We show that \, the basic theory of bounded arithmetic corresponding to the complexity class \, proves the \ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \ can also prove the integer division axiom, and the \-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \ and \. As a side (...) result, we also prove that there is a well-behaved \ definition of modular powering in \\). (shrink)

We study the extension 123) of the theory S21 by instances of the dual weak pigeonhole principle for p-time functions, dWPHPx2x. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S21+dWPHP. We construct a propositional proof system WF , which captures the Π1b-consequences of S21+dWPHP. We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. 256). We prove that (...) dWPHP is equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction in a conservative extension of S21+dWPHP. (shrink)

We prove that all axiomatic extensions of the full Lambek calculus with exchange can be axiomatized by formulas on the level of the substructural hierarchy.

We formalize the construction of Paterson’s variant of the Ajtai–Komlós–Szemerédi sorting network of logarithmic depth in the bounded arithmetical theory , under the assumption of the existence of suitable expander graphs. We derive a conditional p-simulation of the propositional sequent calculus in the monotone sequent calculus.

We study variants of Buss’s theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on \ induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems and \ of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.

This is a reproduction of a book published before 1923. This book may have occasional imperfectionssuch as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed worksworldwide. We appreciate your understanding of the imperfections in the (...) preservation process, and hope you enjoy this valuable book.++++The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Naturwissenschaftliche Plaudereien 3 Emil Arnold Budde. (shrink)

In this article, Emil Sætra examines how teachers and students construct and experience aims and goods in classroom discussions of controversial issues. This study is situated within the emerging tradition of empirical ethics, and the research strategy comprised two main steps. First, Sætra used interview data to analyze, via the experiences of teachers and students, the following two empirical questions: (1) What goods normatively constitute educative discussions of controversial issues? (2) How are these goods constructed in time and space? (...) Second, in the concluding discussion, Sætra makes explicit some normative implications of the empirical findings and discusses theses in relation to previous research. In brief, what emerges from the conjoined experiences of teachers and students is the notion that practice comprises multiple aims and drives. Moreover, because aims and drives are multiple, decisions regarding priorities will sometimes have to be made, and teachers may be subject to conflicting expectations. This means that practitioners must be able to make reflective judgments about what comprises acceptable ends, what projects should be undertaken to achieve those ends, and what actions ought to be performed as part of those projects. (shrink)

The redundant-signals paradigm (RSP) is designed to investigate response behavior in perceptual tasks in which response-relevant targets are defined by either one or two features, or modalities. The common finding is that responses are speeded for redundantly compared to singly defined targets. This redundant-signals effect (RSE) can be accounted for by race models if the response times do not violate the race model inequality (RMI). When there are violations of the RMI, race models are effectively excluded as a viable account (...) of the RSE. The common alternative is provided by co-activation accounts, which assume that redundant target signals are integrated at some processing stage. However, ‘co-activation’ has mostly been only indirectly inferred and the accounts have only rarely been explicitly modeled; if they were modeled, the RSE has typically been assumed to have a decisional locus. Yet, there are also indications in the literature that the RSE might originate, at least in part, at a non-decisional or motor stage. In the present study, using a distribution analysis of sequential-sampling models (ex-Wald and Ratcliff Diffusion model), the locus of the RSE was investigated for two bimodal (audio-visual) detection tasks that strongly violated the RMI, indicative of substantial co-activation. Three model variants assuming different loci of the RSE were fitted to the quantile reaction time proportions: a decision, a non-decision, and a combined variant both to vincentized group as well as individual data. The results suggest that for the two bimodal detection tasks, co-activation has a shared decisional and non-decisional locus. These findings point to the possibility that the mechanisms underlying the RSE depend on the specifics (task, stimulus, conditions, etc.) of the experimental paradigm. (shrink)

Grzegorczyk's modal logic corresponds to the class of upwards well-founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF. We answer his question negatively: Grz corresponds to a class of frames, which does not provably coincide with upwards well-founded posets in ZF alone.

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections (...) in the preservation process, and hope you enjoy this valuable book. (shrink)

We show that VTC0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ VTC ^0$$\end{document}, the basic theory of bounded arithmetic corresponding to the complexity class TC0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {TC}^0$$\end{document}, proves the IMUL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ IMUL $$\end{document} axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the TC0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {TC}^0$$\end{document} iterated (...) multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, VTC0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ VTC ^0$$\end{document} can also prove the integer division axiom, and the RSUV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ RSUV $$\end{document}-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories Δ1b-CR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^b_1\text{- } CR $$\end{document} and C20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0_2$$\end{document}. As a side result, we also prove that there is a well-behaved Δ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _0$$\end{document} definition of modular powering in IΔ0+WPHP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\Delta _0+ WPHP $$\end{document}. (shrink)

We sketch an account according to which the semantic concepts themselves are not pathological and the pathologies that attend the semantic predicates arise because of the intention to impose on them a role they cannot fulfill, that of expressing semantic concepts for a language that includes them. We provide a simplified model of the account and argue in its light that (i) a consequence is that our meaning intentions are unsuccessful, and such semantic predicates fail to express any concept, and (...) that (ii) in light of this it is incorrect to characterize the pathology simply as semantic inconsistency; a more nuanced view of the problem is needed. We also show that the defects of the semantic predicates need not undercut the use of a truth theory in a compositional semantics for a language containing them because the meaning theory per se need not involve commitment to the axioms of the truth theory it exploits. (shrink)