Preface

Curry, Haskell B.; Hindley, J. Roger; Seldin, Jonathan P.

doi:10.1016/s0049-237x(09)70633-0

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“…There is another lengthy proof in the current draft of [2] and the first author has discovered a proof using his axiomatization of strong reduction [3]. The present proof follows the same general line as the latter proof, but it is considerably shorter and simpler.…”

mentioning

confidence: 58%

A short proof of Curry’s normal form theorem

Hindley¹,

Lercher²

1970

Proc. Amer. Math. Soc.

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In Chapter 6 of their book [l] Curry and Feys define a notion of reduction (strong reduction) for the extensional theory of equality in combinatory logic, show [l, Theorem 3, p. 221 ] that strong reduction has the Church-Rosser property, and define a notion of normal form in analogy with the corresponding concept in lambda-conversion. Curry's normal form theorem [l, Theorem 7, p. 230] asserts that if a term ("ob") of combinatory logic is in normal form, it is irreducible, so that if X has normal form X*, then X reduces to X* by a process (namely, strong reduction) that cannot be continued further.Curry's proof of his theorem in [l] is quite long and difficult (see [3, p. 228] for comment). There is another lengthy proof in the current draft of [2] and the first author has discovered a proof using his axiomatization of strong reduction [3]. The present proof follows the same general line as the latter proof, but it is considerably shorter and simpler. Definitions and notation are as in [3], except for the symbol E> which is used here for weak reduction.1. Substitution and abstraction. The first result is essentially from [4, Lemmas 1 and 4]. The proofs, by induction, are easy.Lemma 1. Let P be a redex scheme. Then: (a) P contains at most one occurrence of each meta-variable; (b) If M is a meta-variable occurring in P, there is an N such that NM occurs in P; (c) If P is not basic (i.e. is the result of at least one application of scheme (viii) of [3, p. 233]), then P is weakly irreducible;(d) If P is not basic, then either P = SPiP2 or P=SPi where Pi contains at least one occurrence of an atomic combinator.The hypotheses of the next lemma are, in essence, the properties of redex schemes asserted in Lemma 1(a), (b), and (c).

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confidence: 58%

A short proof of Curry’s normal form theorem

Hindley¹,

Lercher²

1970

Proc. Amer. Math. Soc.

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“…Similarly to the well known canonical reduction form [2] we shall prove here a canonical form theorem. This result makes SC to get close to what in [1] is called combinatory algebra.…”

Section: Canonical Form Theoremmentioning

confidence: 87%

“…The books of Curry, Hindley, and Seldin [2], Curry and Feys [3], Skordev, [12], Ivanov [8], and the dissertation of Zashev [13] as well as the papers of Grzegorczyk [6,7] are particularly relevant to the present work. Now we shall give an example for SC, by a construction which is quite close to the construction of models for A-calculus by the method of Plotkin [9] and Engeler [41.…”

Section: Introductionmentioning

confidence: 99%

Spaces with combinators

Georgieva¹

1993

Arch Math Logic

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“…Quine's preface to Schonfinkel's article in the van Heijenoort collection is a useful primer on the combinators. More advanced treatments may be found in Feys 1958 andCurry et al 1972. Downloaded by [George Mason University] at 11:13 22 December 2014 combinators S and K (the latter of which is distinguished from Fitch's system with the same name) from which the proof of abstraction in pure combinatory logic is a simple affair. 14 Fitch uses the compositionality combinator B, the duplication/contraction combinator W and the distinguished combinators ε and its relational counterpartέ which model membership/exemplification.…”

Section: The System Kmentioning

confidence: 99%

Abstraction in Fitch's Basic Logic

Updike

2012

History and Philosophy of Logic

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Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism.

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Preface

Cited by 27 publications

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A short proof of Curry’s normal form theorem

A short proof of Curry’s normal form theorem

Spaces with combinators

Abstraction in Fitch's Basic Logic

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