4,935 research outputs found
A Refutation of Goodman's Type‐Token Theory of Notation
In Languages of Art, Nelson Goodman presents a general theory of symbolic notation. However, I show that his theory could not adequately explain possible cases of natural language notational uses, and argue that this outcome undermines, not only Goodman's own theory, but any broadly type versus token based account of notational structure.Given this failure, an alternative representational theory is proposed, in which different visual or perceptual aspects of a given physical inscription each represent a different letter, word, or other notational item. Such a view is strongly supported by the completely conventional relation between inscriptions and notation, as shown by encryption techniques etc
An extremal problem on crossing vectors
For positive integers and , two vectors and from
are called -crossing if there are two coordinates and
such that and . What is the maximum size of
a family of pairwise -crossing and pairwise non--crossing vectors in
? We state a conjecture that the answer is . We prove
the conjecture for and provide weaker upper bounds for .
Also, for all and , we construct several quite different examples of
families of desired size . This research is motivated by a natural
question concerning the width of the lattice of maximum antichains of a
partially ordered set.Comment: Corrections and improvement
Nowhere Weak Differentiability of the Pettis Integral
For an arbitrary infinite-dimensional Banach space \X, we construct
examples of strongly-measurable \X-valued Pettis integrable functions whose
indefinite Pettis integrals are nowhere weakly differentiable; thus, for these
functions the Lebesgue Differentiation Theorem fails rather spectacularly. We
also relate the degree of nondifferentiability of the indefinite Pettis
integral to the cotype of \X, from which it follows that our examples are
reasonably sharp.
This is an expanded version of a previously posted paper with the same name
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