Logic in the Tractatus I:Definability∗

Max Weiss

December 5, 2015

Abstract

I present a reconstruction of the logical system of the Tractatus, which dif-fers from classical logic in two ways. It includes an account of Wittgenstein’s“form-series” device, which suffices to express some effectively generated count-ably infinite disjunctions. And its attendant notion of structure is relativized tothe fixed underlying universe of what is named.

There follow three results. First, the class of concepts definable in the systemis closed under finitary induction. Second, if the universe of objects is countablyinfinite, then the property of being a tautology is Π1

1-complete. But third, it isonly granted the assumption of countability that the class of tautologies is Σ1-definable in set theory.

Wittgenstein famously urges that logical relationships must show themselvesin the structure of signs. The results of this paper suggest that it is if and only ifthe number of objects is countable that the conception of logic in the Tractatusmight be coherent.

We have by now a quite systematic and rigorous grasp of the logical work of twoof Wittgenstein’s teachers, Frege and Russell. This is thanks in part to decades offlourishing scholarship, and thanks also to Frege and Russell themselves. In contrast,despite comparably voluminous commentary there is still no received understandingof anything describable as the logical system of the Tractatus (Wittgenstein, 1921). Itis hard to resist the conclusion that the Tractatus did not, despite its professed programand its large reputation, offer any systematic alternative conception of the nature oflogic.

But the conclusion is mistaken. To the contrary, there is a system, or a class ofsimilar systems, which can be understood to explicate the development of logic in theTractatus. They differ rather sharply from those of Frege or Russell, as well as fromclassical first- or second-order logic. Nonetheless they can be exactly described andinvestigated metamathematically, for epistemological and metaphysical evaluation. Inthis paper, I will present one such system, and investigate some of its properties. Theseturn out to be of surprising and indeed independent interest.

∗This is the penultimate version of a paper forthcoming in Review of Symbolic Logic

1

In seeking to understand what logic is supposed to be according to the early Wittgen-stein, we may distinguish two kinds of evidence. First, there are the contours of hisown construction, famously, for example, the truth-functionality thesis and its real-ization through an operation of joint denial. Second, there are in the Tractatus appar-ent declarations of epistemological constraints on the nature of logic: some of these,for example, have been taken to suggest that according to Wittgenstein, logic must bedecidable.

I wish to separate these two strands. In the early decades of the 20th century, itwas entirely possible for a proficient researcher to investigate a computationally in-tractable system under the misapprehension of its decidability. Of course, there maybe substantial interest in the development of systems which now can be seen to res-onate with apparent epistemological declarations of the Tractatus. The question stillremains: what system, if any, did Wittgenstein in fact describe? The undecidabilityof logic being now so deeply rooted in our own understanding, it is hard to take up inimagination the computational intuitions of researchers in the era which preceded itsdiscovery. But in seeking to understand how someone tried to climb a mountain, weshould study the climber’s movements and the mountain’s contours, not transposethe climber to a molehill.

A large body of literature exhibits accelerating progress in our understanding ofthe development of logic in the Tractatus. Thanks especially to Geach (1981) andSoames (1983) and more recently Wehmeier (2004), (2008), (2009), and Rogers andWehmeier (2012), it has become nearly received wisdom that Wittgenstein both in-tended and managed, if only haphazardly, to accommodate the expressive resourcesof first-order logic with equality. However, when the Tractatus was in preparation,first-order logic had not attracted much attention as an autonomous logical system.So it would be surprising to say the least to find that something like first-order logic,as it might be now understood, is the logic of the Tractatus. I want to explore tworespects in which Wittgenstein’s logic differs. They can each be understood to char-acterize a conception of logic which is a kind of intermediate between the logicismsof Frege and Russell on the one hand, and what became classical logic on the other.

First, Wittgenstein’s conception of logic differs from what became classical logicin regard to what is now understood as interpretation. Although the exact height ofmetalogical perspective reached by Frege or Russell has been a matter of some schol-arly controversy, it seems nonetheless safe to deny that anything like the modern no-tion of first-order satisfiability or logical consequence plays a central role in their log-ical writings. Rather, for Frege and Russell, the notion of axiomatic derivability setsfor logic a basic standard of rational organization, and thence also of logical correct-ness. But Wittgenstein denied such a role for proof. So it remains to be said just howthe signs of a formalism could be subject to some standard of logical correctness bythe lights of the Tractatus. Here, Wittgenstein propounds truth-conditional analysesof the notions of logical validity and of logical consequence. These analyses resemblethe now classical reduction of validity and consequence to the notion of truth-in-a-structure. But while classically, a structure for a language may have as its domain anynonempty set, for Wittgenstein all structures relevant to the determination of validityhave an underlying domain in common, the universal collection of all objects. It hasbeen claimed, in commentary on Tarski (1936)—see Mancosu (2010), that this adjust-

2

ment in the conception of structure disrupts the formulation of certain rudiments ofmodel theory. However, Wittgenstein’s divergence runs deeper, since like Russell heholds that each object has its own proper representative in a proposition.

Second, Frege and Russell understood logic to include what we recognize as non-first-order resources. Frege’s Begriffsschrift culminates with what, by Frege’s lights,would be a purely logical analysis of the concept of the ancestral. Likewise, Russell’sintroduction of the axiom of reducibility was motivated in part by the desire to re-duce to logic the principle of mathematical induction. First-order logic not havingbeen isolated as an autonomous logical system, it is natural to suppose that Wittgen-stein inherited the expectation, regarding such non-first-order notions, that they werenonetheless logical. As Geach (1981), Potter (2009), and Ricketts (2012) among oth-ers have observed, he introduced a notion of “form-series” variable, which permitsthe expression, in finite space, of some countably infinite disjunctions. However, theliterature contains no attempt at an exact reconstruction of the device. For example,although it’s agreed that the disjuncts must be generated in some vaguely “effective”manner, it’s not at all clear what this is supposed to mean. Moreover, the logical sig-nificance of the form-series device depends on the intelligibility of quantifying intothe contexts it creates, but this in turn involves subtleties hitherto unaddressed in thesecondary literature.

I will construct a notion of formal series exactly. As an approximation frombelow, my assumption will be that it lies well within the system of constructionsWittgenstein did intend to admit. However, I’ll then prove that the result of addingthis approximation of the form-series device to first-order logic permits the expres-sion of every concept which is definable by finitary induction. Indeed, in a sense tobe made precise, it coincides exactly with the result of adding induction to first-orderlogic. I’ll also give some interpretive evidence to suggest that this was the point.

These explications make it possible to give a mathematically definite and purport-edly exhaustive characterization of the complexity of Wittgenstein’s logical system.Let me quickly summarize the characterization; the technical notions here are stan-dard but will be defined in later sections. It has been widely recognized that Wittgen-stein could respect a commitment to decidability of logic by presupposing that thedomain of quantification is finite. This already implies that the complexity of logicdepends on the cardinality of the domain; in §4 I’ll evaluate the dependence precisely.I will show that if the domain is countably infinite, then the property of being a tautol-ogy in the logical system of the Tractatus is Π1

1-complete in a suitable analogue of theanalytical hierarchy. But moreover, it is only granted that the assumption of count-ability that the concept of tautology isΣ1 definable in the Levy hierarchy of formulasof set theory. Since, in any case, the notions of countable- and of uncountable-domaintautology are Π1

1 and ΠZF1 a priori, these results are just about as strong as possible.

When the form-series device is dropped from the system, then the notion of tautol-ogy returns to its familiar Σ1 position in the arithmetical hierarchy; however, theunderlying notion of consequence remains just as intractable.

There are at least three further respects in which logic as elaborated in the Trac-tatus may seem to differ from classical first-order logic. First, Wittgenstein takes asprimitive not negation or disjunction, but rather an operator N . Second, through anonstandard interpretation of the objectual variables, Wittgenstein tries to eliminate

3

the equality predicate. Third, the role of higher logical types in the Tractatus is a mat-ter of some controversy. The first two of these features do not essentially alter thecomplexity-theoretic situation, but it considerably expedites the mathematics to ab-stract from them. On the other hand, the introduction of higher-order variables canonly raise the complexity of the system. My aim is to establish lower bounds, so Icould simply duck the controversy by introducing no higher types. Nonetheless, thecomplexity of, for example, the full second-order consequence relation is far abovethe lower bounds established here. So, I argue on textual and systematic grounds thathigher-order quantification does not figure centrally in Wittgenstein’s account of theexpressive resources of logic, and that impredicative quantification cannot figure atall. Thus, my contention will not just be that such-and-such are lower bounds, butmoreover that significant strengthenings do not warrant comparable credence.

A project to understand the purported nature of logic in the Tractatus encounterstwo kinds of difficulty. First, there are the well-known exegetical difficulties raised bythe text itself. It is highly compressed, with its use of logical notation somehow bothtelegraphic and inconsistent. Partly because of difficulties of this first kind, pursuitof technical precision has been somewhat rare, though with a few notable exceptions.But a little reflection reveals technical subleties too. Of particular concern here willbe the problem of implementing within quantificational logic Wittgenstein’s proposalto compress some infinitary disjunctions into finite expressions. It would be math-ematically impracticable and interpretively misleading to try, within the bounds ofone paper, both to give a reconstruction which is sufficiently precise to submit tometamathematical analysis, while also presenting the interpretive foundations of theattribution to Wittgenstein of the reconstructed system. So in this paper, I will con-fine exegetical discussion to §1, with the aim there to motivate a handful of fairly weakinterpretive hypotheses. The point of the remainder of the paper is to give a math-ematical explication of the hypotheses and then to investigate their consequences. Ifreely use notation and techniques which did not mature until after Wittgenstein’sdeath.

In outline, the rest of the paper runs as follows. §1 lays out some interpretivebackground. After briefly summarizing the pertinent state of commentary on logic-in-the-Tractatus in general, I introduce the two departures from classical logic underinvestigation, summarizing relevant literature and briefly sketching the importanceof these eccentricities for the philosophical project of the Tractatus. §2 opens themathematical developments. After sketching the formalization of classical logic tobe adopted here, I introduce an approximation of the form-series device, and proposean analysis of quantification into form-series contexts. §2 furthermore presents tworesults fundamental for what follows: first, that the Russellian influence on Wittgen-stein’s conception of structure yields an embedding of the concept of truth in theconcept of consequence, and second, that the result of adding the form-series deviceto first-order logic does enjoy a reasonable form of extensionality. In §3, I turn to aquestion of the power of the form-series device, and show that as explicated here, it co-incides with the result of adding to first-order logic an operator for expressing finitaryinductive definitions. At this point, the stage is set for the investigation of complexity.I present, in §4, an exhaustive characterization of the complexity-theoretic effects ofthe form-series device and the fixed-domain conception of structure both separately

4

and jointly.

1 Interpretive backgroundIn broad outline, Wittgenstein’s conception of logical structure in the Tractatus isstraightforward. The upshot is a collection of sentences, together with a relation onthe collection which might be called “direct denial”. The direct denial relation distin-guishes some sentences as atomic, namely those which directly deny no sentences. Anonatomic sentence is to be true if and only if each of the sentences it directly deniesis not true.

The relation of direct denial is supposed to secure that the truth or falsehood ofeach sentence be determined by the truth or falsehood of the sentences which areatomic. To this end, Wittgenstein prescribes that the collection of sentences and therelation of direct denial must together satisfy a certain structural condition, a so-called“general propositional form”. Accordingly, atomic sentences may be regarded as hav-ing been given initially, and any other sentence must be presented as the joint denialof some sentences presented before it. As Geach (1981, 170) pointed out, the struc-tural condition cannot be understood to require that anybody actually construct allsentences which precede a given sentence in the ordering, because in general, a sen-tence will have infinitely many antecedents. Rather, the condition purports to in-dicate when some finite manipulation of signs could secure for them a sense. Thecrux of the condition is that the relation of direct denial on the class of sentences bewellfounded.

It is clear that if the relation of direct denial exhibited circularities, then it mightnot be intelligible as realizing a logical relationship of denial at all. For example, a sen-tence which appears to deny itself could be so understood only by taking it both tobe both true and to be not true, which for early Wittgenstein would not be an under-standing. But although Russell claimed to locate the origin of paradoxes in circularity,the mere exclusion of circularity does not suffice to secure the coherent interpretabil-ity of direct denial as denial: for example, it would still admit the construction ofa sequence of sentences A0,A1, . . . each of whose terms is the direct denial of all itssuccessors. In practice, Russell enforced the acyclicity of logical dependence by ametaphor of bottom-up construction in the ramified hierarchy of types; in impos-ing the stronger condition of wellfoundedness, this hierarchy is a clear conceptualantecedent of Wittgenstein’s general propositional form.

The existence of a wellfounded direct denial relation is supposed to give a condi-tion on the construction of signs, so that signs so constructed would be capable ofexpressing a sense. This means that wellfoundedness of direct denial should charac-terize the signs themselves, so that the relation can be determined from the mere signswithout reference to their sense (cf. 3.33-3.331). To this end Wittgenstein prescribesthat each nonatomic sentence have two parts. One part, which at 5.501 Wittgensteincalls a “bracket-expression”, serves to present some possibly infinite multiplicity ofother sentences. The other part is the famous operator N , which, attached to thebracket-expression, yields a sentence which directly denies precisely those sentenceswhich the bracket-expression presents. Thus, to determine for any two sentences

5

whether one directly denies the other, it should suffice to check whether the secondis among the sentences presented by the bracket-expression of the first.

It is clear that the complexity of logical dependence exhibited by the system willdepend on the methods available for constructing bracket-expressions. Wittgensteinlists three methods at T5.501. The first is simply to make a list of sentences outright;the resulting expression presents the sentences listed. This first method obviouslyyields an analysis of negation and disjunction. The negation of a sentence is a sen-tence which directly denies just it, and the disjunction of two sentences is the nega-tion of a direct denial of just those two. The second method is intended to yield ananalysis of generalization over objects, and ultimately together with the first methodto recover the expressive power of first-order logic with equality, at least under a cer-tain “Russellian” construal. The third method goes rather farther, and was billed byWittgenstein to yield at least a means of expressing the ancestral, or transitive closure,of expressible relations. In this section I will briefly sketch a somewhat anachronisticand oversimplified account of these two further methods which will suffice for themain purpose of this paper, which, again, is primarily analytical.

1.1 Objectual generalityWittgenstein’s account of quantification is rather sparse, mentioning only the N -operator, its application to finite lists, and its application to the collections of terms ofbracket expressions constructed by a second method. According to T5.501, this sec-ond method consists in “giving a function f x, whose values for all values of x are thepropositions to be described.” Suppose, then, that some bracket expression presentsthe values of f x. Then, the result of attaching the sign of joint-denial to the bracketexpression amounts, in some sense, to the same as the ostensibly Principian formula∼(∃x). f x. Wittgenstein seems to intend this mysterious “function” f x to be some-thing like a propositional function in the sense of Principia. However, it would appearto be only in the 3.31s that there appears anything approaching an explanation: “ifwe turn a constituent of a proposition into a variable, there is a class of propositionswhich are values of the resulting variable proposition” (3.315). Very roughly speak-ing, the second method might now be summarized as follows. The result of turning aconstituent of a proposition into a variable is supposed to determine a propositionalfunction; then an existential generalization is to be analyzed as the negation of thedirect denial of the totality of the function’s values.

1.1.1 The grammar of quantification

The abstractness of this account has abetted some controversy. In particular, Fogelin(1976) argued that since Wittgenstein does not mention any device for indicating thescope of the generalization, therefore he cannot distinguish between, say, the negationof the joint denial of the values of a function, and the joint denial of their negations.However, Soames (1983, 583ff) responded that Wittgenstein does not in his brief re-marks purport to analyze quantification by supplying a notational system with def-inite syntax. Rather, the proposal purports merely to schematize the constructionof one truth-condition from others. Thus, Fogelin’s argument can be taken to show,

6

unsurprisingly, that any particular instance of Wittgenstein’s constructional schemamust include a device for the identification of those subformulas whose semantic roleis to present the totality of values of a propositional function. Geach and Soames con-clude the debate by observing that the schema for construction of truth-conditions issatisfied by a notation tricked out with Principia-like scope indicators. Such a nota-tion can be understood as an “intended model” of the schematic description. Thus,for example, within the model one might distinguish elements NN x f x and N xN f xas respectively truth-conditionally equivalent to the disjunction and conjunction ofthe collection of values of f x. Henceforth, I will simply work in such an intendedmodel. In fact, I will assume that the model includes an image of a classical syntax ofquantificational logic, in a signature which contains at least a few individual constantsand at least a few monadic and dyadic predicate letters. This assumption can actuallybe weakened in some philosophically interesting respects, but I will not explore thepossibility here.

We can now similarly explicate the notion of propositional function. Recall thata procedure of turning a constituent of a proposition into a variable is supposed todetermine a function, which given some argument, returns a proposition. Withoutreference to any particular model of syntax, it is not at all clear just what could bemeant by “constituent” (Bestandteil), let alone by turning one into a variable. Buthaving assumed a classical syntax, then propositions may be identified with closedformulas. The notion of constituent may then be taken to include at least the indi-vidual constants which occur in the formula.

1.1.2 Higher-order logic?

One might wonder if the notion of constituent might not best be explicated to in-clude other syntactically related items as well, for example complex expressions con-structible from a proposition by abstraction. Michael Potter, for example, writes:

Take some sign expressing a proposition and single out part of the sign.Now keep this part fixed and let the rest of the sign vary. All the propo-sitions that can be obtained by varying the sign in this way form a class.The variable used to pick out this class is called a propositional function,but Wittgenstein invariably refers to it simply as a function. (Potter, 2009,269)

Although Potter cites only 5.501 in the vicinity of this proposal, its strongest supportmay come from 3.315. According to the proposal, it is not a singled-out “part” whichis initially allowed to vary, but rather “the rest” of a proposition. Potter does notactually say that the rest is a part at all, nor does he explicitly link his use of the term“part” directly to any usage in the text. But, since Wittgenstein only says that partscan be varied (or really: “turned into variables”), therefore Potter’s gloss appears topresume that for any part of a proposition, the “rest” of the proposition besides itis also a part. However, in general, it is not the case that for any part of a givenproposition, the rest of the proposition is a simple or elemental part. Thus, Potterwould appear to understand the usage of Bestandteil to include parts or aspects ofpropositions which are other than simple parts.

7

Both Ogden-Ramsey (Wittgenstein, 1922) and Pears-McGuiness (Wittgenstein,1961) translate Bestandteil not as “part” but as “constituent”. As with Bestandteil, onemight say that a thing which has constituents is somehow formed by them in a nat-urally privileged way. For example, molecules are constituted by atoms. In contrast,a mere “part” of something can be understood as a result of projection or abstrac-tion, as with the northern hemisphere of the earth. So, one might say that somethingwhich has constituents is already “articulated” in terms of them, for they are simpleor elemental in contrast to it; on the other hand a decomposition into mere partsin general requires choosing one path rather than another. Moreover, one might saythat something depends on its constituents, but its mere parts—like the top half ofthe earth—conversely depend on it.

The heuristic metaphysics of Russell’s ramified hierarchy of propositions providesan historically apposite example. Russell motivates the construction of the ramifiedhierarchy along an ordering of metaphysical dependence, if only as a convenient fic-tion. The ordering begins with the constitution of the universe by various objectsand relations (Whitehead and Russell, 1913, 45); these objects and relations are theconstituents of the earliest propositions (57ff). In contrast, a proposition can be seento fall in the range of various propositional functions, these functions being obtainedfrom the proposition by the abstraction, or “turning into variables” of constituents.Since propositional functions follow their values in the order of dependence, there-fore propositional functions are not constituents of their values. This motivates therequirement in Russell that a propositional function cannot take itself as argument:the function depends on its values, the argument of the function which yields a valueis a constituent of that value, and the value depends on its constituents.

There are a couple of reasons to suppose that at 3.315 and elsewhere, by BestandteilWittgenstein doesn’t just mean part, but part which is simple or elemental. The firstis circumstantial. Wittgenstein himself accepted the rendering by Ogden-Ramsey ofBestandteil as “constituent”. As I’ve just sketched, a usage of the latter term was al-ready established by Russell in the exposition of the ramified theory of types. Andin claiming at 3.333 that “the sign for a function already contains the prototype of itsargument”, Wittgenstein borrows a plank of that theory.

The second reason is internal. Wittgenstein identifies what he refers to as Be-standteile with those expressions whose meaning is chosen, or determined by arbi-trary convention. Thus, at 3.315 he says what might be turned into a variable are“those signs whose meaning is arbitrarily determined” (jene Zeichen, deren Bedeutungwillkürlich bestimmt wurde), and indeed so determined “by arbitrary convention”(willkürlich Übereinkunft). In contrast, the meaning of expressions which are not Be-standteile is determined not by arbitrary convention but by their structure, given thechoice of meanings to the Bestandteile. Thus, at 4.024 he says that to understand a sen-tence it suffices to understand its Bestandteile. And at 4.025 he remarks, heuristically,that in translation only the propositional—or perhaps here sentential—constituents(Satzbestandteile) are translated; presumably Satzbestandteil is supposed to pick outthe sort of thing that is listed in a dictionary, and so is intended, without modifica-tion by any adjective like “simple”, to evoke a contrast with expressions composedof several words. Now, at 4.025 Wittgenstein does mention Bindewörter—translated“conjunctions”—among the Satzbestandteile which are translated. But, this is because

8

4.025 really presents an heuristic analogy of the dependence on convention of propo-sitions with the dependence on convention of unanalyzed sentences of English orGerman. Ultimately, there is only one essential respect in which the identity of aproposition must depend on a conventional choice of meanings, and this is the as-signment of meanings to names. Thus, at 4.5, Wittgenstein proposes to give a generaldescription of symbols, such that everything satisfying the description can express asense given only a suitable choice of meanings of names. And this matters philosophi-cally, because it’s by boiling down the choice of meaning to the choice of meanings ofnames that Wittgenstein works out what’s announced as his “fundamental thought”,that logical constants are not representatives (4.023).

So, Bestandteile are the parts of a proposition whose meaning is chosen, or imme-diately fixed by convention. And the parts of a proposition whose meaning is cho-sen are precisely the names, i.e., the simple propositional constituents. But, at 3.315Wittgenstein describes propositional functions as the results of turning into variablesnot mere parts, i.e., Teile, but Bestandteile, or constituents. Hence, the sort of thingwhich yields a propositional function upon being turned into a variable is a name.This, incidentally, demystifies what in Russell is the obscurity of turning a propo-sitional constituent into a variable: a variable, or “variable name”, is simply what aname becomes upon the local abrogation of that arbitrary convention on which thename depended for its meaning.

Now, it’s true that some kind of higher-type quantification appears to be men-tioned twice in the Tractatus (3.333 and 5.5261). The two mentions can both beglossed with multi-sorted first-order quantification, which entails no increase in logi-cal complexity (Shapiro, 1999, 74). Still, Wittgenstein does leave room for some kindof higher-order generality. For, at 5.501, Wittgenstein introduces his three ways ofpresenting propositional multiplicities without clearly suggesting that the ways areexhaustive. In particular, predicative higher-order generality might be introducedthrough some fourth or fifth method of presentation of propositional multiplicities.

However, the range of any such higher-order generality would be constrained bythe requirement of wellfoundedness of logical evaluation. To see this, consider, forexample, “Napolean has all the properties of a great general.” This sentence can beseen as predicating something, say B , of Napoleon; let’s call the sentence B[n]. Now,further suppose that B[n] generalizes over everything predicable of Napoleon. Then,the truth-value of B[n] would be determined as the joint denial of each negation of asentence C [X ]→ X [n], for X anything predicable of Napoleon. So B itself falls inthe range of X , and the determination of the truth-value of B[n] would depend onthat very determination. In other words: the conception of generality as dependingfor its truth-value on those of its instances rules out higher-order quantification whichis impredicative.

So, while Wittgenstein’s remarks do leave room for some kind of “higher-order”quantification, this would require a predicative interpretation. Precisely what expres-sive power Wittgenstein might’ve sought from such supplementations of the threestated methods of construction remains, to my knowledge, an open question in theliterature.

In sum, a cursory examination of the text suggests that the notion of turning aconstituent into a variable is reasonably explicated on the syntactical model of propo-

9

sitions as replacing a constant term a with a variable term x. Just as propositionsbecome closed formulas, propositional functions are canonically explicated here asformulas containing at least one free variable.

1.1.3 Identity

This syntactical model of propositional functions suggests a natural account of theircourses of values. We might identify the application of a propositional function withthe instantiation of a formula. So the totality of values of a propositional functionwould become the class of closed instances of a formula, and an existential generaliza-tion becomes the disjunction of the elements of such a class. However, the reality isnot quite so simple. As Hintikka (1956) and Wehmeier (op. cit.) have shown, Wittgen-stein introduced a reinterpretation of the bound variable, which helps to sustain acontention that the equality predicate is dispensable. This reinterpretation, whichI’ll call the “sharp” as opposed to the “natural” reading, amounts to requiring that abound variable omit from its range those objects which are mentioned in its scope.Thus, the result of replacing a free variable with an appropriately sorted constant ina formula is a value of the function if and only if the constant does not already occurin the formula. Or equivalently, a proposition is a value of a propositional function ifand only if that function is the result of turning some constant into a variable in theproposition.

This amendation yields a sharp divergence from classical first-order semantics.Hintikka and Wehmeier have shown, in a sense to be made precise, that every truth-condition of a formula of first-order logic with equality under the natural interpre-tation is the truth-condition of a formula of first-order logic without equality underthe sharp interpretation. The translation of a sharply into a naturally read formularequires just two changes. Each existential quantification is disjoined with its omittedinstances. And, each predication of equality is replaced with either a tautology or acontradiction containing the same constants and free variables, according as the twoarguments of the predication, as linguistic expressions, are the same or distinct.

The Hintikka-Wehmeier result certainly helps to justify Wittgenstein’s claim thatthe equality predicate is dispensable. But the justification is not obviously complete,for as Rogers and Wehmeier (2012, 547) point out, the result has a seemingly im-portant qualification: the sharp reading affords an equality-free rephrasing of truth-conditions only when the class of all structures is restricted to those in which nodistinct constants codenote. So, the translations eliminate the equality predicate onlyif there is no nontrivial distinction to be drawn between structures according as theydo or do not assign the same denotation to a given term. In response to this apparentdifficulty, Rogers and Wehmeier (2012, 546) cite Wittgenstein’s remark (5.526) that“the world can be completely described by means of completely generalized proposi-tions”. But this remark could be used to show that there don’t need to be any simplepredications of equality only by being taken to show that there don’t need to be anyelementary propositions. It’s more likely that at 5.52 means just that there are somecompletely generalized propositions such that their truth leaves no further questionhow the world is. It doesn’t follow that generalities could be susceptible to truthand falsehood independently of their instances. For example, waxing psychologistic

10

Wittgenstein says “the understanding of general propositions depends palpably onthat of the elementary propositions” (4.411). It’s unlikely that 5.526 is an offhandremark that the truth-functionality thesis is optional. Rather, the purported elimina-tion of the equality predicate would appear to require some independent justificationfor the claim that simple predications of equality do not distinguish between struc-tures.

Wittgenstein’s position regarding the equality predicate cannot be just that theequality predicate is, like disjunction, not primitive, for its purported uses are not allaccommodated. But nor does Wittgenstein hold that the conventions governing thesign underwrite no symbols at all; one might express by their means a proposition tothe effect that there are at least two authors of Principia Mathematica. The position israther that the conventions do not suit the logicosyntactic role of a predicate. Specif-ically, the basic function of an equality predicate would have to be, completed withsome other occurrences of terms, to yield a proposition which is true iff those termsdenote the same object. But, claims Wittgenstein, there is no such function: whatwould fulfill such a purported function must be either nonsensical or empty (5.5303).

I suggest that the situation should be understood like this. Wittgenstein derivedhis notion of proposition from early Russell, for whom at the end of analysis eachobject has only one representative, which is that very object. Then for anything atall, there is only one true proposition to the effect that it is the same as a given object,namely the proposition that the thing is the same as itself. And as Russell remarked:“when any term is given, the assertion of its identity with itself, though true, is per-fectly futile” (Russell, 1903, 65). Famously, he proceeded in “On Denoting” (1905) toexplain the value of apparent statements of identity on the ground that they aren’tidentities after all. The hypothesis that each object has one and only one represen-tative reduces the truth or falsehood of simple predications of equality to their ownmere identity as propositions, making it plausible that they draw no genuine distinc-tions between structures.

Now, the Russellian conception of propositions also makes it intelligible to sup-pose that “the specification of all true elementary propositions describes the worldcompletely” (4.26). For example, if it is given that there are just two elementarypropositions, that Lisa and Lucy are cats on the mat, then since the propositions them-selves contain cats, the specification that both are true determines that there are twocats on the mat. If, on the other hand, the propositions instead contained proxies notin one-one correspondence with the objects, then knowing the truth-values of ele-mentary propositions would leave open the question of how crowded the mat is. ForWittgenstein maintains that two objects might have all their properties in common(5.5302).

But as is well known, Wittgenstein did pull away from the Russellian conception.On the one hand he maintained that the proposition must by itself determine howthings must stand if it is true. Furthermore, he supposed that this could be the caseonly if the way constituents stand to each other in the proposition were the same as theway that objects are thereby said to stand. Yet if the constituents of the propositionjust are what is mentioned, it follows that every proposition is true. In the Tractatus,Russellian objectual propositional constituents must therefore give way to proxies.

Surely there is no initial plausibility to the suggestion that each object has one

11

and only one representative, though things do not self-represent. But Wittgensteinfully imbibed the doctrine that the world is completely determined by the truth andfalsehood of elementary propositions. So even as objects do not represent themselves,propositions retain a role in Wittgenstein’s logical thinking which requires that eachobject has one and only one representative. Or as Russell himself put it: “there willbe one word and no more for every simple object” (Russell, 1918, 198).

In summary, then, a simple predication of equality could distinguish betweenstructures only if it has both possibilities of truth and falsehood. But it can’t haveboth possibilities, if denotation essentially puts names and objects in one-one cor-respondence. Yet only if names and objects are in one-one correspondence could aproposition be a truth-function of elementary propositions. It’s on pain of breakingthe truth-functionality thesis, then, that Wittgenstein must hold in the Tractatus thatthere is no room for the equality predicate to distinguish nontrivially between struc-tures. In this way, the proviso on the Hintikka-Wehmeier result is discharged. Theseauthors have indeed justified Wittgenstein’s claim to have recovered what could count,in the Tractatus, as the genuine expressive contribution of the equality predicate.

1.1.4 The Russellian constraint and the fixed-domain conception

On the strength of the arguments of Geach, early Soames, and Wehmeier, we canconclude that Wittgenstein manages to accommodate the basic notions of first-orderlogic. But as we’ve seen, the accommodation induces four eccentricities. First, Wittgen-stein rejects the equality predicate. Second, he recaptures some lost expressiveness bystipulating that a variable omits from its range what is mentioned in its scope. Third,he maintains that no further first-order expressiveness remains uncaptured, by stipu-lating that no two constants codenote. Finally, the analysis of quantified propositionsas truth-functions requires that every element of the domain is denoted by a constant.

The point of this paper is to develop a complexity-theoretic analysis of Wittgen-stein’s logical system. From this point of view, the work of earlier interpreters li-censes abstraction from the first two eccentricities. For the translations of Hintikkaand Wehmeier establish that the consequence relations determined by the sharp andnatural semantics of first-order formulas are mutually Turing-reducible (i.e., each isdecidable given an oracle for the other). It is not straightforward to extend these trans-lations to the outer reaches of Wittgenstein’s system; but enough will be clear forpresent purposes. The second pair of eccentricities reflects Wittgenstein’s conceptionof the relationship between names and the universe, which differs in two fundamen-tal respects from the classical model-theoretic treatment of the relationship betweenconstants and the domain of a structure. The argument that the equality predicateis dispensable requires that simple statements of equality draw no nontrivial distinc-tions in the class of all structures, and hence that there is no structure in which distinctconstants codenote. The proposal to analyze an existentially generalized propositionas the disjunction of the values of a propositional function is extensionally adequateonly to structures in which each element of the domain is denoted by a constant.Together these requirements are equivalent to what I’ll call a “Russellian” constrainton structures, that the denotation relation be a one-one correspondence between theclass of names and the domain. The reason for the terminology is simply that the

12

constraint is canonically satisfied by a structure such that every element of its domainis the one and only name of itself.

We’ve seen that the Russellian constraint is well-rooted in the Tractatus, namelyin the doctrine that a proposition is a truth-function of elementary propositions. Tosay that a proposition is a truth-function of some others is to say that, logically, itstruth-value is a function of theirs, so that any maximal consistent choice of elemen-tary propositions and their negations entails either the proposition or its negation.But this claim has counterexamples if distinct constants codenote, and has counterex-amples if not all elements of a domain must be denoted by constants. Hence thetruth-functionality thesis evidently presupposes the Russellian constraint.

It might be wondered whether the truth-functionality thesis somehow obfuscatesthe very notion of a quantifier. However, the truth-functionality thesis follows fromthe Russellian constraint. And the Russellian constraint is only a restriction on theclass of all structures. Hence, the truth-functionality thesis cannot by itself entail anysubstantive change in the theory of truth-conditions. It modifies only the universeof relata to which that theory might be applied, namely the universe of structures.So, the truth-functionality thesis does not affect the meaning of quantifiers, if themeaning of quantifiers depends only on their role in fixing truth-conditions.

On the other hand, the truth-functionality thesis does modify the notion of con-sequence, which Wittgenstein does give a proto- model-theoretic analysis. Say that anelementary truth-possibility is a maximal consistent set of elementary propositionsand negations thereof, and that the truth-grounds of a proposition are the elemen-tary truth-possibilities which entail it. Then a proposition is a consequence of someothers iff each truth-ground of all those others is also a truth-ground of it. So, forexample, a universal generalization becomes a logical consequence of the class of itsinstances. Thus, the truth-functionality thesis does modify the meaning of quanti-fiers if their meaning is taken to depend essentially on their role in constituting theconsequence relation. However, Wittgenstein may well have maintained that the roleof subsentential expressions in constituting the consequence relation is exhausted bytheir contribution to truth-conditions. Thus, the charge that truth-functionality ob-fuscates the quantifiers depends on a substantial philosophical presumption whichWittgenstein would have rejected.

Wittgenstein’s analysis of the consequence relation resembles that of Tarski (1936).The similarity is not just that both analyses generalize over something like logicallypossible distributions of truth and falsehood. For Tarski seems furthermore to havedefined the consequence relation under a “fixed-domain” conception of structure, ac-cording to which they all have the same domain in common (Bays, 2001). The fixed-domain conception and the Russellian constraint are closely related, for they bothrequire that the domains of all structures have the same cardinality. However, thefixed-domain conception of structure is not as strong. For it implies neither that ev-ery object be denoted by a constant, nor that no constants codenote. Conversely,the fixed-domain conception rounds out a development of logic under the Russellianconstraint, since the simplest bijection is the identity, and the generalization to arbi-trary bijections has no complexity-theoretic significance: it amounts only to addingisomorphic copies of structures already admitted.

The two conceptions differ sharply in their effects on the complexity of logic.

13

Although the fixed-domain conception does weaken the classical consequence rela-tion, nonetheless a Lowenheim-Skolem argument shows that, at least on a first-orderlanguage, the completeness theorem applies and the consequence relation remains re-cursively enumerable. In contrast, in §4 the Russellian constraint will be shown, onmere grounds of complexity, to preclude anything like a proof-theoretic analysis ofconsequence which is independent of the cardinality of the domain.

1.2 Formal generalityWe have now considered two of the three methods of presenting propositional mul-tiplicities which Wittgenstein sketches at 5.501. I’ve indicated how they can togetherbe understood to yield something like the expressive resources of first-order logic, butalso argued that this understanding must inflect the underlying proto-model-theoreticanalyses of validity and consequence. Let’s now turn to the third method, whichWittgenstein describes as “giving a formal law” according to which the presented sen-tences are constructed. The sentences so presented constitute the terms of what hecalls a “form-series”. Wittgenstein gives some further explanation in the 5.252s, andan example, though without the notation, appears even earlier:

The general term of the formal series a,O’a,O’O’a, . . . I write like so:“¹a, x,O’xº”. This expression in brackets is a variable. The first term ofthe bracket expression is the beginning of a formal series, the second theform of an arbitrary term x of the series, and the third the form of thatterm of the series which immediately follows x.

Series which are ordered by internal relations I call form-series. Theseries of numbers is ordered not by an external relation, but by an internalrelation. Similarly the series of propositions aRb , (∃x) :aRx . xRb , (∃x, y) :aRx . xRy . yRb , etc. (4.1252; cf. 4.1273.)

As Geach (1981, 171) observed, Wittgenstein here announces an intention, by meansof a so-called formal series, to construct an expression of the ancestral. Frege’s famousmethod of expressing the ancestral involved second-order quantification. In contrast,Wittgenstein begins with the natural idea of constructing the countably infinite dis-junction of all propositions to the effect that b is connected to a by R through thisor that number of steps. However, he then proposes a notation—the third kind ofbracket-expression—by means of which a series of disjuncts would be presented in fi-nite space. The ancestral would be expressed by the negation of the joint denial of thesentences which the bracket expression presents.

So, the form-series variable extends first-order logic to include the simulated pres-ence of some countably infinite disjunctions. Clearly, not every countably infiniteset of formulas will correspond to some such simulated disjunction, for the disjunctsmust be generated by means of what Wittgenstein calls an “operation”. In turn, anoperation should return some sentence B when applied to a sentence A only in virtueof some “internal relation” which B bears to A. Indeed, as Sundholm (1992, 61) pointsout, the apostrophe in Wittgenstein’s notation O’a apparently borrows from the de-scription in Principia of the object to which a bears the relation O. Thus, the questionwhat counts as an operation reduces to the question which relations are “internal”.

14

Wittgenstein’s remarks about technical matters are sketchy in general. But theremarks about quantification borrow some clarity from the analyses of Russell andFrege. And of course quantifiers became a standard part of the logical education ofphilosophers. On the other hand, the origins of the concept of operation are obscure(but see Floyd, 2001), and corresponding extensions of quantificational logic nevergot much traction.

So, it’s acknowledged that the form-series device is supposed to yield an expres-sion of the ancestral, and also generally supposed that it’s not the notion of ances-tral, but some more basic notion of operation. There are few published attempts topropose any general comprehension principles for operations which would suffice toexpress the ancestral. Geach (1981, 170) suggests “the notation [aRb ,aS b ,aR/S b ]gives us the series of propositions aRb ,aR/Rb ,aR/(R/R)b etc. ad inf.. . . .” Here,an expression like aR/Rb is presumably supposed to abbreviate an ordinary first-order formula. However, the disabbreviation for nonelementarily expressed rela-tions is left obscure; and the proposal makes the concept of operation seem tailoredto a particular one of what are presumably various possible uses. In a similar vein,Potter (2009, 272) writes: “the formal series [of 4.1252] is expressed by the variable[aRb , aχ b , (∃x) :aRx . xχ b ]”. The grounds of this declaration aren’t made explicit:it’s not clear why, for example, the third term of the series of indicated propositionsshouldn’t be something like (∃x) : aRx . (∃x) : xRx . xRb . The notation proposed byPotter appears to indicate what’s intended only granted an understanding of what thewhole thing is supposed to mean. So the proposal runs afoul of Wittgenstein’s com-plaint about Russell, that “he had to mention the meaning of signs when establishingthe rules for them” (3.331). Of course, one might question the importance of fidelityto standards which Wittgenstein propounded and didn’t fulfill himself. But the tech-nical details do become pressing when, for example, formulating the volume of a cubein terms of its side length requires that form-series be nested.

There has also been disagreement about what should count as an operation andwhat shouldn’t. For example, Sullivan (2004) contends that no internal relation dis-tinguishes nontrivially between names, or—though this terminology is neither Sulli-van’s nor Wittgenstein’s—that internal relations are “permutation-invariant” (see alsoSundholm, 1992, 69). On the other hand, Ricketts (2012) conjectures that form-seriesdisjunctions might serve to simulate predicative higher-order existential generaliza-tion. This requires enumerating the class of open formulas of the form Rx b for allconstants b , which won’t be underwritten by any permutation-invariant relation be-tween names. Since Wittgenstein’s direct explanations are so sketchy, disagreementslike this probably need to be resolved on systematic grounds, which must be left tofurther work. In this paper, I will adopt a conservative policy: to assume that an op-eration can discriminate between finitely many names, those which occur explicitlyin its own presentation.

1.2.1 Motivations for form-series

Textual evidence suggests that two motives drove Wittgenstein to this supplementa-tion of the system. Wittgenstein seems to have postulated quite early the presence ofsome propositional structure which shares the necessities inherent in a fact. The pos-

15

tulation gets expressed in a remark, at 4.023, that thanks to some “logical scaffolding”,a person can “see” how things must be if a proposition is true. An example from theNotebooks of this sort of visibility might be the following:

For example, perhaps we assert of a patch in our visual field that it is tothe right of a line, and we assume that every patch in our visual field isinfinitely complex. Then if we say that a point in that patch is to the rightof the line, this proposition follows from the previous one, and if thereare infinitely many points in the patch then infinitely many propositionsof different content follow LOGICALLY from that first one. And thisof itself shews that the proposition itself was as a matter of fact infinitelycomplex. That is, not the propositional sign by itself, but it together withits syntactical application.(Wittgenstein, 1979, 18.6.15g)

A year earlier, Wittgenstein had identified a similar complexity in the fact that a chairis brown, and sketched a formal series of propositions which would reproduce it(1979, 19.9.14, pp. 5 and 134).

The second motivation for the form-series device stems from problem of givingthe general propositional form. Although the problem is mentioned already in the1914 notes of Moore (1979, 113) it is not until April 1916 we find a sketch of Wittgen-stein’s eventual approach: “suppose that all simple propositions were given me: thenit can simply be asked what propositions I can construct from them. And these areall propositions and this is how they given” (16.4.16; cf. T4.51). So, the notion ofform-series here finds a second use, in an account of the way in which propositionsare constructed by elementary propositions through repeated application of a formalprocedure. As Avron (2003) argues, this technical task provides a natural motiva-tion for adding to first-order logic a device to express the ancestral. Indeed, Goldfarb(2012) speculates that it was a conviction that logic must comprehend the grounds ofour comprehension of logic, and particularly the underpinnings of recognition of for-mulas and proofs, which sustained in Wittgenstein the view that induction is logical.

The two appearances of formal series in the pre-Tractatus manuscripts lead to twokinds of use in the Tractatus. On the one hand, T4.1252, T5.252 and T5.501 invokeand explain an “immanent” use of formal series in articulating the structure of facts;paradigmatic of such use is the expression of the ancestral. On the other hand, a “tran-scendent” use at T6 purportedly fulfills the promise of 4.51 to say how all propositionscan be constructed from certain simple ones. As Sundholm (1992, 66) remarks, thesecond use is at least superficially more complicated than the first, for it acts on ahigher-order relation which takes as one argument not a single item but a potentiallyinfinite multiplicity.

Sundholm (1992, 70) observes that the use of a form-series variable to specify thetotality of propositions courts seems to flirt with impredicativity. Why couldn’t thatvariable be used to specify the basis of a truth-operation? Here it may help to con-sider the textual history of the “immanent” and “transcendent” versions of the notionof form-series. The Prototractatus (Wittgenstein, 1971) antecedent of the explanationin the T5.252s addresses not the simpler, immanently admitted form but the higher-order form instead. So, in deriving the Tractatus from this earlier draft, Wittgenstein

16

restricted his explanation of formal series to the immanent use. This deliberate rever-sal suggests that Wittgenstein eventually decided to admit only the lower-level formunder constructional procedures described at T6 by a use of the higher-level form.

The separation of “immanent” and “transcendent” form-series variables introducesa couple of responses to the problem about impredicativity raised by Sundholm. First,it might be proposed that there is some fixed type to which all meaningful uses ofthe form-series device must conform, while allowing that the expression at T6 doesnot conform to that type. But, it might be objected that once a system of proposi-tional construction has become surveyable by means of a definite, though higher-levelform-series variable, the higher-level expression should itself become susceptible tosignificant use in a yet broader collection of sentences. On this second proposal, anysuccessful characterization of the entire system of propositions would yield eo ipso afurther method of propositional construction, which the whole would still transcend.In that case, the general form of the truth-function given at T6 would be understoodas in some sense open-ended (cf. Floyd, 2001). The challenge for this second proposalwould be to account for Wittgenstein’s early insistence that every possible form of aproposition must be foreseeable (Wittgenstein, 1979, 21.11.16).

1.2.2 Toward an explication

Luckily, the goals of this paper do not extend to explicating the general propositionalform in detail. But the above gloss would license one helpful assumption, that theform-series method mentioned at 5.501 isn’t intended to handle the apparently morecomplicated induction of T6. Rather, following the 5.2522s, we might naturally con-sider only form-series variables constructed from unary operations. However, eventhe notion of unary operation is of course still unclear, and no purportedly exhaustiveexplication could be uncontroversial. For the sake of establishing lower bounds onthe complexity of the system, then, it would be best to make assumptions about whatcounts as an operation which are as weak as possible. The idea I propose to inves-tigate is roughly this. An operation can be presented by a schematic letter togetherwith a formula containing the letter; the presented operation returns the result ofsubstituting the operand for the letter in the presenting formula.

A bit more concretely, suppose that

ξ 7→Ω(ξ ) (1)

is a procedure which, applied to a formula, returns another formula. Then in some-thing like Wittgenstein’s notation,

¹A,ξ ,Ω(ξ )º (2)

would signify the series of formulas

A,Ω(A),Ω(Ω(A)) . . . . (3)

Now, consider a first-order formula B which is ordinary except in that someplacewhere an atomic formula might have occurred, there occurs instead a schematic letter

17

p. This can be supposed to determine, with respect to p, a function

ξ 7→ B[p/ξ ] (4)

which, applied to the formula A, returns the result B[p/A] of replacing each occur-rence of p in B with A. In turn, the notation

¹A,ξ ,B[p/ξ ]º (5)

can be understood to signify the series of formulas

A,B[p/A],B[p/B[p/A]], . . . . (6)

Of course, there are many other reasonable notions of operation besides those of theform of (4). But I’ll consider just these. This justifies a notational economy: insteadof (5) we may write simply

¹A, p,Bº. (7)

As we’ll see in §3.2, this ostensibly weak reconstruction suffices to express all no-tions which can be defined by induction, including in particular the ancestral. Therewill actually be various ways of doing it, but they’ll all have in common that theycontain only finitely many variables. Thus, it’s clear that they don’t express the an-cestral in quite the way that Wittgenstein envisaged at 4.1252. And more generally,it is certainly not the case that every formal procedure hazarded by Wittgenstein hasthe form A 7→ B[p/A]. Conversely, however, it is plausible that everything of thatform should count as a formal procedure.

On the present reconstruction, the form-series device does not simulate the occur-rence of infinitely many variables in a formula. The form-series Wittgenstein gives at4.1252 to express the ancestral does use infinitely many variables, though of courseinessentially. One might take this to license essential uses too. A familiar conceptwhich might then become expressible is the quantifier “there are infinitely many”.The interpretive grounds for such a strengthened line of reconstruction should besupplemented with some systematic considerations. Wittgenstein does say that num-ber is a “formal” concept (4.1272); perhaps then “there are infinitely many” oughtto be a logical notion too. But if so, then why not “there are uncountably many”or “inaccessibly many”? So far as I know, there isn’t any direct textual evidence thatWittgenstein contemplated using the form-series device to express transfinite cardi-nality concepts. So although the logicality of an infinity quantifier doesn’t seem tobe precluded by the text, the quantifier is not logically definable by induction, yetinduction handles all the envisaged applications of the form-series device for whichthere is textual evidence.

1.2.3 Programmatic background

Let me conclude this section by indicating what I take to be the philosophical signifi-cance of the two logical eccentricities to be investigated in this paper—the restrictionto Russellian structures and the addition of the form-series device. In the preface to

18

the book Wittgenstein promises to solve philosophical problems by clarifying the na-ture of logic. A superficial survey indicates that the centrally organizing task of thebook is to give a general propositional form, a purported common nature of what-ever can be said or thought. It is not clear at first blush how such a task could itselfcomplete the understanding of logic which is needed for the solution to philosophicalproblems.

In the Tractatus, the purported general propositional form gets presented twice.At first, Wittgenstein puts it pretty tersely:

The general propositional form is: es verhalt sich so und so.

But, no amount of squinting through that demonstrative so und so should be supposedby itself to yield the desired clarity.

Thankfully, this isn’t the whole story. Consider, for example, the following re-mark.

A proposition constructs a world with the help of a logical scaffolding,so that one can actually see from the proposition how everything standslogically if it is true. (T4.023e)

Such scaffolding may be supposed to constitute the sort of logic whose misunder-standing is the purported source of philosophical problems. If that’s right, the so-lution to philosophical problems might be seen to require attention to the nature ofthose manipulations of signs through which signs come to have sense: that is, to layout the pieces of scaffolding clearly. Wittgenstein announces such a consideration ofdetails at T5, which leads at T6 to a refinement of the general propositional form:

A proposition is a truth-function of elementary propositions. (T5)

The general form of the truth-function is [ p, ξ ,N ’(ξ )]. That is the gen-eral form of the proposition. (T6)

So, Wittgenstein’s promised solution to philosophical problems would appear todepend, at least in part, on the progress from T4.5 to T6. In particular, the progress issupposed to clarify just how it could be of necessity that if some propositions are true,then some other propositions are true as well. A fundamental problem of logic, then,becomes to clarify how the structure of propositions constrains what distributionsof truth and falsehood are possible for them. Possibilities reappear as distributionsof truth-value over propositions, so that a notion of what might be the case can ariseonly after there has been fixed some notion of what can be said. Moreover, solvingphilosophical problems becomes navigating propositional structure, and ultimatelydelineating the generation of propositions from elementary ones by means of a formalprocedure. For this reason, it becomes natural to suppose that the notion of formalseries, or, more idiomatically, of the “and so on”, belongs among “that with which wehave to do in logic.”

As I’ll show below, this program leads to a realization of logic whose complex-ity turns on the number of objects that exist. In particular, if the number of objectsmight be uncountably infinite, then the complexity of the resulting notion of tautol-ogy seems to imply that there is no reasonable sense in which, after all, logical validity

19

or logical consequence could be mere matter of propositional structure. Under theassumption that validity and consequence must depend only on propositional struc-ture, the significance of the main results of this paper can be stated as follows: it isif and only if the number of objects is countable that the conception of logic in theTractatus might be coherent.

2 FrameworkWe’ve now seen that Wittgenstein’s conception of logic differs from the classical un-derstanding of first-order logic in two ways. First, the notion of structure is subjectedto a Russellian constraint, so that the elements of its domain are in one-one correspon-dence with the constants of its signature. Second, the logical vocabulary contains adevice for the expression in finite space of countably infinite disjunctions. In this sec-tion, I’ll present a framework for the study of these features, and prove some basicfacts about them. But let’s begin by rehearsing a classical understanding of first-orderlogic.

A signature S of a language of first-order logic consists, for all k, of a set of k-ary function symbols and a set of k-ary predicates. A structure M for a signature S

consists of a nonempty set, |M|, plus a mapping which takes k-ary function symbolsto k-ary functions on |M|, and which takes k-ary predicates to k-ary relations on |M|.

A logic maps a signature S to a collection of classes of S-structures, the so-calledelementary classes over S. Typically, the mapping is given by the construction ofa system of formulas, plus a relation of satisfaction defined by induction on theircomplexity. In particular, first-order logic determines a class of formulas over S asfollows. First, there is a set of individual variables x, y, . . . , x0, y0, . . ., which, giventhe function symbols of S, determines the class of terms of the language. Second,the logical vocabulary ¬,∨,∃,= determines the class of formulas, in a way which isassumed to be familiar. The concept of satisfaction of an formula by an assignmentover a structure should also be familiar.

As usual, we will use ∀, ∧, etc., as metalinguistic abbreviations. Each occurrenceof a quantifier ∃x in a formula A has the form ∃xB ; the formula B is called its scope.A variable x in A is said to occur free wherever it occurs in the scope of no quantifica-tion ∃x in A, and it is said to occur bound by any occurrence of ∃x in whose scope itoccurs free. We write A[x/t ] for the result of replacing each free occurrence of x in Awith the term t , unless this replacement introduces new bound occurrences. Vectorsover terms for expressions indicate finite sequences of expressions of the indicatedtype. Then A[~x/~t ] signifies the result of simultaneously substituting each elementof the sequence ~t for the corresponding element of the sequence ~x, relettering vari-ables bound in A where necessary. Finally, A[~a] abbreviates A[~x/~a] where ~x is thecanonically ordered sequence of all variables occurring free in A.

2.1 Russellian first-order logicFor explicating the Tractatus, the notion of signature is somewhat too general. LikeRussell, Wittgenstein allows function symbols only of arity zero, which are also known

20

as constants. Moreover, he may be understood to require that the sets of functionand relation symbols of his system each be nonempty. Let’s say that a signature R isRussellian if it meets both of these conditions. And, let’s say that the restriction offirst-order logic to Russellian signatures is the logic L.

The notion of structure is too general as well. Wittgenstein at least contemplatesa requirement that no two 0-ary function symbols denote the same thing. And hewould appear to require that every element of the domain is the denotation of such asymbol. Let’s say that a structure M for R is Russellian if the domain of M is the setD=DR of constants ofR, and if for each constant a the denotation of a inM is simplya. I will just refer to a Russellian structure over a domain D as a D-structure. Theconvention that constants denote themselves is purely for notational convenience, andall subsequent results will extend transparently to the general case in which denotationis a bijection.

The notion of a D-structure being a mere specialization of the notion of struc-ture, it requires no change in the resulting notion of truth. However, instances of thespecialization enjoy the following feature: that M |= ∃xA iff M |= A[x/a] for somea ∈D. In other words, the specialization toD-structures implies that generalities maybe evaluated by instantiation.

Corresponding to the notion of D-structure we adapt the familiar analyses of va-lidity and consequence.

Definition 1. A sentence A over the signature S is D-valid if M |=A for all D-structuresM for S. Likewise, A is a D-consequence of the set X of sentences provided that there isno D-structure M for S such that M 6|=A while M |= B for all B ∈X .

We write X ⇒D A to mean that A is an D-consequence of X .The notion of D-consequence explicates some basic slogans of the Tractatus. Ac-

cording to 4.26, “the world” is supposed to be completely described by specifyingwhich elementary propositions are true and which false. So, say that the diagram∆(M) of a structure consists of the atomic sentences true in M, plus the negations ofthe atomic sentences false in M.

Proposition 1. Let M and M′ be D-structures. If M′ |=∆(M), then M′ =M.

Proof. Clearly |M|= |M′|, since M and M′ are both D-structures. Moreover, supposeR is anR-predicate. Then (~a) ∈ RM iff R~a ∈∆(M), and R~a ∈∆(M) iff (~a) ∈ RM′

.

When structures are axiomatizable up to identity, consequence becomes a gener-alization of truth. Or in other words, truth is what follows from a diagram.

Proposition 2. Let M be a D-structure. Then M |=A iff∆(M)⇒D A.

Proof. SinceM |=∆(M), therefore∆(M) 6⇒D A ifM 6|=A. Conversely, supposeM |=A. By Proposition 1, if M′ |= ∆(M), then M′ =M, so M′ |= A. Hence ∆(M)⇒D

A.

Of course, it is the left-to-right direction of Proposition 2 which does not hold forthe classical consequence relation. See Martin-Löf (1996) for discussion.

21

Another slogan of the Tractatus is that a proposition is a “truth-function” of ele-mentary propositions. In the present framework, this slogan becomes that each sen-tence or its negation is a Russellian consequence of the diagram of a structure. Itfollows immediately from Proposition 2.

Proposition 3. ∆(M)⇒D A iff∆(M)⇒D ¬A.

Proof. Of course M |=A iff M 6|= ¬A. But by Proposition 2,∆(M)⇒D A iff M |=A,and M 6|= ¬A iff∆(M) 6⇒D ¬A.

Consider, for example, the concept of arithmetical truth. Let R be a Russelliansignature with a couple of three-place predicates S and P , while the set D of constantsof R is countably infinite. Now, let N be an D-structure which is isomorphic to thestructure of the natural numbers under the ternary relations of sum and product.

Proposition 4. Under the D-consequence relation, the set ∆(N) axiomatizes N up toidentity. Thus, the set of arithmetical truths is decidable relative to the set ofD-consequencesof∆(N).

Proof. Immediate from Propositions 1 and 2.

Of course, there’s nothing special about the structure N; Proposition 4 holds forany mathematical structure at all, granted the intelligibility of quantification overarbitrary subclasses of its domain.

2.2 Form-seriesLet’s now introduce an extension of first-order logic to explicate the idea of formalseries. Roughly speaking, the point is to express some countably infinite disjunctions,provided that their disjuncts by the results of applying repeatedly some a finitely pre-sented “operation”. In §1.2.2, I proposed to consider only operations ξ 7→ B[p/ξ ] ofsubstituting the operand for a schematic letter in a formula.

Let’s correspondingly extendL to a logicLF. The underlying notions of signatureand structure remain unchanged. To generate the formulas of LF from a signature,we introduce alongside the atomic formulas an infinite collection of schematic lettersp, q , . . .. Moreover, we now introduce, alongside the usual rules for construction offirst-order formulas, two more rules like this:

• each schematic letter is an atomic formula, and

• the form-series expression∨

¹A, p,Bº is a formula whenever A and B are for-mulas, provided that B contains no form-series expression whose middle termis p.

The rider on the second clause is not essential, but simplifies some syntactical calcu-lations. Let’s call the middle occurrence of p in ¹A, p,Bº a binding of p; the scopeof the indicated binding of p in ¹A, p,Bº is the formula B . By the rider, we can thensay that a binding of p binds every occurrence of p in its scope; an occurrence of p isfree iff not bound. A formula is said to be proper if no schematic letter occurs free in

22

it. The category of improper formulas is somewhat analogous to that of a first-orderformula which is not closed; it serves mainly to streamline the syntactical inductions.We now define an appropriate notion of substitution of formulas for schematic let-ters: the formula B[p/A] is the result of substituting A for every free occurrence of pin B . Finally, if C = ¹A, p,Bº, then we write C 0 for B and C i+1 for B[p/C i ].

To construct a definition of truth for LF it is tempting simply to extend the clas-sical semantics directly, evaluating bound variables by assignment or instantiation.However, this does not work. The reason is that, roughly speaking, we would likeform-series formulas to behave like countably infinite disjunctions—but they’re not.To see the problem, note that the substitution of terms does not commute with theexpansion of form-series formulas into their infinitary counterparts. Consider, forexample, a formula A=

∨

¹F x, p,∃x(Gx ∧ p)º. Applying directly the usual notionof substitution of terms would yield A[x/a] =

∨

¹F a, p,∃x(Gx ∧ p)º, and so, forexample, A[x/a]1 = ∃x(F a ∧Gx). On the other hand, A1 = ∃x(Gx ∧ F x), and soA1[x/a] = ∃x(F x∧Gx). As Kaplan (1986) has observed, quantification doesn’t quiterequire the good behavior of substitution, but here the mischief is deeper: the in-troduction of form-series into first-order logic disrupts the expected notion of freeoccurrence of a variable.

One solution to this problem is simply to run two stages of semantic evaluation.First expand form-series variables into the formal series they present, and only then

evaluate the quantifiers. In more detail, let LF be the result of adding to the for-mation rules of L the construction of countably infinite disjunctions. We wish to

define a mapping · · · from formulas of LF to those of LF so that A =∨

i∈ωAi ifA=∨

¹B , p,Cº, while · · · otherwise passes without effect to the immediate subfor-mulas.

However, we need to make sure that the recursion purportedly definitive of · · ·actually terminates. Define the level of a formula A to be the pair (iA, jA) such thatiA is the number of distinct letters p, q , . . . which occur in A, and jA is the height ofthe formation tree of A. Now say that A has lower level than B iff either iA < iB , oriA= iB and jA< jB . Thanks to the rider on the definition of form-series expressions,it’s clear that each step in the evaluation of · · · pushes its application onto formulas ofstrictly lower level.

We now extend the definition of truth to LF. Let |=∞ be the extension to LF ofthe notion of truth for L.

Definition 2. A sentence A of LF is true in M, or M |=A, iff M |=∞ A.

Since the definition of truth does not proceed by induction on syntactical com-plexity, it would be a pleasant surprise to find that LF enjoys an extensionality prop-erty familiar for first-order logic. To this end, we record a purely syntactical lemma,that the expansion of a form-series formula into its infinitary counterpart commuteswith substitution of formulas, at least in good conditions.

Lemma 1. If∨

¹A, p,Bº is a proper formula, then B[p/A] = B[p/A].

23

We first derive from the lemma the promised extensionality and another result,returning to the lemma’s proof to conclude the section.

Proposition 5. Suppose that the free variables of each of A,B are just ~u, and that C [p/A],C [p/B]are closed and proper. If

M |= ∀~u(A↔ B)

thenM |=C [p/A]↔C [p/B].

Proof. By the definition of |= and · · ·, the supposition implies

M |=∞ ∀ ~v(A↔ B).

Hence by the extensionality of infinitary logic,

M |=∞ C [p/A]↔C [p/B],

so by Lemma 1

M |=∞ C [p/A]↔C [p/B].

The result now follows by the definition of |=.

While in first-order logic, the syntactical complexity of formulas coincides withthat of their logical complexity, in LF the two notions diverge. Lemma 1 lets us

put a number on this. In LF, define a notion ρ of rank as the natural measure oflogical complexity of infinitary formulas: ρ(A) = 0 for A atomic, ρ(¬A) = ρ(∃xA) =ρ(A) + 1, ρ(A∨ B) = max(ρ(A) + ρ(B)) + 1, and ρ(

∨

X ) = supρ(A) : A ∈ X + 1.Now the logical complexity of a formula of LF can be identified with the rank of its

expansion: ρ(A) = ρ(A).

Proposition 6. The supremum of the ranks of LF-formulas is ωω .

Proof. Using Lemma 1,

ρ(C [p/B]) = ρ(C [p/B])

= ρ(C [p/B])

= ρ(B)+ρ(C ).

So

ρ(∨

¹B , p,Cº) = supk∈ω(ρ(B)+ρ(C )× k)+ 1

= (ρ(B)+ρ(C )×ω)+ 1.

Andωω is the least λ satisfying 0< λ, β< λ→β+ 1< λ and β,γ < λ→ (β+ γ ×ω)+ 1< λ.

24

Finally the proof of Lemma 1. It is a hairball, but self-contained.

Proof. We argue by induction on the level of B . Consider the only nontrivial case,where B =∨

¹C , q , Dº. The definition of · · · implies

B[p/A] =∨

k∈ωq[q/D[p/A]]k[q/C [p/A]]︸ ︷︷ ︸

8k

(8)

but alsoB[p/A] =∨

k∈ωq[q/D]k[q/C ][p/A]︸ ︷︷ ︸

9k

. (9)

By (8) and (9), it therefore suffices to establish 8k = 9k, for all k ∈ω.Since C , D have lower level than B , the induction hypothesis implies

q[q/D[p/A]]k = q[q/D[p/A]]k (10)

and similarly

C [p/A] =C [p/A]. (11)

Together (10) and (11) imply

10L︷ ︸︸ ︷

q[q/D[p/A]]k[q/

11L︷ ︸︸ ︷

C [p/A]]︸ ︷︷ ︸

8k

=

10R︷ ︸︸ ︷

q[q/D[p/A]]k[q/

11R︷ ︸︸ ︷

C [p/A]] (12)

Now a couple of tools to bash away at the right-hand side of (12). First, note thatif p does not occur in E , then

E[q/F [p/A]] = E[q/F ][p/A], (13)

for all E and F . Furthermore, if A is proper, it follows that p, q do not occur in A.This implies

E[p/A][q/F ][p/A] = E[q/F ][p/A] (14)

for all E , F .Using induction on k, the observations (13) and (14) give

13L︷ ︸︸ ︷

q[q/D[p/A]]k =

13R︷ ︸︸ ︷

q([q/D][p/A])k︸ ︷︷ ︸

14L

=

14R︷ ︸︸ ︷

q[q/D]k[p/A]. (15)

25

Hence15L︷ ︸︸ ︷

q[q/D[p/A]]k[q/C [p/A]]︸ ︷︷ ︸

12R

=

15R︷ ︸︸ ︷

q[q/D]k[p/A][q/C [p/A]] (16)

Using (13) and (14) again,

q[q/D]k[p/A]

13L︷ ︸︸ ︷

[q/C [p/A]]︸ ︷︷ ︸

16R

= q[q/D]k [p/A]

13R︷ ︸︸ ︷

[q/C ][p/A]︸ ︷︷ ︸

14L

= q[q/D]k

14R︷ ︸︸ ︷

[q/C ][p/A].︸ ︷︷ ︸

9k

(17)

Now 8k = 9k follows by (12), (16), and (17).

3 ApplicationsIn §1.2, we saw that Wittgenstein introduced the device of formal series to expressthe ancestral and other inductively definable notions. And in §2.2, we found an ap-proximation of this device which appears to simulate some countably infinite disjunc-tions. However, the terms of such a simulated infinitary disjunction are generated bythe iterative application of a procedure chosen from a quite narrowly defined class.So it is not obvious a priori that this reconstruction actually increases the expressivepower of the logic: does it, for example, suffice to express the ancestral? Moreover,the reconstruction seems to be motivated mainly from considerations of syntacticalmanageability. So even if some new things can be said, it ought still to be wonderedwhether they have any naturally unifying semantical characterization. In this section,I’ll show that adding the form-series device to first-order logic coincides exactly, fromthe point of view of expressiveness, with the result of adding a capacity to form fini-tary inductive definitions. The essential idea, attributed by Barwise (1977) to ArthurRubin, is that a finitary inductive definition can be regarded as a countably infinitedisjunction which contains only finitely many variables.

3.1 Definable operatorsThe expressive capacity of a logic can be understood to consist in its power to draw dis-tinctions in the class of all structures. By determining a notion of truth of a formula,the logic determines the notion of truth-condition, that is, of a mapping from struc-tures to truth-values. Now a logic L1 can be said to be at least as expressive as logic L2if every truth-condition of a formula of L2 is the truth-condition of a formula of L1,or in other words L2 ≤L1. So understood, the notion of expressive capacity depends

26

on the notion of structure. And we’ve just seen that the notion of structure appro-priate to the Tractatus is nonstandard, in restricting the class of all structures to thosewhich are Russellian with respect to a given signature. So, to each domain of Russel-lian signature there corresponds a comparison ≤D. Since the D-truth-condition ofa formula is simply the intersection of its classical truth-condition with the class ofall D-structures, it follows that L1 ≤ L2 implies L1 ≤D L2 but not conversely, so ≤draws finer distinctions. In this section, I’ll compare logics via the classical notion ofstructure, not the restricted one. However, it will be clear that all noninclusions tobe established here carry over to ≤D for all infinite D, while none do for D finite.

Formally, truth can be seen as a 0-ary relation, and satisfaction, in turn, can be seenas a 0-ary mapping on the class of relations. Thus, the comparison of expressivenessvia truth-conditions can be generalized. An open formula A in k free variables issaid to define, over a structure M, the k-ary relation which holds of those ~a suchthat M |= A[~a]. By generating a collection of formulas over a signature, a logic itselfdetermines a class of satisfaction conditions, or mappings from structures to classesof k-tuples. The resulting comparison is more fine-grained than the comparison viatruth-conditions. For example, while sharp first-order logic expresses all of the truth-conditions expressible by natural first-order logic, this is not the case for satisfaction-conditions.

Taking another step back, mappings on relations are sometimes called “operators”(e.g., Aczel, 1977), though there is no reason to infer anything from this terminolog-ical coincidence with the Tractatus. Suppose R is a k-ary predicate not in the under-lying signature, let A be a formula maybe containing R and with n free variables ~x,and write M,X for the result of expanding M to interpret R as the k-ary relation X .Over a structure M, say that the formula A defines through R, ~x the k , n-ary operator

|A|M : X 7→ (~a) : M,X |=A[~a],

where X is an arbitrary subset of |M|k . If furthermore

|A|M(X ) = Γ (X )

for all M,X , then let’s say that A defines absolutely, through R, ~x, the operator Γ .In first-order logic, there are a couple of ways that the notion of definable relation

gets applied: first, through the concept of satisfaction, in evaluating generalizations ofthe defining formula, and second, through immediately evaluating the instantiationof the defining formula by some constants. In a logic without second-order generality,the definability of operators can be applied only through instantiation of schematicletters to open formulas; but that is always possible according to the following conse-quence of Proposition 5.

Proposition 7. The class ofLF-definable relations is closed under application of definableoperators.

Proof. Suppose the formula A in the free variables ~x defines the relation |A|M, andthat B defines through R, ~y the operator |B |M, where the length of ~x is the arity of R.Then

M, |A|M |= B[~a] iff (~a) ∈ |B |M(|A|M). (18)

27

We can assume B to have the form B ′[p/R~x] where R doesn’t occur in B ′. NowProposition 5 implies

M |= B ′[p/A][~a] iff M, |A|M |= B[~a], (19)

and the claim follows by (18) and (19).

In a broader classification of operators, application would be a special case of com-position. This suggests the following generalization.

Proposition 8. The class of LF-definable operators is closed under composition.

Proof. Suppose the formula A defines through R, ~x the operator |A|M, and that Bdefines through S, ~y the operator |B |M, where the length of ~x is the arity of S. Then

M, |A|M(X ) |= B[~a] iff (~a) ∈ |B |M(|A|M(X )). (20)

We can assume B to have the form B ′[p/S~x] where S doesn’t occur in B ′. By Propo-sition 5,

M,X |= B ′[p/A][~a] iff M, |A|M(X ) |= B[~a] (21)

and the claim follows by (20) and (21).

3.2 InductionInductive definition can be seen as a transformation on operators themselves. That isto say, consider a k , k-operator Γ . Since a result of Γ has the same arity as its basis, it canbe applied to its own results. Write I0

Γ (X ) = X , and IαΓ (X ) =⋃

β<α IβΓ ∪ Γ (⋃

β<α IβΓ )

for α > 0. Since β < α implies IβΓ (X ) ⊆ IαΓ (X ), there must be some λ such thatIλ+1Γ (X ) = IλΓ (X ). Then IΓ (X ) = IλΓ (X ) is the relation which Γ is said to define by

induction at X , and the earliest such λ is the ordinal of Γ at X .An inductive definition may be said to be “finitary” if its ordinal is ω. Many

familiar inductions are finitary. For example, let X be the set of all triples of the form(a, 0,a). Let Γ (Y ) be the result of adding to Y all triples (a, b , c) such that (a, d , e) ∈Xfor some d , e such that b = d + 1 and c = e + 1. The ordinal of Γ at X is ω, and thisdefines inductively the class of triples of the form (a, b ,a+ b ).

We can now say that a logic expresses induction provided that whenever an op-erator Γ is definable over a structure, the operator IΓ is definable over that structuretoo. Of course, here there is only a question of expressing induction which is fini-tary. Thus, for all finitary induction to be expressible, it suffices that whenever Γ isdefinable over some structure, then so is IωΓ .

Proposition 9. LF expresses all finitary induction.

Proof. Suppose that A defines through R, ~x the k , k-ary operator |A|M. Without lossof generality, we can assume A to have the form A′[p/R~x], where R doesn’t occur inA′. So,

M,X |=A[~a] iff (~a) ∈ |A|M(X ) (22)

28

for all ~a,X ,M.Now define IA=

∨

¹R~x, p,A′º. We claim that I kA defines Ik

|A|M through R, ~x for all

k, so that IA itself will define Iω|A|M . This is clear for I 0A = R~x. So, suppose I k

A defines

Ik|A|M . Then M,X |= I k

A[~a] iff (~a) ∈ Ik|A|M(X ). And so, using Proposition 5 and the

induction hypothesis,

M,X |= I k+1A [~a] iff M,X |=A′[p/I k

A][~a]

iff M, Ik|A|M(X ) |=A[~a]

iff (~a) ∈ |A|M(Ik|A|M(X )) = Ik+1

|A|M(X ).

So I k+1A defines Ik+1

|A|M(X ), as desired.

As a first application of Proposition 9, let’s consider the ancestral.

Proposition 10. In the logic LF, the ancestral operator is definable.

Proof. Let A be the formula ∃w(S xw∧Rwy) for new predicates R, S, and write M,Xfor an expansion of M to interpret R as X . Then

|A|M,X (X k ) = (a, b ) : (a, c) ∈X k and (c , b ) ∈X for some c=X k+1.

So,Iω|A|M,X (X ) =⋃

k∈ωX k .

By Proposition 9, some formula IA defines through S, x, y the operator Iω|A|M,X . More-

over, |Rxy| defines X over M,X . So

|IA|M,X (|Rxy|M,X ) = Iω|A|M,X (X ).

By Proposition 7, there’s a formula which defines |IA|M,X (|Rxy|M,X ) over M,X .Over M itself, this formula—call it Ancestral— defines the ancestral operator; butsince the formula is given independently of M, it follows that Ancestral defines theancestral operator absolutely.

Toward a second illustration, note that the concepts of addition and multiplicationcan be seen as 2,4-ary operators which take a notion of “successor” to relations of sumand product on the chain of successors so determined. More generally, a k-ary relationρ on the natural numbers becomes understandable as the 2, k+1-ary operator Γ ρ suchthat

(a, b1, . . . , bk ) ∈ Γρ(X ) iff (a, b1) ∈X n1 , . . . , (a, bk ) ∈X nk . . .

for some (n1, . . . , nk ) ∈ ρ. (23)

Proposition 11. The addition and multiplication operators are LF-definable.

29

Proof. Since these definitions are supposed to handle arbitrary relational structures,they work a bit differently than usual. They’ll allow any particular object to play therole of zero, but will insist that only things connected to the zero object can play therole of other numbers relative it.

For the case of addition, begin with a formula A0 which recognizes any quadruple(a,a,a,a) as an instance of the fact that the sum of an initial ordinal with itself is itself,namely w = x ∧ x = y ∧ y = z. Next take a formula A0, which recognizes additionto be closed under the operation of at once taking successor in the left summand andin the sum, namely ∃x1∃x3(Sy0x1y2x3 ∧ Rx1y1 ∧ Rx3y3). Finally let A2 recognizeaddition to be closed under taking successors at once in the right summand and in thesum, namely as the formula ∃y2∃y3(S z0z1y2y3 ∧Ry2z2 ∧Ry3z3). Then

|A0|M,X = (a,a,a,a) : a ∈ dom(M)

Iω|A1|M,X (|A0|M,X ) = (a, b ,a, c) : (a, b ) ∈X k ∧ (a, c) ∈X k

for some k

Iω|A2|M,X (Iω|A1|M,X (|A0|

M,X )) = (a, b , c , d ) : (a, b ) ∈X j ∧ (a, c) ∈X k

∧ (a, d ) ∈X j+k for some j , k.

By Propositions 7 and 8, it follows that some formula, call it Plus defines the relationIω|A2|M,X (I

ω|A1|M,X (|A0|M,X )) over M,X ; this formula therefore defines the addition op-

erator over M itself. But M was arbitrary, so Plus deserves its name.The treatment of multiplication is similar. After treating the case of a zero mul-

tiplicandum separately and taking the multiplication of one by itself as the base case,simply close the class of nonzero multiplications under the operations of at once tak-ing successor in one multiplicand and adding the other to the product. Call the re-sulting absolute definition of the multiplication operator Times.

Proposition 12. Ifρ is an arithmetically definable relation on the natural numbers, thenΓ ρ is an LF-definable operator.

Proof. It suffices to show that for any arithmetically definable relation ρ, there is anoperator Γ ρ satisfying (23) above. We argue by induction on the complexity of for-mulas in a relational language of arithmetic 0,′ ,+,×.

The proof is trivial. Let φ? be the image of φ under the translation

• 0(x) 7→ x = w; ′(x, y) 7→ Rxy;+(x, y, z) 7→ Plus(w, x, y, z);×(x, y, z) 7→ Times(w, x, y, z);x = y 7→ x = y;

• ¬φ 7→φ?; φ∨ψ 7→φ? ∨ψ?; ∃xφ 7→ ∃x(Ancestral(w, x)∧φ?).

Now define Aφ to be the formula Ancestral(w, x1)∧ . . .∧Ancestral(w, xk )∧φ?, wherex1, . . . , xk are the variables free in φ. Clearly, if φ defines ρ, then Aφ defines Γ ρ.

From a given relation X , the addition and multiplication operators determinerelations which look as much addition as X looks like successor. For example, if

30

X looks like the successor relation on the hours of a clock, then the correspondingaddition and multiplication relations look like those of arithmetic modulo twelve.However, in the odd event thatω itself happens to be lying around somewhere, thenthat can certainly be reported. This observation is essentially due to Goldfarb (2012).

Proposition 13. Suppose that the domain of R is infinite, and that R contains a dyadicpredicate. There is a single formula A of LF such that M |= A iff M contains a corre-sponding copy of the natural numbers under the successor relation. Moreover, if M |= A,then any counterpart of an arithmetically definable relation is definable on M.

Proof. Immediate from Propositions 10 and 12.

3.3 ComparisonsWe’ve got a lower bound on the expressiveness of LF, and hence of the result ofadding to first-order logic the form-series device as explicated here. I will now showthat this is also an upper bound. That is to say, on the present explication, and mod-ulo first-order logic, the “immanent” form-series device exactly coincides, in pointof expressiveness, with finitary inductive definition. So somewhat surprisingly, thesimplistic, syntactically motivated idea of iterated formula-nesting receives a naturalsemantic characterization.

There are various implementations of the idea of “adding induction”. For presentpurposes, a convenient approach is an extension of first-order logic, call it here LI,which appears in Gurevich and Shelah (1986, 272). Begin by adding to the logicalvocabulary an infinite collection of schematic predicate-letters of each arity. Furtheradd a symbol I such that (IR,~x A) is a k-ary predicate whenever A is a formula possiblycontaining k-ary predicate R, and ~x has length k. Write M,X for the expansion ofM which assigns R the extension X . Now, define |I0

R,x A|M = and |Ik+1R,~x

A|M =|Ik

R,~x A|M ∪ (~a) ∈M : M, |IkR,~x A|M |= A[~a]. Finally, we extend the notion of truth

so that M |= (IR,~x A)~a iff (~a) ∈⋃

k∈ω |IkR,~x A|M.

Proposition 14. LI is exactly as expressive as LF.

Proof. It suffices to find translations τ,τ1 between the formulas of LI and those ofLF which preserve satisfaction conditions. The translations need only change theform-series disjunctions and the inductive predications.

In one direction, suppose that A is a formula of LI; we can assume A has the form|IR,~x B |. Suppose Γ is the operator B defines over M. By Proposition 9, there is aformula B ′ such that M |= B ′[~a] iff (~a) ∈ Γω, but by induction hypothesis and thedefinition of truth for LI, this is precisely iff M |= (IR,~x Bτ)~a. Conversely, supposethat A has the form

∨

¹B , p,Cº. Let D = (¬∃~xR~x ∧ Bτ1) ∨ (∃~xRx ∧C [p/R~x]τ1).It’s then routine to verify by induction on k that |Ik+1

R,~xD |M =⋃

j≤k(~a) : M |=∨

¹B , p,Cºk[~a] for all k.

Let’s now compare LF with some other extensions of first-order logic. First, itis obvious from the semantics that LF is a subsystem of the infinitary logic Lω1ω

31

which results by adding to first-order logic countably infinite disjunctions. However,even L∞ω has no wellfoundedness quantifier (Lopez-Escobar, 1966). Indeed, LF is asubsystem of the fragment Lωω1ω

in which no formula contains infinitely many vari-ables (Barwise, 1977); and Lωω1ω

lacks a quantifier “there are infinitely many”. Thesystem LF is also related to some subsystems of second-order logic. For example,the Π1

1 fragment of second-order logic (Π11SOL, as defined in Heck (2011)) expresses

finitary inductive definitions, so it is at least as expressive as LF. But it also has aninfinity quantifier, so it is strictly more expressive. The situation is different withmonadic second-order logic (MSOL), which a priori expresses finitary inductive def-initions only of monadic properties. And indeed, since the monadic second-ordertheory of the successor relation is decidable (Büchi, 1960), it follows that unlike LF,monadic second-order logic cannot define both addition and multiplication from suc-cessor. Conversely, the monadic fragment, like the Π1

1 fragment, has a wellfounded-ness quantifier, which is not available even in full Lω1ω

. Thus monadic second-orderlogic and LF are incomparable. Strictly weaker than all of these systems is the resultLA of adding an ancestral operator to first-order logic. In sum, we have

Proposition 15. Write X → Y to mean that logic Y is strictly more expressive thanlogic X . Then

Lω1ωΠ1

1SOL M SOL

Lωω1ω

OO

LQ∞

bb ;;

LQWF

cc ;;

LF

ii

OO

LA.

jj

OO

4 DefinabilityOne way to characterize the consequence relation in first-order logic is through auniversal generalization over structures. Since a structure consists of a maybe infinitecollection of objects plus some relations on the collection, this analysis makes the con-sequence relation look unmanageable: to verify that the condition is fulfilled, it looksas though we’d have to run through an infinity of in general infinite structures andcheck whether each is a countermodel. However, first-order logic admits a completenotion of proof: accordingly, whenever some formula is a first-order consequence ofsome others, some finite pattern of formulas gives an effective witness. So, to deter-mine that the consequence relation obtains, it turns out to suffice to enumerate thefinite patterns of formulas until a proof appears. This collapse in the complexity ofthe consequence relation is a fairly special property of first-order logic. Slight enrich-

32

ments of the logic tend to complicate the consequence relation and outrun any systemof finite witnesses.

In this section, we turn to the problem of characterizing the complexity-theoreticeffects of the Russellian semantics and of the form-series device. On the one hand,the Russellian constraint winnows the class of countermodels, so that the answers toold questions may change. On the other hand, the form-series device introduces newformulas, hence raising more questions.

The results of this section can be summarized as follows. By itself, Russellian con-straint broadens but does not significantly complicate the class of validities. However,on classical semantics, it’s compactness which ensures that if validity is witnessed byfinite proofs, then so is consequence. And D-consequence is not compact, because forexample a universal generalization is a D-consequence of the class of its instances. SoD-consequence might turn out to be more complicated than validity. And indeed, theeffects are as strong as possible: D-consequence permits no simplification of the anal-ysis of consequence through universal generalization over structures. On the otherhand, in the case of D countably infinite, the form-series device makes the concept ofvalidity already that bad regardless of the Russellian constraint. But, the form-seriesdevice also has the effect of concentrating, into the notion of validity, the complica-tions of the consequence relation which follow from the Russellian constraint. Thisimplies, in the general case, that the concept of tautology, or of logically valid formula,is not even Σ1 definable in set theory. So in the general case, it is not just that we can-not replace the search through the collection of structures on some infinite domainwith an enumeration of finite proofs. Rather, validity cannot in general be witnessedby any system of mathematical constructions which are identifiable by properties in-trinsic to the constructions themselves. Rather, it depends essentially on the extrinsicmatter of which bijections happen to exist in the mathematical universe.

4.1 Measuring definitionsLet’s start with a simple framework for measuring the complexity of logical notions,which we can then apply to the notions of D-validity and D-consequence for L andLF. The oldest measure of logical complexity is given through the theory of the the-ory of computability of functions on the natural numbers. Its application to logic de-pends on the technique of arithmetization, according to which formulas are “coded”as natural numbers. In the present context, arithmetization is somewhat annoying,since it prejudges some nontrivial interpretive questions. First, it applies only if thesignature is countable. Since a Russellian signature includes the domain of its Rus-sellian structures, this excludes structures with uncountable domains. Second, arith-metization yields the definability of, e.g., the class of names coded by the even in-tegers. But as indicated in §1.2, it is a matter of interpretive dispute whether, forexample, there should be a definable enumeration of an infinity of names-in-the-sense-of-Tractatus. Even if the answer were actually yes, it needn’t be yes in exactly the wayarithmetization requires.

A solution to this problem appears in Barwise (1975, 78ff): code syntactical con-structions not arithmetically but set-theoretically, treating logical vocabulary as puresets, and the terms of the signature as urelements. In a little more detail, suppose S

33

is a structure. Let HF(|S|) be the class of hereditarily finite sets over the domain |S|of S; thus the elements of HF(|S|) are generated, given the elements of |S| initially,by repeatedly forming all finite sets of what’s obtained already. This determines afirst-order structureHF(S)whose domain is HF(|S|), together with the natural mem-bership relation HF, the property of belonging to the domain |S| of urelements, andeach of the relations and functions baked into S itself.

We’ll be specially concerned with HF(R) for R not any old structure, but a Rus-sellian signature considered as a structure. So understood, the domain of R consists ofits nonlogical vocabulary, and the relations of R identify the logical types of the ele-ments of its domain. But, this can be simplified. By stipulating thatR contains exactlyone predicate of each arity, the domain of R reduces to the collection D of its names.Mathematically, this is no essential restriction, since arity-index pairs can be coded asarities. Since for present purposes we need not consider distinctions of logical typebetween the elements of the domain of R, there is no need for R, as a structure, tocontain its own type-theoretical vocabulary. Thus, we may simply identify R with D

altogether.The classHF(D) can now be regarded as the universe of possible syntactical con-

structions from the initially given assortment of nonlogical vocabulary. Atomic for-mulas are coded in HF(D) as finite sequences of elements of D. Logical vocabularyshould be coded by pure elements of HF. Nonatomic formulas now result from theatomic formulas through this or that natural, pure, and “effective” set-theoretic con-struction.

The structureHF(D) is of course a structure for a first-order language whose twononlogical predicates are those of membership and urelementhood, or ∈ and D. Thecomplexity of a class on HF(D) can now be measured by the logical complexity ofthe simplest formulas which define it. I’ll just sketch the portion of the frameworkwe’ll need; the details can be found in Barwise (1975). A formula is∆0 if it is built upfrom atomic formulas by negation, disjunction, and bounded existential quantifica-tion ∃x ∈ y . . .. A formula isΣ1 (orΠ1) if it’s the result of prefixing a∆0 formula witha sequence of existential (universal) quantifiers. Now the complexity of a subclass ofHF(D) is given by the complexity of the simplest formula which defines it.

The definability-theoretic characterization of complexity of classes of hereditarilyfinite sets generalizes naturally the recursion-theoretic measures of complexity on thenatural numbers. Note that each natural number is uniquely represented as a sumof powers of two, and so it can be taken to code the set of whatever is coded by thepowers (Ackermann, 1937). This gives a bijection between the naturals and HF. It’snow straightforward to verify that a set of integers is recursive iff it’s the image of a∆0 class on HF (Barwise, 1975, 47ff), likewise for r.e. and Σ1, and so on. As Kirby(2009) argues,HF can therefore be seen as a natural home of finitary constructions.

Moreover, the extension ofHF toHF(D) preserves the alignment of definability-theoretic and computability-theoretic classifications. Perhaps the clearest way to bringthis out is to suppose that D is countably infinite, so that there is a bijection f fromHF’s copy of ω to D. Now HF contains a copy of HF(D, f ) in terms of whichquestions about pure sets posed to HF(D, f ) can be rephrased without any jump inquantifier complexity. So, every class of pure sets which is∆0 onHF(D, f ) is already∆0 on HF; and of course everything ∆0 on HF(D) is ∆0 on HF(D, f ). Since the

34

property of being a pure set is∆0 onHF(D), everything∆0 onHF is∆0 onHF(D),and the chain is closed.

It should be clear that the simplifying assumptions underlying this particular de-velopment can be discharged at the expense of some extra notation. With a little morefootwork, one can take as primitive not the collection of names, but the collection ofelementary propositions together with a system of relations of sameness of predica-tion (cf. Fine, 2000, 20ff). Such an approach can be seen to integrate the semantics ofWehmeier Wehmeier (2004, cf. §1.1.3), but the development of this idea must be leftto future work.

4.2 Finite controllabilityLet’s now apply this framework to analyzing various notions of the theory of L andLF. The first couple of results should be reassuring.

Proposition 16. Suppose D is finite. Then D-validity for LF is ∆0 on HF(D), andD-consequence for L is Σ1.

Proof. Consider the infinitary expansion A of a formula A. Since D is finite, we maytake B to be the result of inductively replacing each existentially quantified subfor-

mula of A with the finite disjunction of its D-instances. Let C be a countable dis-junction of B which has no countable disjunctions as subformulas. Then C has theform∨

(D , E[p/D], E[p/E[p/d ]], . . .), where D and E are constructed from some katomic formulas by negation and finite disjunction, though E may also contain p. Soby Proposition 5, C must be equivalent to the disjunction of its first 22k

disjuncts, andwe can just drop the rest. Let A↓ be the formula, constructed from atomic sentencesby negation and finite disjunction, which results by eliminating from B in this wayeach of its infinitary subformulas. Clearly there is a Σ1-definable function onHF(D)associating A↓ to A. But validity for finitary formulas is∆0.

If D is finite, then D-consequence is compact. So, A is an D-consequence of Xiff there’s a conjunction B of elements of X such that B → A is valid; this can beexpressed as a Σ1 formula onHF(D).

Let’s now turn to the case of an infinite domain. The second reassuring result isthat, if we drop form-series from the logic, and consider only the notion ofD-validity,then this is no more complicated than usual.

Proposition 17. D-validity for L is Σ1 onHF(D).

Proof. It suffices to consider the case where D is infinite. Let I be a collection of L-formulas to the effect that “there is at least one thing, there are at least two things,. . . ”. For A a formula of L, Let TA be an L-formula a 6= b ∧ a 6= c ∧ . . . to the effectthat no two names in A denote the same thing.

We now argue that A is D-valid iff A is a classical consequence of I ∪TA. In onedirection, suppose that A is not D-valid, so that M 6|= A for some D-structure M.Clearly M |= TA since A is an D-structure, and M |= I since D is infinite. Hence Ais not a classical consequence of I ∪ TA. Conversely, suppose that M |= I ∪ TA but

35

M 6|= A, the classical signature of M consists just of the nonlogical vocabulary of A.Since M |= T , it follows by the Lowenheim-Skolem theorems that M is elementarilyequivalent to a structure M′ whose domain is D. Since M′ |= TA, therefore M′ isisomorphic to an M′′ such that aM

′′= a for each constant a which occurs in A. In

turn, M′′ may be expanded to an D-structure M′′′ such that M′′′ 6|=A.The completeness theorem for first-order logic implies that A is a classical conse-

quence of T ∪IA iff there is a proof of A from T ∪IA. Formally, however, the property“being a proof of A from T ∪IA” is∆0 onHF(D), so that “having a proof from T ∪IA”is Σ1. From the previous paragraph, it follows that the collection of D-valid formulasis Σ1 onHF(D).

So, D-validity is never more complicated than classical validity. Does this alsohold for D-consequence? By the compactness theorem, classical consequence is nomore complicated than classical validity. But if D is infinite, then D-consequence isclearly not compact. For example, a universal generalization is a D-consequence ofthe set of its instances, but not of any finite subset. In other words, the complexity ofthe D-consequence relation remains to be determined.

4.3 Countability and categoricityThe second batch of results characterize the case in which the domain is countablyinfinite. Recall, from §2.1 and §3.2, that the Russellian constraint and the form-seriesdevice independently lead to categorical axiomatizations of the standard model ofarithmetic. Of course, this capacity extends to similar countably infinite structures aswell—including, if D is countable,HF(D) itself. This leads to an amusing applicationof Tarski’s theorem, which shows that validity is not first-order definable onHF(D).By a similar argument we then conclude that in fact validity is Π1

1-complete.

Proposition 18. Suppose D is countably infinite. Then (i) HF(D) is characterizable upto isomorphism by a single formula of LF, and (ii) there’s a set of L-formulas whose onlyD-model is isomorphic toHF(D). Likewise forHF(D, f ).

Proof. Write D and E for a monadic and dyadic predicate of L; these can serve, insideL or LF, to express the counterparts of the properties “x is an urelement” and “xbelongs to y”.

For (i), note that HF(D) can be seen as the smallest set containing all urelementsand closed under the operation x, y → x ∪ y. This suggests that the class of D-structures which are isomorphic toHF(D) can be defined by a single formula of LF.Write Adjunction[x, y, z] = ∀w(Ew z ↔ Ew x ∨ w = y). And now define the L-axiomatization ofHF(D) like this:

Z = ∀x(D x↔¬∃zE z x) . . .∧∀x∃z(D z ∧¬E z x) . . .∧∀x∀y∃!zAdjunction[x, y, z]) . . .∧∀x(IF ,x∃u∃v(F u ∧ F v ∧Adjunction[u, v, x]))x. (24)

36

The first conjunct says that urelements are like so many “empty sets”. Given theinfinitude of the underlying domain and the other conjuncts, the second conjunctensures that the extension of the L-predicate D is infinite and so has the same cardi-nality asD. The third conjunct says that the universe is closed under adjunction. Andthe fourth conjunct uses induction, as provided by Proposition 14, to say that all setsare obtained from the urelements by repeated adjunction, so that only hereditarily fi-nite sets exist. Clearly any L-structure isomorphic toHF(D) satisfies Z . Conversely,supposeM |= Z , and let h be a bijection from DM ontoD. To extend h ′ to an isomor-phism of M onto HF(D), first note by an induction justified by the fourth conjunctthat if t belongsM to s , then there’s a set s − t such that s is the adjunctionM of t tos− t ; moreover s− t is unique if not inM the class DM. Using the axiom of countablechoice, this justifies the inductive extension of h to h ′ : s 7→ h ′(s− t )∪h ′(t ), whichis the isomorphism desired.

For (ii), suppose that MHF(D) is an D-structure isomorphic toHF(D) itself. Let

∆(MHF(D)) = the diagram of MHF(D). (25)

From Proposition 1 it follows that M |=∆(MHF(D)) iff M=MHF(D).In both of cases (i) and (ii), the extension toHF(D, f ) replaces the function f with

its graph and adds the obvious axioms, either (i) saying that the graph mapsω one-oneonto the extension of D , or (ii) enumerating the graph.

Thus, for each formula φ of the metalanguage, a formula ðφñ of LF results fromφ by replacing ∈ with E and D with D . So if (|M|, DM, EM) is isomorphic to(HF(D),D,∈), then M |= ðφñ iff HF(D) |= φ. Let’s also mix variables A,B ,C overLF-formulas into ð. . .ñ too, so that e.g. ðA→ φñ is the LF-conditional whose an-tecedent is A and whose consequent is ðφñ. Note that since φ contains no constants,each object ðφñ can be assumed to belong to HF.

Now Tarski’s theorem has the following form:

Fact 1. Suppose that HF(D) |= ψ[ðφñ,X ]↔ φ for all φ. Then X is not first-orderdefinable onHF(D). Likewise forHF(D, f ).

From Proposition 18 and Fact 1, first-order indefinability is straightforward.

Proposition 19. Suppose that D is countably infinite. (i) The class of D-valid LF-formulas is not first-order definable on HF(D). (ii) Nor is the D-consequence relationfor L-formulas.

Proof. (i) Expand HF(D) to HF(D)′ so as to interpret VALID as the set of D-validformulas of LF. Let Z be as in (24). Then M |= Z iff M is isomorphic to HF(D), soZ→ ðφñ is D-valid iffHF(D) |=φ. Thus,

HF(D)′ |=VALID(ðZ→φñ)↔φ, (26)

and so (i) follows by Fact 1.

37

(ii) ExpandHF(D, f ) toHF(D, f )′ so as to interpret IMPLIES as theD-consequencerelation onL-formulas. Clearly, someL-diagram∆(MHF(D, f )) is first-order definableonHF(D, f ). So,

HF(D, f )′ |= IMPLIES(∆(MHF(D, f )),ðφñ)↔φ. (27)

By Fact 1, it follows that D-consequence is not first-order definable on HF(D, f ),hence not onHF(D) either.

In fact, something stronger is true. A formula is said to be Π11 if it is the result

of prefixing a first-order formula with a string of universal second-order quantifiers.The official definitions of D-validity and of D-consequence have the form “for allD-structures. . . ”. Moreover, D-structures can be coded as functions from finite se-quences of urelements to truth-values. And the underlying notion of truth-conditioncan be defined using a universal second-order quantifier over LF-formulas. So theofficial definitions of D-validity and D-consequence are naturally rendered as Π1

1 for-mulas overHF(D). Now the question returns: can this be simplified at all? I’ll arguethat the answer is no.

A class X is said to be Π11-complete over some structure if for every Π1

1 formulaφ, there’s a first-order formula ψ such that φ[~a]↔ ψ[X , ~a] holds for all ~a in thedomain. It clearly suffices to show that D-validity on LF and D-consequence on L

are each Π11-complete onHF(D).

Fact 2. Suppose that HF(D) |=ψ[ðφñ,X ] iff |φ|HF(D) is wellfounded, for all φ. ThenX is Π1

1-complete onHF(D). Similarly forHF(D, f ).

Proof. Let WF = ðφñ : |φ|HF is wellfounded. By a famous result of Kleene (1955),plus the arithmetization of finite set theory, the class WF is Π1

1-complete onHF.Recall that HF contains a copy of HF(D). So, HF(D) contains a pure copy of

itself; call the domain of this copy D′. Say that a purification of the sets ~s is a functionf which, in a one-one manner, swaps urelements in the transitive closures of the sets~s for elements of D′; note that the domain of f is finite and so f is a set inHF(D).

Suppose that φ is Π11. Then there’s a Π1

1 formula φ′ such that

HF(D) |=φ[~s] iffHF |=φ′[ f (~s)]

for any purification f of ~s . By Kleene’s result, there’s a first-order formula ψ suchthat

HF |= ∀~x(φ′↔ψ[WF]).

Let ψ′ be the result of restricting all quantifiers in ψ to HF. Then

HF(D) |= ∀~x(φ↔∃ f ( f is a purification of ~x ∧ψ′[WF, f (~x)])),

as desired.The claim about HF(D, f ) is proved similarly, except the built-in f works as a

purification.

38

Proposition 20. Suppose that D is countably infinite. Then (i) the consequence relation⇒D on L is Π1

1-complete onHF(D), and (ii) so is the class of validities of LF.

Proof. Let Suc be a formula ofLwhich, relative to the axiomatizations (25) and (24) ofHF(D) inside LF or L, formalizes “y is the finite ordinal which succeeds x”, relativeto the standing choice of D , E for urelementhood and membership. For some newdyadic predicate Q, there is an L-formula which says “Q enumerates an infinite chainof satisfiers of φ”, namely

Aφ = “Q is a many-one relation” . . .

∧∀x∀y(Suc[x, y]→∃v∃w(Q xv ∧Qyw ∧ ðφñ[v, w])).

Expand HF(D) to HF(D)′ so as to interpret VALID by the class of D-valid formulasof LF. Then

HF(D)′ |=VALID(ðZ→¬Aφñ) iff |φ|HF(D) is wellfounded. (28)

Moreover, ðZ→¬Aφñ is of course a first-order definable function of ðφñ. So by (28),it follows that the class of codes of formulas defining wellfounded relations is first-order definable relative to the concept of D-validity. So D-validity is Π1

1-complete byFact 2.

(ii) This is essentially similar, though we need to work in HF(D, f ). If HF(D, f )is expanded to HF(D, f )′ so as to interpret IMPLIES by the D-consequence relationfor L, then

HF(D, f )′ |= IMPLIES(∆(MHF(D, f )),Aφ) iff |φ|HF(D) is wellfounded.

Note that a priori, the definition of D-consequence is Π11, since quantification

over structures is essentially just second-order quantification overHF(D). So, Propo-sition 20 is as strong as possible. It shows that if the underlying universe is countablyinfinite, then there is no notion of proof which, as in the case of first-order logic,manifests any collapse in the complexity of the consequence relation.

It might be noted, however, that if D is countable, then the quantifiers of L orLF can be replaced with countable disjunctions, and the systems become fragmentsof countably infinite truth-functional logic. So, they do have a complete notion ofproof (Lopez-Escobar, 1965). Unlike the notion of proof for classical logic, this doesnot show the consequence relation to be any simpler than what’s given by its a prioricharacterization. Nonetheless, it might be claimed to demonstrate that the corre-sponding consequence relation is appropriately grounded in propositional structure.The evaluation of this proposal is a difficult problem, which must be left to furtherwork.

4.4 Uncountability and nonabsolutenessFinally, let’s consider what happens when the assumption of countability is dropped.Recall that the concept of being an LF-tautology, relative to a signature D, can be

39

expressed by a formula using second-order quantification over HF(D). But the defi-nition can also be phrased as a formula which uses a single first-order universal quan-tification ranging over the power set of HF(D). We can now ask whether, using theaxioms of set theory, it is possible to prove that the universal quantification can be re-placed with an existential one. By Proposition 20, the existential quantifier couldn’tjust range overHF(D), but we might hope to find some infinitary notion of proof towitness the concept of D-tautology on LF, or of D-consequence on L.

As before, it’s natural to avoid assuming that D has been coded as some pure set.But it is straightforward to adapt ZFC to handle D as a collection of urelements. Addto the language of set theory a primitive predicate D which corresponds to the prop-erty of belonging to the classD of urelements. Add an axiom to the effect that nothingin D has an element, and replace the usual axiom of extensionality with an axiom thatthings outside of D are the same if they have the same elements. It simplifies mattersto assume that the elements of D form a set, although the result established here canbe extended to the case in which the assumption is dropped.

In the resulting set theory ZFCU, the theory of logics L and LF easily results byformalizing the earlier construction which took place in HF(D). In particular, theassumption that D forms a set means that the definition of validity and consequencebecome expressible by universal generalization over subsets of D. We conclude witha result which indicates that there is no hope of finding a system of infinitary proof-theoretic witnesses to the basic metalogical notions of the Tractatus. Roughly speak-ing, the completeness of such a system of witnesses would yield a method of recog-nizing that the domain is uncountable. By the following lemma, no such method canbe provably expressed by a Σ1 formula, assuming, of course, that ZF is consistent.

Lemma 2. There is no Σ1 formula φ such that ZFCU ` Dℵ0↔φ.

Proof. Assume there were such a φ. Let V(D∩m) formalize “the class of sets x suchthat m contains every element of D which falls in the transitive closure of x.” Then

ZFCU ` m ´ℵ0→ (D´ℵ0)V(D∩m)

andZFCU ` (D´ℵ0)

V(D∩m)→¬φV(D∩m),

soZFCU ` m ´ℵ0→¬φ

V(D∩m). (29)

On the other hand, since we assumed that ZFCU ` D ℵ0→ φ, a Lowenheim-Skolem argument shows that

ZFCU ` Dℵ0 → ∃m(m ´ℵ0 ∧m is transitive∧φm).

But since φ is Σ1, therefore

ZFCU ` Dℵ0→∃m(m ´ℵ0 ∧φV(D∩m)). (30)

From (29) and (30), it follows that

ZFCU ` Dℵ0 → ⊥

which leads to a contradiction in ZF.

40

Now, let VALID and IMPLIES formalize in ZFCU the definitions of D-tautologyand D-consequence from §2. Say that a formula is Σ1-definable in ZFCU if it is prov-ably equivalent to a Σ1 formula. Then

Proposition 21. Neither VALID nor IMPLIES is Σ1-definable in ZFCU.

Proof. By Proposition 10, there is an LF-formula A which says that if R is a one-onerelation, then not everything is connected by R. It’s clear that A is a tautology iff Dis uncountable. Of course, this argument can be made in ZFCU, so that for some Σ1term α we have

ZFCU `VALID(α)↔ Dℵ0. (31)

On the other hand, suppose D is infinite, and let X be some countably infinitesubset of D. For some monadic predicate G, suppose GX is the collection of all for-mulas Ga for a ∈X , together with all formulas ¬Ga for a 6∈X . And, for some dyadicpredicate R, let B be a formula to the effect that R doesn’t map the Gs onto D. ThenB is an D-consequence of GX iff D is uncountable. Moreover, this fact is recognizablein ZFCU. That is, there is a Σ1 function symbol γx in the free variable x, and a Σ1term β, such that

ZFCU ` ∃x(x ≈ℵ0 ∧ IMPLIES(γx ,β))↔ Dℵ0. (32)

By lines (31) and (32), a Σ1 definition of either VALID or IMPLIES would yield aprovable Σ1 criterion for Dℵ0. But this is impossible by Lemma 2.

ReferencesAckermann, W. (1937). Die widerspruchsfreiheit der allgemeinen mengenlehre. Math-

ematische Annalen, 114:305–315.

Aczel, P. (1977). An introduction to inductive definitions. In Barwise, J., editor,Handbook of Mathematical Logic, pages 739–781. North-Holland.

Avron, A. (2003). Transitive closure and the mechanization of mathematics. In ThirtyYears of Automating Mathematics, pages 149–171. Kluwer Academic Publishers.

Barwise, J. (1975). Admissible Sets and Structures. Springer-Verlag, Berlin.

Barwise, J. (1977). On Moschovakis closure ordinals. Journal of Symbolic Logic,42(2):292–296.

Bays, T. (2001). On Tarski on models. Journal of Symbolic Logic, 66(4):1701–1726.

Büchi, J. (1960). On a decision method in restricted second-order arithmetic. InInternational Congress for Logic, Methodology and Philosophy of Science, pages 1–11.

Fine, K. (2000). Neutral relations. The Philosophical Review, 109(1):1–33.

Floyd, J. (2001). Number and ascriptions of number in Wittgenstein’s Tractatus. InFloyd, J. and Shieh, S., editors, Future Pasts. Harvard University Press, Cambridge,MA.

41

Fogelin, R. (1976). Wittgenstein. Routledge and Kegan Paul.

Geach, P. (1981). Wittgenstein’s operator N . Analysis, 41(4):178–81.

Goldfarb, W. (2012). Wittgenstein against logicism. unpublished.

Gurevich, Y. and Shelah, S. (1986). Fixed-point extensions of first-order logic. Annalsof Pure and Applied Logic, 32:265–280.

Heck, R. (2011). A logic for Frege’s theorem. In Frege’s Theorem. Oxford UniversityPress, New York.

Hintikka, J. (1956). Identity, variables, and impredicative definitions. Journal of Sym-bolic Logic, 21(3):225–245.

Kaplan, D. (1986). Opacity. In Hahn, E. and Schilpp, P., editors, The Philosophy ofW. V. Quine. Open Court.

Kirby, L. (2009). Finitary set theory. Notre Dame Journal of Formal Logic, 50(3):227–244.

Kleene, S. C. (1955). Arithmetical predicates and function quantifiers. Transactionsof the American Mathematical Society, 79(2):312–340.

Lopez-Escobar, E. G. K. (1965). An interpolation theorem for denumerably longformulas. Fundamenta Mathematicae, LVII:253–272.

Lopez-Escobar, E. G. K. (1966). On defining well-orderings. Fundamenta Mathemat-icae, 59(1):13–21.

Mancosu, P. (2010). Fixed- versus variable-domain interpretations of tarski’s accountof logical consequence. Philosophy Compass, 5(9):745–759.

Martin-Löf, P. (1996). On the meanings of the logical constants and the justificationsof the logical laws. Nordic Journal of Formal Logic, 1(1):11–60.

Potter, M. (2009). The logic of the Tractatus. In Gabbay, D. M. and Woods, J., editors,From Russell to Church, volume 5 of Handbook of the History of Logic. Elsevier.

Ricketts, T. (2012). Logical segmentation and generality in Wittgenstein’s Tractatus.In Potter, M. and Sullivan, P., editors, Wittgenstein’s Tractatus: History and Inter-pretation. Oxford University Press.

Rogers, B. and Wehmeier, K. (2012). Tractarian first-order logic: identity and theN -operator. Review of Symbolic Logic, 5(4):538–573.

Russell, B. (1903). The Principles of Mathematics. Allen and Unwin, London.

Russell, B. (1905). On denoting. Mind, 14(56):479–493.

Russell, B. (1918). The philosophy of logical atomism. In Marsh, R. C., editor, Logicand Knowledge. Allen and Unwin.

42

Shapiro, S. (1999). Foundations without Foundationalism: A Case for Second-OrderLogic. Oxford University Press, New York.

Soames, S. (1983). Generality, truth-functions, and expressive capacity in the Tracta-tus. Philosophical Review, 92(4):573–589.

Sullivan, P. (2004). The general form of the proposition is a variable. Mind,113(449):43–56.

Sundholm, G. (1992). The general form of the operation in Wittgenstein’s Tractatus.Grazer Philosophische Studien, 42:57–76.

Tarski, A. (1936). O pojeciu wynikania logicznego. Przeglad Filozoficzny, 39:58–68.

Wehmeier, K. (2004). Wittgensteinian predicate logic. Notre Dame Journal of FormalLogic, 45:1–11.

Wehmeier, K. (2008). Wittgensteinian tableaux, identity, and codenotation. Erkennt-nis, 69(3):363–376.

Wehmeier, K. (2009). On Ramsey’s ‘silly delusion’ regarding Tractatus 5.53. In Prim-iero, G. and Rahman, S., editors, Acts of Knowledge - History, Philosophy and Logic.College Publications.

Whitehead, A. N. and Russell, B. (1910-1913). Principia Mathematica. CambridgeUniversity Press.

Wittgenstein, L. (1921). Logische-philosophische abhandlung. Annalen der Natur-philosophie, XIV:168–198.

Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Kegan Paul.

Wittgenstein, L. (1961). Tractatus Logico-Philosophicus. Routledge & Kegan Paul.

Wittgenstein, L. (1971). Prototractatus. Routledge & Kegan Paul, London.

Wittgenstein, L. (1979). Notebooks 1914-1916. Blackwell, Oxford.

43