Journal•ISSN: 0049-237X

# Studies in logic and the foundations of mathematics

Elsevier BV

About: Studies in logic and the foundations of mathematics is an academic journal. The journal publishes majorly in the area(s): Axiom & Mathematical proof. It has an ISSN identifier of 0049-237X. Over the lifetime, 1163 publications have been published receiving 31852 citations.

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TL;DR: This chapter discusses the several classes of sentence-generating devices that are closely related, in various ways, to the grammars of both natural languages and artificial languages of various kinds.

Abstract: Publisher Summary This chapter discusses the several classes of sentence-generating devices that are closely related, in various ways, to the grammars of both natural languages and artificial languages of various kinds. By a language it simply mean a set of strings in some finite set V of symbols called the vocabulary of the language. By a grammar a set of rules that give a recursive enumeration of the strings belonging to the language. It can be said that the grammar generates these strings. The chapter discusses the aspect of the structural description of a sentence, namely, its subdivision into phrases belonging to various categories. A major concern of the general theory of natural languages is to define the class of possible strings; the class of possible grammars; the class of possible structural descriptions; a procedure for assigning structural descriptions to sentences, given a grammar; and to do all of this in such a way that the structural description assigned to a sentence by the grammar of a natural language will provide the basis for explaining how a speaker of this language would understand this sentence.

819 citations

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TL;DR: The theory of types as mentioned in this paper is a full-scale system for formalizing intuitionistic mathematics as developed, which allows proofs to appear as parts of propositions so that the propositions of the theory can express properties of proofs.

Abstract: Publisher Summary The theory of types is intended to be a full-scale system for formalizing intuitionistic mathematics as developed. The language of the theory is richer than the languages of traditional intuitionistic systems in permitting proofs to appear as parts of propositions so that the propositions of the theory can express properties of proofs. There are axioms for universes that link the generation of objects and types and play somewhat the same role for the present theory as does the replacement axiom for Zermelo–Fraenkel set theory. The present theory is based on a strongly impredicative axiom that there is a type of all types in symbols. This axiom has to be abandoned, however, after it has been shown to lead to a contraction. This chapter discusses Normalization theorem, which can be strengthened in two ways: it can be made to cover open terms and it can be proved that every reduction sequence starting from an arbitrary term leads to a unique normal term after a finite number of steps. The definition of the notion of convertibility and the proof that an arbitrary term is convertible can no longer be separated because the type symbols and the terms are generated simultaneously.

725 citations

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TL;DR: This chapter discusses that relating constructive mathematics to computer programming seems to be beneficial, and that it may well be possible to turn what is now regarded as a high level programming language into machine code by the invention of new hardware.

Abstract: Publisher Summary This chapter discusses that relating constructive mathematics to computer programming seems to be beneficial. Among the benefits to be derived by constructive mathematics from its association with computer programming, one is that you see immediately why you cannot rely upon the law of excluded middle: its uninhibited use would lead to programs that one did not know how to execute. By choosing to program in a formal language for constructive mathematics, like the theory of types, one gets access to the conceptual apparatus of pure mathematics, neglecting those parts that depend critically on the law of excluded middle, whereas even the best high level programming languages so far designed are wholly inadequate as mathematical languages. The virtue of a machine code is that a program written in it can be directly read and executed by the machine. The distinction between low and high level programming languages is of course relative to the available hardware. It may well be possible to turn what is now regarded as a high level programming language into machine code by the invention of new hardware.

618 citations

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TL;DR: In this paper, a notational system for lambda calculus is developed, where occurrences of variables are indicated by integers giving the "distance" to the binding λ instead of a name attached to that λ. This convention is known to cause considerable trouble in cases of substitution.

Abstract: In ordinary lambda calculus the occurrences of a bound variable are made recognizable by the use of one and the same (otherwise irrelevant) name at all occurrences. This convention is known to cause considerable trouble in cases of substitution. In the present paper a different notational system is developed, where occurrences of variables are indicated by integers giving the “distance” to the binding λ instead of a name attached to that λ. The system is claimed to be efficient for automatic formula manipulation as well as for metalingual discussion. As an example the most essential part of a proof of the Church-Rosser theorem is presented in this namefree calculus.

583 citations

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TL;DR: In this paper, a semantical analysis of intuitionistic logic I is presented and a model theory for intuitionistic predicate logic is presented. Butler et al. present a decision procedure for logic I.

Abstract: Publisher Summary The chapter discusses a semantical analysis of intuitionistic logic I. The chapter presents a semantical model theory for Heyting's intuitionist predicate logic and proves the completeness of that system relative to the modeling. The semantics for modal logic that is announced and developed together with the known mappings of intuitionistic logic into the modal system, S4, inspired the present semantics for intuitionist logic. It is important to develop the semantics of intuitionistic logic independently of that of S4; this procedure helps to obtain somewhat more information about intuitionistic logic, including the mapping into S4 as a consequence thereof. In addition to giving a simple decision procedure for Heyting's propositional calculus, the chapter presents the undecidability of monadic intuitionistic quantification theory. The proof is based on the semantics previously developed. Beth semantic tableaux for intuitionistic logic is developed in the chapter. The chapter describes consistency property: in a standard formalization of Heyting's predicate calculus, the axioms are all valid, and the rules preserve validity.

571 citations