall focus on the notion of information in distributed systems, analysing the paradigm of information hiding in dependent type theories. (shrink)

In this book, Zalta attempts to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for (...) producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

In this paper it is shown how the DRT (Discourse Representation Theory) treatment of temporal anaphora can be formalized within a version of Montague Semantics that is based on classical type logic.

In this paper it is shown how simple texts that can be parsed in a Lambek Categorial Grammar can also automatically be provided with a semantics in the form of a Discourse Representation Structure in the sense of Kamp [1981]. The assignment of meanings to texts uses the Curry-Howard-Van Benthem correspondence.

This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it (...) is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of (...) this relation must be given in the form of a logic program. The resulting system does not only throw light on the properties of sense and their relation to computation, but also shows circular behaviour if some ingredients of the Liar Paradox are added. The connection is natural, as algorithms can be inherently circular and the Liar is explained as expressing one of those. Many ideas in the present paper are closely related to those in Moschovakis (1994), but receive a considerably lighter formalization. (shrink)

This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910-1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's (...) Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Church's own simply typed λ-calculus (λ → C) and we finish by comparing RTT, STT and λ → C. (shrink)

In Russell''s Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Church''s simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT. Despite the disappearance of orders which have a strong correlation with predicativity, predicative (...) logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löf''s Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russell's orders, (2) the placing of the historical system RTT underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (Martin-Löf''s type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS. (shrink)