## Course Notes in Typed Lambda Calculus (1998)

Citations: | 2 - 0 self |

### BibTeX

@TECHREPORT{Coquand98coursenotes,

author = {Thierry Coquand},

title = {Course Notes in Typed Lambda Calculus},

institution = {},

year = {1998}

}

### OpenURL

### Abstract

this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of proposition, propositional function, and variable, from those contexts in which we are dealing with completely arbitrary, logical general propositions . . . To examine this possibility more closely and to pursue it would be valuable not only from the methodological point of view that enjoins us to strive for the greatest possible conceptual uniformity but also from a certain philosophic, or if you wish, aesthetic point of view."

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Citation Context ...ddle [4]. This system was however shown to be inconsistent by his students Kleene and Rosser [16] 1 . Church formulated then an elegant formulation of higher-order logic, using simply typed -calculus =-=[5]-=-, which can be seen as a simplification of the type system used in Principia Mathematica, but also is in some sense a return to Frege. It seems that Wittgenstein noticed first [30] that it was possibl... |

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Citation Context ...ral induction.) This corollary gives an elegant way of proving the operational equivalence of part of the programs. 8 4 Where does this come from? It is extremely instructive to read Reynolds' papers =-=[20, 21]-=- to understand one motivation of the introduction of type variables and logical relations. One wants to represent the idea of type definitions and type abstractions. The idea is that one can define a ... |

339 | Theorems for free
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Citation Context ...ten a little before Reynolds found its proof [22], and where it is explained why it may seem at first that there does exist a set-theoretic model of polymorphism. This will be also an introduction to =-=[28]-=-, which is a nice application of the technique of logical relations. Let us look at simple type, like \Piff:ff!ff or \Piff:ff!(ff!ff)!ff. A type like \Piff:ff!ff has no direct set-theoretic interpreta... |

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Citation Context ...he continuum. Actually, a rigourous and satisfactory description of these domains is best done by an explicit construction of each basis, and this is achieved by the notion of information system, see =-=[29]. 2.4.3 Re-=-cursive Model I refer here to [18]. Intuitively, this model keeps only the functions and functionals that are "computable". It is important to realise that the meaning of computable is quite... |

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Citation Context ...mply typed -calculus [5], which can be seen as a simplification of the type system used in Principia Mathematica, but also is in some sense a return to Frege. It seems that Wittgenstein noticed first =-=[30]-=- that it was possible to represent natural numbers as formulae, or -terms. In the tractacus [30], one can find the definition of what is now called Church numerals, as well as the representation of ad... |

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Citation Context ...ke D ff = T ff and tae is obtained by substituting x to be ae(x) in t. This term model is a nice illustration of Skolem "paradox" since all types are interpreted by a countable set. In Henki=-=n's paper [13]-=-, where such a construction is first defined, it is explicitely noted that the existence of a countable model is not clear a priori, given the apparent impredicativity or circularity of the notion of ... |

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Citation Context ...sfactory description of these domains is best done by an explicit construction of each basis, and this is achieved by the notion of information system, see [29]. 2.4.3 Recursive Model I refer here to =-=[18]. Intuitiv-=-ely, this model keeps only the functions and functionals that are "computable". It is important to realise that the meaning of computable is quite subtle however. 5 At type '!', it coincides... |

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Citation Context ...ch thought first it would be possible to avoid Russell's paradox without introducing types, but by staying within an intuitionistic logic that use only some limited form of the law of excluded middle =-=[4]-=-. This system was however shown to be inconsistent by his students Kleene and Rosser [16] 1 . Church formulated then an elegant formulation of higher-order logic, using simply typed -calculus [5], whi... |

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Citation Context ...x. It is simple to extend the notion of fi-reduction for this calculus, and the proof the this reduction is Church-Rosser (the argument of Tait-MartinL of was indeed presented first for such a system =-=[17]-=-.) A context will be a sequence of type assignements of the form x 1 : a 1 ; : : : ; x k : a k , since types will be represented as special terms, namely terms of type : Using the letters \Gamma; \Del... |

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Citation Context ...ical definition of identity. Exercice: Prove the reflexivity, symmetry and transitivity of the relation Eq: Higher-order logic is ordinary presented with extensionality. In this case, as explained in =-=[2]-=-, it is better to take the (extensional) equality as primitive and it is then possible to define all other logical connectives from it. It is thus important to relate the intensional presentation of h... |

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Citation Context ...ied out directly here. Indeed, the first derivation of a paradox in this system by Girard was a variation of Burali-Forti paradox (which considers roughly an ordinal of all ordinals, see for instance =-=[3]-=-). I found out later that it was also possible to represent Reynolds' result of the non existence of a set-theoretic model of polymorphism in this system, and that this gives a different paradox, with... |

22 |
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Citation Context ...es, but by staying within an intuitionistic logic that use only some limited form of the law of excluded middle [4]. This system was however shown to be inconsistent by his students Kleene and Rosser =-=[16]-=- 1 . Church formulated then an elegant formulation of higher-order logic, using simply typed -calculus [5], which can be seen as a simplification of the type system used in Principia Mathematica, but ... |

16 |
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Citation Context ...resentation of addition and multiplication. It is not clear how much Church was inspired by this in his formulation of arithmetic in -calculus. Kleene showed how to represent the predecessor function =-=[11]-=-, that was not at all obvious at the beginning, and could show that more and more functions were actually representable. This leads, rather surprisingly if one thinks that the starting point was the p... |

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Citation Context ...ly universally quantified types, this extension is rather intuitive. To understand Reynolds' derivation, it is useful first to explain [21], which was written a little before Reynolds found its proof =-=[22]-=-, and where it is explained why it may seem at first that there does exist a set-theoretic model of polymorphism. This will be also an introduction to [28], which is a nice application of the techniqu... |

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Citation Context ...a term is definable. A related question is the definability problem: is there an algorithm that computes when a given object a 2 D ff built from given finite sets is definable? It was shown by Sieber =-=[26]-=- that the answer is positive if ff is of orders2 (where the order of a ground type is 0 and the order of ff!fi is the maximum of the order of fi and the successor of the order of ff:) A surprising res... |

6 |
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Citation Context ...o define all other logical connectives from it. It is thus important to relate the intensional presentation of higher-order logic given above to an extensional one. Such an interpretation was done in =-=[8]-=- and is an early appearance of the notion of logical relation. The idea is to define for every type an extensional equality by induction on the type. On the type o we take the logical equivalence as e... |

5 | The discovery of my completeness proofs
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Citation Context ...at all types by the technique or logical relations. In this way, we can interpret extensional higher-order logic in intensional higher-order logic. Henkin has an enjoyable and important discussion in =-=[14]-=- describing how he found his completeness theorem for type theory and first-order logic. I found it quite interesting that the starting point was an analysis of the definable elements of the set-theor... |

4 |
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Citation Context ...in some sense, these proofs of paradoxes get more and more complicated when one tries to understand them! Suprisingly in this context, a quite short derivation of a paradox can actually be found, see =-=[15]-=-. In this presentation of the system which uses untyped abstraction, one gets furthermore a term that reduces eventually to itself. 9.2 Martin-Lof Type Theory Because of this paradox, Martin-Lof intro... |

3 |
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Citation Context ...ral induction.) This corollary gives an elegant way of proving the operational equivalence of part of the programs. 8 4 Where does this come from? It is extremely instructive to read Reynolds' papers =-=[20, 21]-=- to understand one motivation of the introduction of type variables and logical relations. One wants to represent the idea of type definitions and type abstractions. The idea is that one can define a ... |

2 |
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Citation Context ...in-Lof is the use of a sigma type !x : a? b in some sense dual to proudct types for representing existence. This allows for a natural derivation of the axiom of choice and the epsilon choice operator =-=[19]. 17 -=-10 A Brief Survey of Paradoxes in Formal Systems We can cite [27]. "The list of logicians who have tried to construct substantial interpreted formal languages adequate for sizable parts of mathem... |

1 |
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Citation Context ...sentation of the predecessor function, that was historically an important problem, and is solved in a conceptual way in polymorphic lambda calculus. I will follow the presentation in the survey paper =-=[6], notice t-=-hat the exponential is not definable without "shifting" types, and explain why there cannot be a good representation of exponential in simply typed lambda calculus. By this, we mean the foll... |

1 | A Realizability Semantics of the Theory of Species - Tait |

1 |
On the building blocks of mathematical logic. In "From Frege to Godel", van Heijenoort
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Citation Context ...ation f a instead of f(a): More generally, when more than two terms occur, association shall be to the left, so that f a b denotes (f a) b. This notation was introduced already in Schonfinkel's paper =-=[24], who-=- introduces the concept of combinator. The goal of this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the... |

1 |
Domains and Logics, extended abstract
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Citation Context ... continuously on f . There is something quite remarkable about the cardinality of the domains D ff ; which is indeed described by Dana Scott as his "first original discovery" in the theory o=-=f domains [25]-=-. If we start from domains D ' such that there exists a countable basis B ' ` D ' ; that is a subset such that any element can be written as a sup of an increasing chain of elements in B ' , then it c... |

1 |
The General From of Operations
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Citation Context ...dct types for representing existence. This allows for a natural derivation of the axiom of choice and the epsilon choice operator [19]. 17 10 A Brief Survey of Paradoxes in Formal Systems We can cite =-=[27]. &qu-=-ot;The list of logicians who have tried to construct substantial interpreted formal languages adequate for sizable parts of mathematics, but who failed in their early attempts, is impressive: it compr... |