Notes on the Riemannian Geometry of Lie Groups

Lie groups occupy a central position in modern differential geometry and physics, as they are very useful for describing the continuous symmetries of a space. This paper is an expository article meant to introduce the theory of Lie groups, as well as survey some results related to the Riemannian geometry of groups admitting invariant metrics. In particular, a non-standard proof of the classification of invariant metrics is presented. For those unfamiliar with tensor calculus, a section devoted to tensors on manifolds and the Lie derivative is included

The Meaning of Meaning in the Law

The Court uses the words mean and meaning in a number of ways. The difficulty with this is that there are two senses of these words that can, and often are confused, and these are what we have been calling conventional meaning and contextual significance. Locutions containing mean that are particularly problematic, for mean that is sometimes equivalent to \u27entail\u27 and sometimes to \u27implicate.\u27 Clearly it is in the interest of the Court to be more explicit about what it means by mean and meaning. Unfortunately, its practice of adding modifiers like common, ordinary, natural, every day, and plain not only does not make its meaning clearer, it further exacerbates the problem. The Supreme Court would be well-advised to adopt the analytic techniques of linguists, specifically semanticians and pragmaticians. Rather than the question-begging usage of dictionaries, the Court could employ entailment tests of the sort used here to determine the conventional meaning of the language of disputed texts. Arguably, it would come up with results that not only are more empirically sound, but provide a stronger basis for its real task-to determine the applicability or import, i.e., the contextual significance, of the legal texts that the Court interprets

On Invited Inferences

This work was supported in part by the 1970 MSSB Advanced Research Seminar in Mathematical Linguistics, sponsored by the National Science Foundation through a grant to the Center for Advanced Study in the Behavioral Sciences, Stanford, and held at the Ohio State University