Modern Type Theories Research Articles

Should Type Theory Replace Set Theory as the Foundation of Mathematics?

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative.

Identity Criteria of Common Nouns and dot-types for Copredication

Copredication, especially when combined with quantification, provides interesting examples to support the idea that common nouns have their own identity criteria, as once argued for by Geach and subsequently studied by others. In this paper, revisiting the use of dot-types in modern type theories to model copredication, we show that, when both copredication and quantification are involved, CNs are not just types but should better be interpreted as types associated with their own identity criteria. In other words, formally, CNs are setoids – pairs whose first component is a type that interprets the domain of a CN and whose second component gives the identity criterion for that CN. For copredication with quantification, identity criteria play an essential role in giving a proper treatment of individuation and counting and hence constructing appropriate semantics to facilitate reasoning correctly. With CNs being setoids, the dot-type approach provides an adequate theory for copredication in general and for copredication with quantification in particular. It is further explained that the CNs-as-types approach is still the appropriate characterisation of our approach to interpreting CNs since, in phenomena that do not involve the interaction of copredication with quantification, the identity criteria of related CNs are essentially the same and can be safely ignored.

Type Theories and Lexical Networks: using Serious Games as the basis for Multi-Sorted Typed Systems

In this paper, we show how a rich lexico-semantic network which has been built using serious games, JeuxDeMots, can help us in grounding our semantic ontologies as well as different sorts of information in doing formal semantics using rich or modern type theories (type theories within the tradition of Martin Löf). We discuss the domain of base types, adjectival and verbal types, hyperonymy/hyponymy relations as well as more advanced issues like homophony and polysemy. We show how one can take advantage of this wealth in a formal compositional semantics framework. This is a way to sidestep the problem of deciding how your type ontology should look like once you have made a move to a many sorted type system. Furthermore, we show how this kind of information can be extracted from JeuxdeMots and inserted into a proof-assistant like Coq in order to perform reasoning tasks using modern type theoretic semantics.

Adjectival and Adverbial Modification: The View from Modern Type Theories

In this paper we present a study of adjectival/adverbial modification using modern type theories (MTTs), i.e. type theories within the tradition of Martin-Löf. We present an account of various issues concerning adjectival/adverbial modification and argue that MTTs can be used as an adequate language for interpreting NL semantics. MTTs are not only expressive enough to deal with a range of modification phenomena, but are furthermore well-suited to perform reasoning tasks that can be easily implemented (e.g. in proof-assistants) given their proof-theoretic nature. In MTT-semantics, common nouns are interpreted as types rather than predicates. Therefore, in order to capture the semantics of adjectives adequately, one needs to meet the challenge of modeling CNs modified by adjectives as types. To explicate that this can be done successfully, we first look at the mainstream classification of adjectives, i.e. intersective, subsective and non-subsective adjectives. There, we show that the rich type structure available in MTTs, along with a suitable subtyping framework, offers an adequate mechanism to model these cases. In particular, this modelling naturally takes care of the characterising inferences associated with each class of adjectives. Then, more advanced issues on adjectival modification are discussed: (a) degree adjectives, (b) comparatives and (c) multidimensional adjectives. There, it is shown that the use of indexed types can be usefully applied in order to deal with these cases. In the same vein, the issue of adverbial modification is discussed. We study two general typings for sentence and VP adverbs respectively. It is shown that the rich type structure in MTTs further provides useful organisational mechanisms in giving formal semantics for adverbs. In particular, we discuss the use of varSigma -types to capture the veridicality/non-veridicality distinction and further discuss cases of intensional adverbs using the type theoretic notion of context (i.e. without resorting to intensional typing). We also look at manner, subject and speech act adverbials and propose solutions using MTTs. Finally, we show that the current proof technology can help mechanically check the associated inferences. A number of our proposals concerning adjectival and adverbial modification have been formalised in the proof assistant Coq and many of the associated inference patterns are checked to be correctly captured.

Type-theoretical semantics with coercive subtyping

In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as functional subsets of entities as in Montague grammar. 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 functional subsets. The theory of coercive subtyping adequately extends the modern type theories with a notion of subtyping and, as shown in this paper, plays a very useful role in making type theories more expressive for formal semantics. In particular, it gives a satisfactory treatment of the type-theoretic interpretation of modified common nouns and allows straightforward interpretations of interesting linguistic phenomena such as copredication, whose interpretations have been found difficult in a Montagovian setting. We shall also study some type-theoretic constructs that provide useful representational tools for formal lexical semantics, including how the so-called dot-types for representing logical polysemy may be expressed in a type theory with coercive subtyping.

Spanish change of state verbs in composition with atypical theme arguments: Clarifying the meaning shifts

For many instances of verb-object combinations involving change of state verbs, different kinds of internal arguments seem to trigger distinct interpretations of the verb phrase. These are usually divided into literal uses, such as romper la ventana ‘break the window’ or cortar el papel ‘cut the paper’ and figurative uses such as romper el desarrollo ‘interrupt the development’ or cortar la circulación ‘cut off the circulation’.Based on an extensive manual annotation of corpus (the used corpus is the El País Corpus consisting of all the El País newspaper issues from 1976 to 2007 and is hosted at the Insitut Universitari de Linguistica Aplicada (IULA) at the University Pompeu Fabra) data involving verb-object combinations with Spanish change of state verbs, I argue that combinations like romper el desarrollo or cortar la circulación, far from representing frozen idiom chunks, exemplify very productive compositional patterns. The frequency and naturalness with which change of state verbs take both physical and abstract entities as objects raises the question of how verbs express their meaning and makes this kind of data especially relevant for a theory of the lexicon as well as of composition.I provide a clear inventory of the typical combinatorial patterns of romper and cortar and I show that their combinatorial behaviour is much more diverse than usually acknowledged. I then argue that these facts need to be addressed by the compositional system, rather than by postulating homophonic lexical entires (Dowty, 1979; Alonso Ramos, 2011) or contextualist accounts (Recanati, 2005).For a proposal I turn to Modern Type Theories, which allow me to incorporate a richer notion of lexical semantics within compositional semantics. These theories thus allow me provide an insightful compositional account of what has long been considered non-compositional, namely combinations of change of state verbs with objects denoting abstract entities.

Natural Language Inference in Coq

In this paper we propose a way to deal with natural language inference (NLI) by implementing Modern Type Theoretical Semantics in the proof assistant Coq. The paper is a first attempt to deal with NLI and natural language reasoning in general by using the proof assistant technology. Valid NLIs are treated as theorems and as such the adequacy of our account is tested by trying to prove them. We use Luo's Modern Type Theory (MTT) with coercive subtyping as the formal language into which we translate natural language semantics, and we further implement these semantics in the Coq proof assistant. It is shown that the use of a MTT with an adequate subtyping mechanism can give us a number of promising results as regards NLI. Specifically, it is shown that a number of inference cases, i.e. quantifiers, adjectives, conjoined noun phrases and temporal reference among other things can be successfully dealt with. It is then shown, that even though Coq is an interactive and not an automated theorem prover, automation of all of the test examples is possible by introducing user-defined automated tactics. Lastly, the paper offers a number of innovative approaches to NL phenomena like adjectives, collective predication, comparatives and factive verbs among other things, contributing in this respect to the theoretical study of formal semantics using MTTs.

Formal semantics in modern type theories with coercive subtyping

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.

Type-theoretical semantics with coercive subtyping

In the formal semantics based on modern type theories, common nouns are interpreted as types, rather than as functional subsets of entities as in Montague grammar. 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 functional subsets. The theory of coercive subtyping adequately extends the modern type theories with a notion of subtyping and, as shown in this paper, plays a very useful role in making type theories more expressive for formal semantics. In particular, it gives a satisfactory treatment of the type-theoretic interpretation of modified common nouns and allows straightforward interpretations of interesting linguistic phenomena such as copredication, whose interpretations have been found difficult in a Montagovian setting. We shall also study some type-theoretic constructs that provide useful representational tools for formal lexical semantics, including how the so-called dot-types for representing logical polysemy may be expressed in a type theory with coercive subtyping.

A Refinement of de Bruijn's Formal Language of Mathematics

We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence: – WTT is faithful to the mathematician's language yet is formal and avoids ambiguities. – WTT is close to the usual way in which mathematicians express themselves in writing. – WTT has a syntax based on linguistic categories instead of set/type theoretic constructs. More so than MV however, WTT has a precise abstract syntax whose derivation rules resemble those of modern type theory enabling us to establish important desirable properties of WTT such as strong normalisation, decidability of type checking and subject reduction. The derivation system allows one to establish that a book written in WTT is well-formed following the syntax of WTT, and has great resemblance with ordinary mathematics books. WTT (like MV) is weak as regards correctness: the rules of WTT only concern linguistic correctness, its types are purely linguistic so that the formal translation into WTT is satisfactory as a readable, well-organized text. In WTT, logico-mathematical aspects of truth are disregarded. This separates concerns and means that WTT – can be easily understood by either a mathematician, a logician or a computer scientist, and – acts as an intermediary between the language of mathematicians and that of logicians.

A New Implementation of Automath

This paper presents aut, a modern Automath checker. It is a straightforward re-implementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT-68 and AUT-QE dialects of Automath. This program was written to restore a damaged version of Jutting's translation of Landau's Grundlagen. Some notable features: • It is fast. On a 1 GHz machine it will check the full Jutting formalization (736 K of nonwhitespace Automath source) in 0.6 seconds. • Its implementation of λ-terms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. • The program can compile an Automath text into one big ‘Automath single line’-style λ-term. It outputs such a term using de Bruijn indices. (These λ-terms cannot be checked by modern systems like Coq or Agda, because the λ-typed λ-calculi of de Bruijn are different from the Π-typed λ-calculi of modern type theory.) The source of aut is freely available on the Web at the address .

A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems

In Russell's Ramified Theory of Types RTT, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of RTT is predicative. The concept of order however, is almost dead since Ramsey eliminated it from RTT. This is why we 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 based on Martin-Lof's Type Theory which uses type universes. Those type universes, and also degrees of expressions in AUTOMATH, are closely related to orders. 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-Lof's type theory and PTSs), and (3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.