A Note on Prototypes,Convexity and Fuzzy Sets

The work on prototypes in ontologies pioneered by Rosch [10] and elaborated by Lakoff [8] and Freund [3] is related to vagueness in the sense that the more remote an instance is from a prototype the fewer people agree that it is an example of that prototype. An intuitive example is the prototypical “mother”, and it is observed that more specific instances like ”single mother”, “adoptive mother”, “surrogate mother”, etc., are less and less likely to be classified as “mothers” by experimental subjects. From a different direction Gärdenfors [4] provided a persuasive account of natural predicates to resolve paradoxes of induction like Goodman’s “Grue” predicate [5]. Gärdenfors proposed that “quality dimensions” arising from human cognition and perception impose topologies on concepts such that the ones that appear “natural” to us are convex in these topologies. We show that these two cognitive principles — prototypes and predicate convexity — are equivalent to unimodal (convex) fuzzy characteristic functions for sets. Then we examine the case when the fuzzy set characteristic function is not convex, in particular when it is multi-modal. We argue that this is an indication that the fuzzy concept should really be regarded as a super concept in which the decomposed components are subconcepts in an ontological taxonomy.     

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches. This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

The Comparative Set Fallacy

This paper argues for the validity of inferences that take the form of: A is more X than B; therefore A and B are both X. After considering representative counterexamples, it is claimed that these inferences are valid if and only if the comparative terms in the inference are taken from no more than one comparative set, where a comparative set is understood to be comprised of a positive, comparative, and superlative, represented as {X, more X than, most X}. In all instances where arguments appearing to be of this form are invalid, it is the case that the argument has fallaciously taken terms from more than one comparative set. The fallacy of appealing to more than one comparative set in an inference involving comparative terms is shown to be analogous to the fallacy of equivocation in argumentation. The paper concludes by suggesting a conflation of logical issues with grammatical issues is the core difficulty leading some to consider inferences in the form of A is more X than B; therefore A and B are X to be invalid.     

Event-related EEG oscillations to semantically unrelated words in normal and learning disabled children

Learning disabilities (LD) are one of the most frequent problems for elementary school-aged children. In this paper, event-related EEG oscillations to semantically related and unrelated pairs of words were studied in a group of 18 children with LD not otherwise specified (LD-NOS) and in 16 children with normal academic achievement. We propose that EEG oscillations may be different in LD NOS children versus normal control children that may explain some of the deficits observed in the LD-NOS group. The EEGs were recorded using the 10/20 system. EEG segments were edited by visual inspection 1000ms before and after the stimulus, and only correct responses were considered in the analysis. Time-frequency (1-50Hz) topographic maps were obtained for the increases and decreases of power after the event with respect to the pre-stimulus average values. Significant differences between groups were observed in the behavioral responses. LD-NOS children show less number of correct responses and more omissions and false alarms than the control group. The event-induced EEG responses showed significant differences between groups. The control group showed greater power increases in the frequencies 1-6Hz than the LD-NOS group from 300 to 700ms. These differences were mainly observed in frontal regions, both to related and non-related words. This was interpreted as a deficit in attention, both to internal and external events, deficits in activation of working memory and deficits in encoding and memory retrieval in the LD-NOS children. Differences between groups were also observed in the suppression of alpha and beta rhythms in the occipital regions to related words in frequencies between 8 and 17Hz from 450 to 750ms. LD-NOS children showed shorter durations of the decreases in power than the control group. These results suggest also deficits in attention and memory retrieval. It may be concluded that LD-NOS children showed physiological differences from normal children that may explain their cognitive deficiencies.