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On Operator N and Wittgenstein's Logical Philosophy JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY ON OPERATOR N AND WITTGENSTEIN’S LOGICAL VOLUME 5, NUMBER 4 PHILOSOPHY EDITOR IN CHIEF JAMES R. CONNELLY KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS EDITORIAL BOARD In this paper, I provide a new reading of Wittgenstein’s N oper- ANNALISA COLIVA, UnIVERSITY OF MODENA AND UC IRVINE ator, and of its significance within his early logical philosophy. GaRY EBBS, INDIANA UnIVERSITY BLOOMINGTON I thereby aim to resolve a longstanding scholarly controversy GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES concerning the expressive completeness of N. Within the debate HENRY JACKMAN, YORK UnIVERSITY between Fogelin and Geach in particular, an apparent dilemma SANDRA LaPOINte, MCMASTER UnIVERSITY emerged to the effect that we must either concede Fogelin’s claim CONSUELO PRETI, THE COLLEGE OF NEW JERSEY that N is expressively incomplete, or reject certain fundamental MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT tenets within Wittgenstein’s logical philosophy. Despite their ANTHONY SKELTON, WESTERN UnIVERSITY various points of disagreement, however, Fogelin and Geach MARK TEXTOR, KING’S COLLEGE LonDON nevertheless share several common and problematic assump- AUDREY YAP, UnIVERSITY OF VICTORIA tions regarding Wittgenstein’s logical philosophy, and it is these RICHARD ZACH, UnIVERSITY OF CALGARY mistaken assumptions which are the source of the dilemma. Once we recognize and correct these, and other, associated ex- REVIEW EDITORS pository errors, it will become clear how to reconcile the ex- JULIET FLOYD, BOSTON UnIVERSITY pressive completeness of Wittgenstein’s N operator, with sev- CHRIS PINCOCK, OHIO STATE UnIVERSITY eral commonly recognized features of, and fundamental theses ASSISTANT REVIEW EDITOR within, the Tractarian logical system. SEAN MORRIS, METROPOLITAN STATE UnIVERSITY OF DenVER DESIGN DaNIEL HARRIS, HUNTER COLLEGE JHAPONLINE.ORG © 2017 JAMES R. CONNELLY ON OPERATOR N AND WITTGENSTEIN’S LOGICAL ical philosophy that are not fully appreciated by either Geach PHILOSOPHY or Fogelin. These are (1) Wittgenstein’s conception of infinity as actual as opposed to potential; (2) Wittgenstein’s provision JAMES R. CONNELLY of the N operator for purely philosophical as opposed to prac- tical purposes; (3) Wittgenstein’s characterization of elementary propositions as endowed with structure but nevertheless logi- cally independent; and (4) Wittgenstein’s characterization of N 1. INTRODUCTION as a sentential as opposed to quantificational operator. Because they fail to appreciate (1) and (2), it seemed obvious to both Concerns about the expressive completeness of Wittgenstein’s Fogelin and Geach (in opposition to (4)) that Wittgenstein in- operator N have led some scholars to consider or even suspect tended the N operator to be deployed in consort with some sort that Wittgenstein may have been guilty of an elementary logical of quantificational device, whether it be an open sentence, or a blunder (Soames 1983, 578; Fogelin 1987, 82). Other scholars class forming operator, in order to facilitate quantification over (McGray 2006, 150; Geach 1981, 170) have noted the low proba- an infinite domain. However, careful attention to Wittgenstein’s bility that, if Wittgenstein had committed an elementary blun- symbolism and explanations shows that he only ever intended der, exceptionally competent reviewers of the day, such as Rus- N to apply to semantically atomic, elementary propositions, and sell and Ramsey, would have let it pass. Within the interchanges to N-expressed1 truth-functions thereof. As per (3), above, ele- between Fogelin and Geach, in particular, a dilemma emerged mentary propositions have structure, but are nevertheless log- whereby it seems we must either accept Fogelin’s claim that N ically independent. This entails that N may be applied, in the is expressively incomplete, or reject certain fundamental theses basis case, to semantically atomic sentence letters which lack within Wittgenstein’s logical philosophy, such as his repudia- any internal structure, just as Wittgenstein indicates within his tion of set theory, and his claim that all meaningful propositions symbol for the general form of a truth-function: [p¯, ξ,¯ N(ξ¯)] may be expressed via a finite number of successive operations. (TLP 6).2 N may then subsequently be applied to N-expressed In either case, Wittgenstein would have committed a fairly sig- nificant logical blunder. In this paper, I will instead argue that 1To say that a truth-function is “N-expressed” means simply that it appears there was no blunder and thus that there was nothing for Russell as expressed using the N operator, rather than as it would appear as expressed or Ramsey to let pass. Once we identify and recognize the sig- via any other combination of truth-functional connectives, including the am- persand (conjunction), the tilde (negation), or the wedge (disjunction). Thus, nificance of several interrelated and underappreciated features the truth-function known as joint negation may be expressed using the am- of Wittgenstein’s logical philosophy, and correct a number of as- persand and the tilde as “∼p & ∼q”, or using the N operator as “N(p, q).” In sociated expository errors, it will become clear how to address the latter case, the truth-function is said to be “N-expressed” in the intended Fogelin’s concerns about expressive completeness without un- sense. 2When I wish to refer to Wittgenstein’s Tractatus with indifference to any dermining any of the commonly recognized and fundamental distinctions between the 1922 Odgen translation, and the 1961 Pears and theses which characterize the Tractarian logical system. McGuiness translation, I will simply use “TLP” followed by a proposition In addition to several subsidiary features, there are four pri- number (e.g., TLP 5.501). By contrast, I will use TLP 1922 followed by a mary, significant and interrelated elements of Wittgenstein’s log- proposition number, to refer specifically to the Odgen translation, and TLP JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY VOL. 5 NO. 4 [1] truth-functions of the elementary propositions symbolized by Because most contributors to this debate are, like me, unper- these semantically atomic sentence letters, in order to express suaded that Wittgenstein is guilty of an elementary blunder, all truth-functions, including the truth-functional expansions3 they do not in general accept, as does Fogelin, the idea that the which correspond to quantified propositions. It can then read- N operator is expressively incomplete. Many, as we shall see, ily be seen to be expressively complete with regards to classical nevertheless do not fully appreciate these integral, and excul- predicate and propositional logic, provided we recognize and patory features of the Tractarian logical system. In a valiant appreciate the significance of (1) and (2). effort to rescue Wittgenstein from the appearance of incompe- tence, commentators have been led to devise many ingenious, 1961 followed by a proposition number, to refer specifically to the Pears and if ultimately unnecessary technical, notational innovations. An McGuiness translation. unfortunate consequence of these commendable efforts, how- 3Throughout this paper I will deploy truth-functional expansions ex- pressed using classical truth-functions such as conjunction and disjunction, for ever, is that such technical, notational innovations have come illustrative purposes, as aids to understanding the content of truth-functional to appear to be much more relevant to the understanding of expansions expressed using N. Despite the fact that Wittgenstein is critical of Wittgenstein’s proposal than they actually are. A subsidiary Frege and Russell, at TLP 5.521, for introducing generality in association with goal of this paper will thus to be to recover the deeper and ulti- conjunction and disjunction, our use of disjunction and conjunction to eluci- date truth-functional expansions, and generality, will be harmless provided mately philosophical meaning of the N operator, which has less to the reader recognizes that these connectives are ultimately to be eliminated do with concerns of notational innovation, as we shall see, than from our expressions in favour of a single truth-function, specifically N. No- has sometimes been supposed. tably, at 5.521 Wittgenstein claims merely that understanding generality in In order to more fully appreciate these important lessons it terms of these truth-functions is associated with difficulties; he does not ex- plicitly proscribe it. Indeed, when Wittgenstein also says at 5.521, that he will be helpful to delve first, in Section 2, into the nature of “dissociates the concept all from truth-functions,” he cannot be proscribing Wittgenstein’s N operator and to clear up some misconceptions such an elucidatory use of conjunction and disjunction, since N is also a concerning its application. In that context, focus will be placed “truth-function” and in that case he would also be proscribing the use of on the three different methods of selecting elementary proposi- N to elucidate quantification. And aside from being prima facie implausible, tions for presentation to the N operator, identified by Wittgen- this would make it very hard to explain why Wittgenstein defines N such that it may take an indefinite number of arguments (see Section 2). It is stein at TLP 5.501. Special attention will be paid to explicating thus more probable that, by this remark, Wittgenstein means simply that like Wittgenstein’s intended interpretation of the second
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