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Logic in the Tractatus Logic in the Tractatus by Max Weiss A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Philosophy) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2013 © Max Weiss 2013 Abstract What is logic, in the Tractatus? There is a pretty good understanding what is logic in Grundgesetze, or Principia. With respect to the Tractatus no comparably definite answer has been received. One might think that this is because, whether through incompetence or obscurantism, Wittgenstein simply does not propound a definite conception. To the contrary, I argue that the text of the Tractatus sup- ports, up to a high degree of confidence, the attribution of a philosophically well-motivated and mathematically definite answer to the question what is logic. ii Preface Warum hier plötzlich Worte? (5.452) It is something of an accident that I ended up writing a dissertation about the Tractatus. In 2009, I was casting around for a thesis topic and somehow found myself in Carl Posy’s seminar on Brouwer at The Hebrew University. He seemed like a pretty good person to ask. When we met, he took out his copy of the Critique of Pure Reason and read out a passage in which he discerned insights later conveyed in certain early 1970s lectures at Princeton on the philosophy of language. History—or, better, long conversation—brings a structure that he thought I needed. It also helped that Posy clearly enjoys talking about Brouwer. Then it did come down to early Wittgenstein.1 In my first year as an under- graduate, my favorite course had been an introduction to logic taught by San- ford Shieh. I asked him what to read next, and he said, “well, the Grundlagen of course. Then then you might want to look at the Tractatus”. So I went out, found the Tractatus, and carried it around for a year or two. It ended up on my book- shelf and remained there for about ten years. Some feeling of personal failure lingers around that episode, which maybe a whole dissertation begins to redress. When I first began to read the Tractatus as an adult, then, the first thing that struck me was the extreme implausibility of its characterization as a kind of logi- cal atomism. It seemed just incomprehensible to me that Hume or Russell could say that each object has written into its nature all its possibilities of occurrence in states of affairs. But it also struck me as unpalatable to respond by arguing that the Tractatus is not atomist but holist, since that seems to amount to little more 1Or Spinoza. So it could have been worse. iii than consigning it to the opposing role in some Kantian antinomy. Anyway, this initial response was enough for a dissertation prospectus, but that was about it. Then in 2012 I had the good luck to end up in Cambridge, MA, where, as it happened, Warren Goldfarb was giving a seminar on the Tractatus.2 A few weeks into the seminar, he pointed out that Wittgenstein tried to develop resources for analysis of the ancestral of a relation. Goldfarb said he was working on the ques- tion what is the complexity of the property of being a tautology in the Tractatus system. I was intrigued by Goldfarb’s question, and spent much of that year trying to assign it a definite mathematical content. The basic idea behind the construc- tion of propositions in the Tractatus is a process of inductive generation. Some elementary propositions are initially given. Furthermore, for any given bunch of propositions, some proposition is the joint denial of the propositions in that bunch. This construction has some fairly obvious mathematical and philosoph- ical similarities with the cumulative hierarchy of sets. For the cumulative hier- archy of sets is described as the result of repeatedly applying, “as much as possi- ble”, an operation of set-formation to multiplicities of items “already” obtained. Moreover, neither construction is an ordinary inductive definition which serves to single out objects from some antecedently given class of objects: rather, they are each inductions of the sort that Kleene (1952, 258) calls “fundamental”, pur- porting to generate the entities of the domain in the first place. However, this resemblance, while suggestive, should not be taken too flatfootedly. In particu- lar, it is implausible and boring to suppose that for any multiplicity of proposi- tions whatsoever, a proposition can be constructed which denies exactly them. What, by Wittgenstein’s lights, actually is the extent of the capacity to survey multiplicities? At 5.501 Wittgenstein says that one can specify multiplicities of propositions by “stipulating the range of a propositional variable.” He gives “three ways” of fixing such ranges.3 The first two ways basically correspond to the syntax of first- 2He and a student couldn’t quite agree about whether it had been ten or eleven years since its previous offering. 3References to entries of the Tractatus (Wittgenstein 1989) will take this standard form, sole exception being single-digit entries which will be prefaced by the letter T. (I will not refer to its preface.) Reference to entries of the Prototractatus will take the same form but prefixed by PT. iv order logic. But the third way allows fixing the range of a variable by constructing a so-called “form-series”—roughly, an inductive specification of a multiplicity of formulas. Sadly, Wittgenstein says almost nothing about which operations can be used in such inductions. Frustrated by Wittgenstein’s vagueness on this point, I decided to explore the hypothesis that one can form the joint denial of any effectively enumerable col- lection of formulas. Under this hypothesis, there then followed an analogy of the Tractatus system with the hyperarithmetical hierarchy, a technical construction which arose in mid 20th-century mathematical logic. Goldfarb wasn’t impressed. He urged, in particular, that according to Wittgenstein, we need to be able to “see” which things are subformulas of a given formula. So I tossed aside the arith- metization and tried just to say very carefully how Wittgenstein’s construction seemed to work. Rather than imposing external ideas about computability onto the collection of propositional signs, I tried instead to think about operations that one might think of as native to the realm of propositional signs itself. The idea that immediately suggested itself was that of substitution of a formula for a propositional variable in a formula. It quickly became clear that such a simple idea actually did the required technical work. That is, by forming a disjunction of the results of iterated substitution, it is fairly straightforward to express the ancestral of a given relation. In this way, I settled on what seems to me to be a plausible approximation of the system of logic developed in the Tractatus. It is an “approximation from below” in the sense that everything expressible in it is claimed to be expressible in the Tractatus. No converse claim is intended, but I doubt that significant strengthenings are interpretively defensible. My original formulation of this analysis used a bizarre metalanguage: in par- ticular Wittgenstein’s talk of variable-ranges was handled with plural variables. This made the formulation pretty hard to understand (according to people who tried to read it). So I recast the whole thing as a construction inside a particular standard model of the axiomatic set theory KPU. The result is a mathematically definite explication of the property of being a tautology which Goldfarb’s origi- References to the Notebooks (Wittgenstein 1979) will always take the form: α.β.γ with the numer- als α,β,γ respectively denoting the day, month, and year. A Roman letter at the end of such a reference indicates particular paragraphs of the denoted passage. v nal question addressed. This notion is definite despite avoiding any prejudgment whatsoever about the number and form of elementary propositions. By risking such prejudgment, one can instantiate the abstract notion to more concrete ones. Anyway, all this appears in Chapter 5, which embodies the main claim of the thesis: “here is what Wittgenstein thinks logic is”. vi Table of Contents Abstract ........................................... ii Preface ............................................ iii Table of Contents ..................................... vii Introduction ........................................ 1 1 The general propositional form ......................... 12 1.1 Analysis ..................................... 12 1.1.1 Logic and analysis .......................... 15 1.1.2 What about reality? ......................... 26 1.1.3 Sign and symbol ........................... 38 1.1.4 Propositions as truth-functions . 45 1.2 Workings of the GPF ............................. 48 1.2.1 The one-many problem ...................... 49 1.2.2 Some early-Russellian technology . 52 1.2.3 Variables in the Tractatus ..................... 55 1.2.4 Propositional variables: some details . 60 1.2.5 Insufficiency of local constraints . 64 1.2.6 The general propositional form . 67 1.3 The GPF in action ............................... 71 1.3.1 A guide to analysis ......................... 71 1.3.2 Origins of the independence criterion . 81 2 Objectual generality ................................ 89 2.1 What is the issue? ............................... 93 vii 2.2 Nonelementary propositions ........................ 97 2.3 Fogelin ...................................... 102 2.4 Geach-Soames ................................. 107 2.5 Wehmeier and Ricketts ............................ 111 2.6 Pictoriality ................................... 118 2.7 The picture ................................... 124 3 Formal generality .................................. 130 3.1 Two problems with higher-order generality . 137 3.2 Wittgenstein’s alternative . 142 3.3 Form-series in analysis in the NB . 147 3.4 Another role for form-series? . 156 3.5 T5.252 and the evolving conception of hierarchy . 158 3.6 The riddle of T6 ................................ 165 3.7 An alternative reading ............................ 171 4 Illustrations ...................................... 180 4.1 Ways One and Two, a sketch .
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