LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the meta-theory of LEGO's type systems leading to a machine-checked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...

Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. For example, intersection types distinguish between call-by-name and call-by-value functions, because the subtyping law (A → B) ∩ (A → C) ≤ A → (B ∩ C) is unsound for the latter in the presence of effects. In this paper we develop a proof-theoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered through linear logic, but we propose a fresh origin in Dummett’s program of justifying the logical laws through alternative verificationist or pragmatist “meaning-theories”, which include a bias towards either introduction or elimination rules. We revisit Dummett’s analysis using the tools of Martin-Löf’s judgmental method, and then show how to extend it to a unified polarized logic, with Girard’s “shift ” connectives acting as intermediaries. This logic safely combines intuitionistic and dual intuitionistic reasoning principles, while simultaneously admitting a focusing interpretation for the classical sequent calculus. Then, by applying the Curry-Howard isomorphism to polarized logic, we obtain a single programming language in which evaluation order is reflected at the level of types. Different logical notions correspond directly to natural programming constructs, such as pattern-matching, explicit substitutions, values and call-by-value continuations. We give examples demonstrating the expressiveness of the language and type system, and prove a basic but modular type safety result. We conclude with a brief discussion of extensions to the language with additional effects and types, and sketch the sort of explanation this can provide for operationally-sensitive typing phenomena. 1

Kleene introduced the notion of slash to investigate the disjunction and existence properties under implication for intuitionistic arithmetic. In this paper Kleene's slash is translated to type theory. Besides translations of Kleene's results, the main application of the slash in type theory is that conditions are given for a typable term, containing free variables, to have a normal form beginning with a constructor. 1 Introduction The disjunction and existence properties, that is, ` AB implies ` A or ` B and ` 9xA(x) implies ` A(t) for some term t , respectively, were first proved for intuitionistic arithmetic by Kleene [9] using a modification of recursive realizability. Harrop [8] extended Kleene's result by also considering derivations depending on assumptions. Harrop proved C ` A B implies C ` A or C ` B (ED) C ` 9xA(x) implies C ` A(t) for some term t (EE) where C is a closed formula not containing any strictly positive occurrences of and 9 ; such a formula is called a Har...

This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore ✸Kϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘✸Kϕ therefore ϕ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK-thesis in order to get around the Cartesian restriction on applications of the knowability principle. ∗ To appear in Joseph Salerno, ed., All Truths are Known: New Essays on the Knowability Paradox, Oxford University Press. This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder.

Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the underlying operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. To give an example, intersection types distinguish between call-by-name and call-by-value functions because the subtyping rule (A → B) ∩ (A → C) ≤ A → (B ∩ C) is valid for the former but not the latter in the presence of effects. I propose to develop a unified, proof-theoretic approach to analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered and developed through linear logic, but I seek a fresh origin in Dummett’s program of justifying the logical laws through alternative “meaning-theories, ” essentially hypotheses as to whether the verification or use of a proposition has a canonical form. In my preliminary work, I showed how a careful judgmental analysis of Dummett’s ideas may be used to define a system of proofs and refutations, with a Curry-Howard interpretation as a single programming language in which the duality between call-by-value and call-by-name is realized as one of types. After extending its type system with (both positive and negative) union and intersection operators and a derived subtyping relationship, I found that many operationally-sensitive typing phenomena (e.g., alternative CBV/CBN subtyping distributivity principles, value and “covalue” restrictions) could be logically reconstructed. Here I give the technical details of this work, and present a plan for addressing open questions and extensions.

It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. 1.