Proof Irrelevance in Type-Theoretical Semantics (original) (raw)
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On the strength of proof-irrelevant type theories
Logical Methods in Computer Science, 2008
We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.
Formal Semantics in Modern Type Theories: Is It Model-Theoretic, Proof-Theoretic, or Both?
Lecture Notes in Computer Science, 2014
In this talk, we contend that, for NLs, the divide between model-theoretic semantics and proof-theoretic semantics has not been well-understood. In particular, the formal semantics based on modern type theories (MTTs) may be seen as both model-theoretic and prooftheoretic. To be more precise, it may be seen both ways in the sense that the NL semantics can first be represented in an MTT in a modeltheoretic way and then the semantic representations can be understood inferentially in a proof-theoretic way. Considered in this way, MTTs arguably have unique advantages when employed for formal semantics. ⋆ This paper is associated with the invited talk of mine in Logical Aspects of Computational Linguistics 2014. Its published version in LACL proceedings (LNCS 8535) contains some typos which have been corrected here. ⋆⋆ This work is partially supported by the research grant F/07-537/AJ of the Leverhulme Trust in U.K. 1 In philosophy, different kinds of philosophical semantics have been proposed, including the conceptual role semantics such as semantic inferentialism that has been advocated by Brandom [4,5] and others. Note that such philosophical semantic studies have rather different assumptions and more ambitious objectives as compared to formal semantics, although they have interesting influences on the latter.
Formal Semantics in Modern Type Theories: Theory and Implementation
2014
Formal semantics based on Modern Type Theories (MTTs) provides us with not only a viable alternative to Montague Grammar, but potentially an attractive full-blown semantic tool with advantages in many respects. We shall introduce the MTT-based semantics and then study several issues such as adjectival modification, co-predication and coordination. Key comparisons to Montague Grammar are done all along, discussing the advantages of MTTs over simple type theory. For example, subtyping is crucially needed for proper semantic treatments of some linguistic features but has proven difficult in a Montagovian setting; coercive subtyping is adequate for MTTs and has become a key foundation for the MTTbased semantics. MTTs have been implemented in proof assistants, which can be used to implement MTT-based semantics and hence conduct computer-assisted reasoning in natural language. We shall consider NL inference implemented in Coq to show that such activities can be supported effectively.
A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations
Lecture Notes in Computer Science, 2007
A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl's predicative mathematics and an impredicative one in the verification of security protocols.
Modern Type Theories for Natural Language Semantics ( Introductory Course in Language and Logic )
2016
Modern Type Theories (MTTs) provide us with a new framework for formal semantics with attractive advantages as compared to Montague Grammar. First, MTTs have rich type structures that can be employed effectively to capture various linguistic features that have proved difficult in the Montagovian setting. Second, MTTs are prooftheoretically specified and can hence be usefully implemented in proof assistants such as Coq, where the MTT-semantics has been implemented for computer-assisted reasoning. These two respects may be characterised as saying that the MTT-semantics is both modeltheoretic and proof-theoretic. They offer unique features unavailable in traditional logical systems that have proved very useful in formal semantics. We shall introduce MTTs and how they can be used for formal semantics. The lectures will be informal and explanatory. They will be rigorous but contain a lot of examples, to illustrate the use of MTTs, on the one hand, and to compare the MTT-semantics with Mo...
Introduction: Modern Perspectives in Type Theoretical Semantics
2017
Type theories, from the early days of Montague Semantics (Montague 1974) to the recent work of using rich or modern type theories, have a long history of being employed as foundational languages of natural language semantics. In this introductory chapter, we will describe and discuss the development of type theories as foundational languages of mathematics, as well as their applications as foundational languages for formal semantics. In the end, a brief description of each chapter in the volume will follow.
MTT-Semantics Is Model-Theoretic As Well As Proof-Theoretic∗
2020
In this paper, we argue that formal semantics based on modern type theories (MTT-semantics) is both model-theoretic and proof-theoretic, and hence has unique advantages as a semantic framework. Being model-theoretic, it provides a wide coverage of various linguistic features partly because the rich type structure in MTTs can be used effectively to represent various collections, playing a role as sets do in Montague’s model-theoretic semantics. Being proof-theoretic, its foundational languages have a proof-theoretic meaning theory and provide a solid foundation for natural language reasoning using proof assistants. After presenting the basic arguments, we shall then focus on further development of the first, and arguably less understood, aspect: MTTsemantics is model-theoretic. We shall develop a notion of signature to allow new forms of subtyping and definitional entries and show that such formal contextual tools support useful ways of representing incomplete possible worlds in sema...
Synthese Library
The intensional paradoxes present a continuing challenge to any theory of concepts (properties, attributes, propositional functions). In his seminal paper on Russell, Gödel expressed sympathy for the strategy of limited ranges of significance, which is derived from but logically independent of Russell's theory of types. According to this strategy, every predicate determines a concept, but applying a concept to certain arguments may take us outside the range of meaningfulness. Gödel's idea is most naturally implemented in a logic that admits truth-value gaps. Unfortunately, attempts in this direction often result in deductively weak theories. Although Gödel rejected the theory of types, one can make a case that a satisfactory type-free system needs to be able to recover the deductive strength of classical simple type theory. Ordinary fixed-point theories à la Kripke fail to satisfy this desideratum. Based on Gödel's ideas, we present a naïve theory of concepts, formulated over Weak Kleene logic, which preserves the deductive strength of classical simple type theory. Our language contains some novel restricted quantifiers, but no additional conditional or recapture operators.
Classical Predicative Logic-Enriched Type Theories
A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.