The Writing of Introduction to Metamathematics

Kleene, S. C.

doi:10.1007/978-0-8176-4769-8_11

Cited by 312 publications

(471 citation statements)

References 5 publications

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“…In his well known textbook [35] (also [36]) Kleene introduces Hilbert system as basic but soon proves a lot of secondary rules of introduction and elimination which together provide adequate formalization of classical logic. In contrast to Gentzen's system and to most other described above, Kleene uses intuitonistic sequents but with antecedent as sets of formulae.…”

Section: Kleene and Sequential Representation Of Standard Natural Dedmentioning

confidence: 99%

A Survey of Nonstandard Sequent Calculi

Indrzejczak

2014

Stud Logica

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Abstract. The paper is a brief survey of some sequent calculi (SC) which do not follow strictly the shape of sequent calculus introduced by Gentzen. We propose the following rough classification of all SC: Systems which are based on some deviations from the ordinary notion of a sequent are called generalised; remaining ones are called ordinary. Among the latter we distinguish three types according to the proportion between the number of primitive sequents and rules. In particular, in one of these types, called Gentzen's type, we have a subtype of standard SC due to Gentzen. Hence by nonstandard ones we mean all these ordinary SC where other kinds of rules are applied than those admitted in standard Gentzen's sequent calculi. We describe briefly some of the most interesting or important nonstandard SC belonging to the three abovementioned types.Keywords: Sequent Calculi, Natural Deduction, Proof Methods. IntroductionThe notion of sequent calculus (SC) is commonly, and rightly, connected with the name of Gerhard Gentzen. Quite automatically, we think of calculi with structural and logical rules introducing constants either to antecedent or to succedent of a sequent. However, one should be aware that nowadays this term may be applied not only to such calculi like original Gentzen's LJ, LK or their variants, but also to a variety of calculi which use sequents in a significantly different way than they are used in Gentzen's like calculi. Some of them are even older than Gentzen's LK (or LJ) -like Hertz's calculus -but most were invented after Gentzen, mainly for providing more flexible or natural systems for actual proof search than original Gentzen's calculus constructed rather for theoretical purposes. In this paper we survey some of the most important or interesting proposals. The paper is based on the 10th chapter of Indrzejczak [31].Let us define an ordinary sequent calculus (SC) as a finite collection of (schemata) of (primitive) sequent rules of the form: Sequents on the left of / are premises of the rule, whereas S n+1 is its conclusion. Note that n, i.e. the number of premises, may be 0 -in this case we will say on (primitive) axiomatic sequent; in case n > 0 we will simply say on rules. Note that every SC have a nonempty set of primitive sequents and a nonempty set of rules 1 .So far, the only restriction we put on the notion of a sequent calculus is that rules have unique conclusions. The next restriction is connected with the very notion of a sequent. We define it in a standard way as an ordered pair Γ ⇒ Δ, where Γ is the antecedent and Δ the succedent of a sequent. Hence we exclude from our considerations such calculi which operate on sequents having more arguments 2 or being structures of different character 3 or using more types of sequents in one system 4 .One should also specify what kind of structures are denoted by arguments of a sequent. We allow sequences, multisets or sets of formulae but exclude from further considerations other kinds of structures. Hence we do consider neither display calculi ...

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Section: Kleene and Sequential Representation Of Standard Natural Dedmentioning

confidence: 99%

A Survey of Nonstandard Sequent Calculi

Indrzejczak

2014

Stud Logica

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“…On the contrary, to define the subsets, one had to add a number of sentences to the Report to make sure that no recursive procedure activations are possible. These sentences disallow recursion by stating that a procedure identifier has no meaning before the end of the procedure declaration, 15 or disallow mutual recursion either by requiring that procedures are declared before they are used 16 or by limiting the occurrences of identifiers in block heads, 17 or disallow self-application by requiring that all the parameters 14 See for example [29], modification to Section 5.2 and modification (e) to Section 5.4.…”

Section: The Consequences Of the Call By Name Mechanismmentioning

confidence: 99%

On the Origin of Recursive Procedures

Hove¹

2014

The Computer Journal

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We investigate the origin of recursive procedures in imperative programming languages. We attempt to set the record straight, and to identify the trend that led to recursive procedures, by means of an analysis of the related concepts and of the most reliable available documents, as far as known to us. We show that not all of those who were involved in defining these concepts in these documents were fully aware of the implications of their proposals. Our aim is not primarily historical, but to contribute to a clarification of some of the concepts related to recursion. In particular, we demonstrate that recursive procedure declarations and recursive procedure activations are logically disjoint concepts.

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“…For the interpretation in partial structures [ϕ] 3 M e, we follow Berezin et al [5]: intuitively, the interpretation of a function is defined if and only if the interpretations of all parameters are defined and the vector of parameters belongs to the function domain. The interpretation of logical operators and quantifiers is defined according to Kleene logic [20].…”

Section: Structures and Interpretationsmentioning

confidence: 99%

Checking Well-Formedness of Pure-Method Specifications

Rudich

Darvas

Müller

Lecture Notes in Computer Science

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Abstract. Contract languages such as JML and Spec# specify invariants and pre-and postconditions using side-effect free expressions of the programming language, in particular, pure methods. For such contracts to be meaningful, they must be well-formed: First, they must respect the partiality of operations, for instance, the preconditions of pure methods used in the contract. Second, they must enable a consistent encoding of pure methods in a program logic, which requires that their specifications are satisfiable and that recursive specifications are well-founded. This paper presents a technique to check well-formedness of contracts. We give proof obligations that are sufficient to guarantee the existence of a model for the specification of pure methods. We improve over earlier work by providing a systematic solution including a soundness result and by supporting more forms of recursive specifications. Our technique has been implemented in the Spec# programming system.

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The Writing of Introduction to Metamathematics

Cited by 312 publications

References 5 publications

A Survey of Nonstandard Sequent Calculi

A Survey of Nonstandard Sequent Calculi

On the Origin of Recursive Procedures

Checking Well-Formedness of Pure-Method Specifications

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