Some Commutative Algebra

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

TRACTARIAN FIRST-ORDER LOGIC: IDENTITY AND THE N-OPERATOR

In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of theTractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in theTractatuswhen taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.5.503

At the Margin of Discourse: Footnotes in the Fictional Text

The referential and marginal features of footnotes serve different functions in criticism and literature: scholarly footnotes shore up the text by enclosing it and limiting its claims; in fiction, footnotes extend textual authority by enlarging the fictional context. Both inner- and outer-directed, these two kinds of notations display a self-conscious anxiety about the critical and creative acts they annotate. Scholarly notes mask this ambivalence by claiming extratextual authority; literary notes highlight the ambivalence by consciously dividing the text against itself. This essay examines the ways footnotes in Tom Jones, Tristram Shandy, and Finnegans Wake parody the notational convention and draw attention to the faulted authority of its discourse by flouting scholarly claims to objectivity and neutrality, by calling into question the relations of author and reader on textual grounds, and by using self-reflexive narrative methods to illustrate the rhetorical double bind that keeps all language at the margin of discourse.