Algebraic Aspects of Cut Elimination

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].     

Cut Elimination for S4C: A Case Study

S4C is a logic of continuous transformations of a topological space. Cut elimination for it requires new kind of rules and new kinds of reductions     

Cut Elimination inside a Deep Inference System for Classical Predicate Logic

Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations.     

Algorithmic Proof Methods and Cut Elimination for Implicational Logics Part I: Modal Implication

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].     

Metaphysical Infinitism and the Regress of Being

This article offers a limited defense of metaphysical “infinitism,” the view that there are, or might be, infinite chains of ontological dependence. According to a widespread presupposition, there must be an ultimate ground of being—most likely, a plurality of fundamental atoms. Contrary to this view, this article shows that metaphysical infinitism is internally coherent. In particular, a parallel with the debate concerning infinitism about epistemic justification is suggested, and an “emergence model” of being is put forward. According to the emergence model, the being of any given entity gradually arises out of an infinite series of progressively less dependent entities—it is not wholly transmitted, as it were, from a basic, ungrounded level to all the dependent ones in a step‐by‐step fashion. Some objections are considered and rebutted.     

Regress and the Doctrine of Epistemic Original Sin

Existing solutions to the epistemic regress problem, and the theories of justification built upon them, are inadequate, for they fail to diagnose the root source of the problem. The problem is rooted in our attachment to a pernicious dogma of modern epistemology: the idea that a judgement must be supported by some kind of reason or evidence to be justified. The epistemic analogue of the doctrine of original sin, this idea renders every judgement in need of redemption – guilty until shown to be innocent – distorting our understanding of reason at a very deep level, and preventing us from conceptualizing a satisfactory solution to the problem. If we opt for a more context–sensitive mechanism for assigning default epistemic status, however, we get a more plausible picture of justification, an epistemology more in tune with epistemic practice, and an elegant solution to the regress problem.     

Internalist Foundationalism and the Problem of the Epistemic Regress*

I provide a construal of the epistemic regress problem and I take issue with the contention that a foundationalist solution is incompatible with an internalist account of warrant. I sketch a foundationalist solution to the regress problem that respects a plausible version of internalism. I end with the suggestion that the strategy that I have presented is not available only to the traditional versions of foundationalism that ascribe foundational status to experiential beliefs. It can also be used to generate a version of internalist foundationalism based on reliabilist principles.     

Using Hope in the Counseling Process

This article presents a framework for the development of hope in the counseling relationship. Although hope has often been viewed as an impediment to growth, the authors believe that realistic hope is an effective tool in a counseling relationship. The authors also examine the role of the counselor in the formulation of hope.     

The Epistemic Regress Problem,the Problem of the Criterion,and the Value of Reasons

There are important similarities between the epistemic regress problem and the problem of the criterion. Each turns on plausible principles stating that epistemic reasons must be supported by epistemic reasons but that having reasons is impossible if that requires having endless regresses of reasons. These principles are incompatible with the possibility of reasons, so each problem is a paradox. Whether there can be an antiskeptical solution to these paradoxes depends upon the kinds of reasons that we need in order to attain our epistemic goals. This article explains the problems and considers the ways in which two different conceptions of human flourishing support the value of different kinds of reasons. One conception requires reasons that allow an easy solution to these paradoxes. The other—rational autonomy—requires reasons that depend upon endless regresses. So we cannot have the kinds of fully transparent reasons required for rational autonomy.     

Using the Analytic Hierarchy Process to Analyze Multiattribute Decisions

Since its introduction a decade ago, the Analytic Hierarchy Process (AHP) has been used by decision makers in many contexts. The vast majority of these applications has involved allocating resources or making choices from among alternatives. This article discusses how AHP may be used in a different manner: to assist researchers in the analysis of decisions. AHP is briefly compared with other decision-analysis techniques, such as multiattribute utility measurement, conjoint measurement, and general linear models (regression and analysis of variance). Some of the possible insights into decision processes that can be obtained using AHP are illustrated with data gathered from practicing auditors of financial statements.     

Testing Process Factor Analysis Models Using the Parametric Bootstrap

Process factor analysis (PFA) is a latent variable model for intensive longitudinal data. It combines P-technique factor analysis and time series analysis. The goodness-of-fit test in PFA is currently unavailable. In the paper, we propose a parametric bootstrap method for assessing model fit in PFA. We illustrate the test with an empirical data set in which 22 participants rated their effects everyday over a period of 90 days. We also explore Type I error and power of the parametric bootstrap test with simulated data.     

Process timing and its relation to the coding of tonal harmony

Advances in auditory research suggest that gamma-band synchronization of frequency-specific cortical loci could be responsible for the integration of pure tones (harmonics) into harmonic complex tones. Thus far, evidence for such a mechanism has been revealed in neurophysiological studies, with little corroborative psychophysical evidence. In six experiments, we observed a rate- and time-specific response-time advantage for a sequence of target pips when the defining frequency of the target was a fractional multiple of a priming frequency. The effect was only observed when the prime and target sequences were presented at 33 pips per second and when the interstimulus interval was approximately 100 and 250 ms. This evidence implicates oscillatory gamma-band activity in the representation of harmonic complex tones and suggests that synchronization with precise temporal characteristics is important for disambiguating related harmonic templates. An outline of a model is presented, which accounts for these findings in terms of fast resynchronization of relevant neuronal assemblies.     

Compound conditioning: Elimination of the blocking effect

Prior conditioning employing one element of a compound stimulus as the CS blocked the acquisition of a Conditioned Emotional Response (CER) to the second element of a simultaneous compound stimulus that was subsequently used as the CS in further conditioning trials. Presentation of a brief “surprise” stimulus 3 or 5 sec after the occurrence of the US during the compound trials eliminated the blocking effect.