Michiel Van Lambalgen

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others.

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation

Gaisi Takeuti has recently proposed a new operation on orthomodular latticesL, $\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $ :P(L)»L. The properties of $\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $ suggest that the value of $\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} $ (A) (A) $ \subseteq $ L) corresponds to the degree in which the elements ofA behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular latticesL and the existence of two-valued homomorphisms onL

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed-world reasoning of particular interest for human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine-grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3-valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning.

In this paper we present a semantic analysis of the imperfective paradox based on the Event Calculus, a planning formalism characterizing a class of models which can be computed by connectionist networks. We report the results of a questionnaire that support the semantic theory and suggest that different aspectual classes of VPs in the progressive give rise to different entailment patterns. Further, a processing model is outlined, combining the semantic analysis with the psycholinguistic principle of immediacy in the framework of recurrent networks. The model is used to derive predictions concerning the electrophysiological correlates of the computations described by the Event Calculus.

Executive function has become an important concept in explanations of psychiatric disorders, but we currently lack comprehensive models of normal executive function and of its malfunctions. Here we illustrate how defeasible logical analysis can aid progress in this area. We illustrate using autism and attention deficit hyperactivity disorder (ADHD) as example disorders, and show how logical analysis reveals commonalities between linguistic and non-linguistic behaviours within each disorder, and how contrasting sub-components of executive function are involved across disorders. This analysis reveals how logical analysis is as applicable to fast, automatic and unconscious reasoning as it is to slow deliberate cogitation.

ADHD is a psychiatric disorder characterised by persistent and developmentally inappropriate levels of inattention, impulsivity and hyperactivity. It is known that children with ADHD tend to produce incoherent discourses, e.g. by narrating events out of sequence. Here the aetiology of ADHD becomes of interest. One prominent theory is that ADHD is an executive function disorder, showing deficiencies of planning. Given the close link between planning, verb tense and discourse coherence postulated in van Lambalgen and Hamm (The proper treatment of events, 2004), we predicted specific deviations in the verb tenses produced by children with ADHD. Here we report on an experiment corroborating these predictions.