Untersuchungen über das logische Schließen. II
TL;DR: In this paper, a process for sizing cellulose fibers or cellulose fiber containing materials and a composition for carrying out the process are described, and a method for sizing according to the general formula of R1 is presented.
Abstract: The present invention relates to a process for sizing cellulose fibers or cellulose fiber containing materials and to a composition for carrying out the process. More particularly the invention relates to a process for sizing according to which cellulose fibers or cellulose fiber containing materials in a manner known per se are brought into contact with compounds having the general formula WHEREIN R1 is an organic, hydrophobic group having 8 to 40 carbon atoms and R2 is an alkyl group having 1 to 7 carbon atoms or has the same meaning as R1.
Citations
More filters
TL;DR: An effective rule (or algorithm) for distinguishing sentences from nonsentences is obtained, which works not only for the formal languages of interest to the mathematical logician, but also for natural languages such as English, or at least for fragments of such languages.
Abstract: (1958). The Mathematics of Sentence Structure. The American Mathematical Monthly: Vol. 65, No. 3, pp. 154-170.
1,432 citations
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
Dissertation•
01 Jan 2007
TL;DR: This thesis is concerned with bridging the gap between the theoretical presentations of type theory and the requirements on a practical programming language.
Abstract: Dependent type theories have a long history of being used for theorem proving One aspect of type theory which makes it very powerful as a proof language is that it mixes deduction with computation This also makes type theory a good candidate for programming---the strength of the type system allows properties of programs to be stated and established, and the computational properties provide semantics for the programs
This thesis is concerned with bridging the gap between the theoretical presentations of type theory and the requirements on a practical programming language Although there are many challenging research problems left to solve before we have an industrial scale programming language based on type theory, this thesis takes us a good step along the way
693 citations
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
TL;DR: The Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms, showing that the expressive power of each first-order system is rounded out, or the system is functionally complete.
Abstract: One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system.A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T1)…(∃Tk)A or (T1)…(Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC⊿ of sets of models, investigated by Tarski in [12] and [13].
608 citations