Michiel Van Lambalgen

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

We sketch four applications of Marr's levels‐of‐analysis methodology to the relations between logic and experimental data in the cognitive neuroscience of language and reasoning. The first part of the paper illustrates the explanatory power of computational level theories based on logic. We show that a Bayesian treatment of the suppression task in reasoning with conditionals is ruled out by EEG data, supporting instead an analysis based on defeasible logic. Further, we describe how results from an EEG study on temporal prepositions can be reanalyzed using formal semantics, addressing a potential confound. The second part of the article demonstrates the predictive power of logical theories drawing on EEG data on processing progressive constructions and on behavioral data on conditional reasoning in people with autism. Logical theories can constrain processing hypotheses all the way down to neurophysiology, and conversely neuroscience data can guide the selection of alternative computational level models of cognition.

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.

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.

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.

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.

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.

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of different gradation, which makes vision at each of them approximate. On the logical side, our task is therefore twofold • to formalise the layers and the ways in which they may refine each other, and • to develop logical means to let description vary with such degrees of refinement. The first task is formalised by means of an inverse systems of first order models, with reality appearing as its inverse limit. The second task is formalised by means of so-called conditional quantifiers, a new form of generalised quantification which can best be described as resource bounded quantification. We show that the logic provides for a semantics and pragmatics of direct perception reports. In particular, direct perception reports have a possibly nonveridical, approximative semantics, which becomes veridical only by virtue of our pragmatic expectation that what is perceived would continue to be the case, were we to perceive more accurately. It is a general feature of resource bounded logics that the underlying logics are weak, but that stronger principles can be obtained pragmatically, by strengthening the resource. For the logic of vision this feature is clarified by showing how changes in the resource capture different notions of partiality, and by studying how the perception verb interacts with connectives and quantifiers in different visual contexts. The inference Veridicality, which is now viewed rather as a nonmonotonic inference, is also studied in depth. We end with an attempt to buttress the proposed model by comparing it with suggestions put forward in Cognitive or Conceptual Semantics, in the literature on evidentials, and in Husserl's philosophy of perception.

In the late summer of 1998, the authors, a cognitive scientist and a logician, started talking about the relevance of modern mathematical logic to the study of human reasoning, and we have been talking ever since. This book is an interim report of that conversation. It argues that results such as those on the Wason selection task, purportedly showing the irrelevance of formal logic to actual human reasoning, have been widely misinterpreted, mainly because the picture of logic current in psychology and cognitive science is completely mistaken. We aim to give the reader a more accurate picture of mathematical logic and, in doing so, hope to show that logic, properly conceived, is still a very helpful tool in cognitive science. The main thrust of the book is therefore constructive. We give a number of examples in which logical theorizing helps in understanding and modeling observed behavior in reasoning tasks, deviations of that behavior in a psychiatric disorder (autism), and even the roots of that behavior in the evolution of the brain.

Compositionality remains effective as an explanation of cases in which processing complexity increases due to syntactic factors only. It falls short of accounting for situations in which complexity arises from interactions with the sentence or discourse context, perceptual cues, and stored knowledge. The idea of compositionality as a methodological principle is appealing, but imputing the complexity to one component of the grammar or another, instead of enriching the notion of composition, is not always an innocuous move, leading to fully equivalent theories. Compositionality sets an upper bound on the degree of informational encapsulation that can be posited by modular or component-based theories of language: simple composition ties in with a strongly modular take on meaning assembly, which is seen as sealed off from information streams other than the lexicon and the syntax.