Although it has few adherents today, logical atomism was once aleading movement of early twentieth-century analytic philosophy.Different, though related, versions of the view were developed byBertrand Russell and Ludwig Wittgenstein. Russell’s logicalatomism is set forth chiefly in his 1918 work “The Philosophy ofLogical Atomism” (Russell 1956), Wittgenstein’s in hisTractatus Logico-Philosophicus of 1921 (Wittgenstein 1981).The core tenets of Wittgenstein’s logical atomism may be statedas follows: (i) Every proposition has a unique final analysis whichreveals it to be a truth-function of elementary propositions(Tractatus 3.25, 4.221, 4.51, 5); (ii) These elementarypropositions assert the existence of atomic states of affairs (3.25,4.21); (iii) Elementary propositions are mutually independent —each one can be true or false independently of the truth or falsity ofthe others (4.211, 5.134); (iv) Elementary propositions are immediatecombinations of semantically simple symbols or “names”(4.221); (v) Names refer to items wholly devoid of complexity,so-called “objects” (2.02 & 3.22); (vi) Atomic statesof affairs are combinations of these objects (2.01).

Although these doctrines are recognizably atomist in spirit, the term‘logical atomism’ is not used by Wittgenstein. It wasintroduced by Russell in his 1911 lecture to the French PhilosophicalSociety,Le Réalisme Analytique (Russell 1911).^[1] Russell had advertised “The Philosophy of LogicalAtomism” as being “very largely concerned with explainingcertain ideas which [he had] learnt from [his] friend and former pupilLudwig Wittgenstein” (Marsh, 177). No doubt partly as a resultof this description, the term ‘logical atomism’subsequently became associated with Wittgenstein’s earlyphilosophy. The term is now standardly used to apply to a vaguelydefined set of doctrines centered on theses (i)–(vi). InRussell’s opinion, what makes it appropriate to speak oflogicalatomism is that the atoms in question are to bearrived at by logical rather than physical analysis (Russell 1956,179). For Wittgenstein too, the ultimate constituents of reality areto be revealed by a process of logical analysis; so, to that extent,the label seems apt. It is not, however, uncontroversial (see Floyd1998 and 2007).^[2]

• 1. Names and Objects
• 2. Linguistic Atomism
  • 2.1 Wittgenstein’s Early Conception of Analysis
• 3. Metaphysical Atomism
  • 3.1 Objects as the Substance of the World
  • 3.2 The Argument for Substance
• 4. The Epistemology of Logical Atomism
• 5. The Dismantling of Logical Atomism
  • 5.1 First phase: The colour-exclusion problem
  • 5.2 Second phase: Generality and Analysis
• Bibliography
  • Primary Sources
  • Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. Names and Objects

The “names” spoken of in theTractatus are notmere signs (i.e., typographically or phonologically identifiedinscriptions), but rather signs-together-with-their-meanings —or “symbols.” Being symbols, names are identified andindividuated only in the context of significant sentences. A name is“semantically simple” in the sense that its meaning doesnot depend on the meanings of its orthographic parts, even when thoseparts are, in other contexts, independently meaningful. So, forexample, it would not count against the semantic simplicity of thesymbol ‘Battle’ as it figures in the sentence“Battle commenced” that it contains the orthographic part,“Bat,” even though this part has a meaning of its own inother sentential contexts. For Wittgenstein, however, something elsedoes count against this symbol’s semantic simplicity, namely,that it is analyzable away in favour of talk of the actions of people,etc. This point suggests that in natural language Tractarian nameswill be rare and hard to find. Even apparently simple singular termssuch as ‘Obama,’ ‘London,’ etc., will not becounted as “names” by the strict standards of theTractatus since they will disappear on further analysis.(Hereafter, ‘name’ will mean “Tractarian name”unless otherwise indicated.)

It is a matter of controversy whether theTractatus reservesthe term ‘name’ for those semantically simple symbols thatrefer to particulars, or whether the term comprehends semanticallysimple symbols of all kinds. Since objects are just the referents ofnames, this issue goes hand in hand with the question whether objectsare one and all particulars or whether they include properties andrelations. The former view is defended by Irving Copi (Copi 1958) andElizabeth Anscombe (Anscombe 1959 [1971, 108 ff]), among others. It issupported byTractatus 2.0231: “[Material properties]are first presented by propositions — first formed by theconfiguration of objects.” This might seem to suggest thatsimple properties are not objects but rather arise from the combiningor configuring of objects. The Copi-Anscombe interpretation has beentaken to receive further support fromTractatus 3.1432:

We must not say, “The complex sign ‘aRb’says ‘a stands in relationR tob;’” but we must say, “That‘a’ stands in a certain relation to‘b’ saysthat aRb.”

This has suggested to some commentators that relations are not,strictly speaking, nameable, and so not Tractarian objects (see, forexample, Ricketts, 1996, Section III). It may, however, be intendedinstead simply to bring out the point that Tractarian names are notconfined to particulars, but include relations between particulars; sothis consideration is less compelling.

The opposing view, according to which names include predicates andrelational expressions, has been defended by Erik Stenius and Merrilland Jaakko Hintikka, among others (Stenius, 1960, 61–69;Hintikka and Hintikka, 1986, 30–34). It is supported by aNotebooks entry from 1915 in which objects are explicitlysaid to include properties and relations (NB, 61). It isfurther buttressed by Wittgenstein’s explanation to Desmond Lee(in 1930–1) ofTractatus 2.01:“‘Objects’ also include relations; a proposition isnot two things connected by a relation. ‘Thing’ and‘relation’ are on the same level.” (LK,120).

The Anscombe-Copi reading treats the forms of elementary propositionsas differing radically from anything we may be familiar with fromordinary — or even Fregean — grammar. It thus respectsWittgenstein’s warning to Waismann in 1929 that “Thelogical structure of elementary propositions need not have theslightest similarly with the logical structure of [non-elementary]propositions” (WWK, 42).

Going beyond this, Wittgenstein seems once to have held that there can be no resemblance between the apparent or surface formsof non-elementary propositions and the forms of elementarypropositions. In “Some Remarks on Logical Form” (1929) hesays: “One is often tempted to ask from an a priori standpoint:What, after all, can be the only forms of atomic propositions, and toanswer, e.g., subject-predicate and the relational propositions withtwo or more terms further, perhaps, propositions relating predicatesand relations with one another, and so on. But this, I believe, is amere playing with words” (Klagge and Nordman, 1993, 30). Asimilar thought already occurs in a more compressed form in theTractatus itself: “There cannot be a hierarchy of theforms of the elementary propositions. Only that which we ourselvesconstruct can we foresee” (5.556).

It is possible, then, that the options we began with represent a falsedichotomy. Perhaps Wittgenstein simply did not have an antecedentopinion on the question whether Tractarian names will turn out to benames of particulars only, particulars and universals, or whatnot. Andperhaps he believed that the final analysis of language would (ormight) reveal the names to defy such classifications altogether. Thisbroader range of interpretive possibilities has only recently begun toreceive the attention it deserves (See Johnston 2009).

2. Linguistic Atomism

By “Linguistic atomism” we shall understand the view thatthe analysis of every proposition terminates in a proposition all ofwhose genuine components are names. It is a striking fact that theTractatus contains no explicit argument for linguisticatomism. This fact has led some commentators — e.g., PeterSimons (1992) — to suppose that Wittgenstein’s positionhere is motivated less by argument than by brute intuition. Andindeed, Wittgenstein does present some conclusions in this vicinity asif they required no argument. At 4.221, for example, he says:“It is obvious that in the analysis of propositions wemust come to elementary propositions, which consist of names inimmediate combination” (emphasis added). Nonetheless, some basicobservations about theTractatus’s conception ofanalysis will enable us to see why Wittgenstein should have thought itobvious that analysis must terminate in this way.

2.1 Wittgenstein’s Early Conception of Analysis

A remark from thePhilosophical Grammar, written in 1936,throws light on how Wittgenstein had earlier conceived of the processof analysis:

Formerly, I myself spoke of a ‘complete analysis,’ and Iused to believe that philosophy had to give a definitive dissection ofpropositions so as to set out clearly all their connections and removeall possibilities of misunderstanding. I spoke as if there was acalculus in which such a dissection would be possible. I vaguely hadin mind something like the definition that Russell had given for thedefinite article (PG, 211).

One of the distinctive features of Russell’s definition is thatit treats the expression “thex such thatFx” as an “incomplete symbol.” Such symbolshave no meaning in isolation but are given meaning by contextualdefinitions that treat of the sentential contexts in which they occur(cf.PM, 66). Incomplete symbols do, of course,havemeaning because they make a contribution to the meanings of thesentences in which they occur (cf.Principles, Introduction,x). What is special about them is that they make this contributionwithout expressing a propositional constituent. (For more on thenature of incomplete symbols, see Pickel 2013)

Russell’s definition is contained in the following clauses (Forthe sake of expository transparency, his scope-indicating devices areomitted.).

[1]G(thex:Fx) =∃x(∀y(Fy↔y=x)&Gx)    Df.
(cf. Russell 1905b; Russell 1990, 173)

[2] (thex:Fx) exists =∃x∀y(Fy↔y=x)   Df.
(cf. Russell 1990, 174)

The fact that existence is dealt with by a separate definition showsthat Russell means to treat the predicate ‘exists’ asitself an incomplete symbol.

One can understand why Wittgenstein should have taken there to be anaffinity between the theory of descriptions and his own envisioned“calculus,” for one can extract from his remarks in theTractatus and elsewhere two somewhat parallel proposals foreliminating what he calls terms for “complexes”:

[3]F[aRb] iffFa &Fb &aRb

[4] [aRb] exists iffaRb

Clauses [1] to [4] share the feature that any sentence involvingapparent reference to an individual is treated as false rather than asneither true nor false if that individual should be discovered not toexist.

Wittgenstein’s first contextual definition — our [3]— occurs in aNotebooks entry from 1914 (NB,4), but it is also alluded to in theTractatus:

Every statement about complexes can be analysed into a statement abouttheir constituent parts, and into those propositions which completelydescribe the complexes (2.0201).

In [3] the statement “about [the complex’s] constituentparts” is “Fa &Fb,” while theproposition which “completely describes” the complex is“aRb.” If the propositions obtained by applying[3] and [4] are to be further analysed, a two-stage procedure will benecessary: first, the apparent names generated by the analysis —in the present case ‘a’ and‘b’ — will need to be replaced^[3] with symbols that are overtly terms for complexes, e.g.,‘[cSd]’ and ‘[eFg];’ second,the contextual definitions [3] and [4] will need to be applied againto eliminate these terms. If there is going to be a unique finalanalysis, each apparent name will have to beuniquely pairedwith a term for a complex. So the program of analysis at whichWittgenstein gestures, in addition to committing him to somethinganalogous to Russell’s theory of descriptions, also commits himto the analogue of Russell’s “description theory ofordinary names” (cf. Russell 1905a). The latter is the idea thatevery apparent name not occurring at the end of analysis is equivalentin meaning to some definite description.

Wittgenstein’s first definition, like Russell’s, strictlyspeaking, stands in need of a device for indicating scope, forotherwise it would be unclear how to apply the analysis when wechoose, say, “~G” as our instance of“F.” In such a case the question would arisewhether the resulting instance of [3] is [5]:“~G[aRb] = ~Ga & ~Gb&aRb,” which corresponds to giving the term for acomplex wide scope with respect to the negation operator, or whetherit is: [6] “~G[aRb] = ~[Ga &Gb &aRb],” which corresponds to givingthe term for a complex narrow scope. One suspects thatWittgenstein’s intention would most likely have been to followRussell’s convention of reading the logical operator as havingnarrow scope unless the alternative is expressly indicated (cf.PM, 172).

Definition [3] has obvious flaws. While it may work for suchpredicates as “x is located in England,” itobviously fails for certain others, e.g., “x is greaterthan three feet long” and “x weighs exactly fourpounds.” This problem can hardly have escaped Wittgenstein; soit seems likely that he would have regarded his proposals merely astentative illustrations, open to supplementation and refinement.

Although Wittgenstein’s second contextual definition — our[4] — does not occur in theTractatus, it is implied bya remark from theNotes on Logicthat seems to anticipate2.0201:

Every proposition which seems to be about a complex can be analysedinto a proposition about its constituents and … the propositionwhich describes the complex perfectly;i.e., that propositionwhich is equivalent to saying the complex exists (NB,93; emphasis added)^[4]

Since the proposition that “describes the complex,”[aRb], “perfectly” is just the proposition thataRb, Wittgenstein’s clarifying addendum amounts to theclaim that the proposition “aRb” is equivalent tothe proposition “[aRb] exists.” And thisequivalence is just our [4].

It turns out, then, that existence is defined only in contexts inwhich it is predicated of complexes. Wittgenstein proposal thusmirrors Russell’s in embodying the idea that it makes no senseto speak of the existence of immediately given (that is, named)simples (cf.PM, 174–5). This is why Wittgenstein waslater to refer to his “objects” as “that for whichthere is neither existence nor non-existence” (PR, 72).His view seems to be that when ‘a’ is aTractarian name, what we try to say by uttering the nonsense string“a exists” will, strictly speaking, beshown by the fact that the final analysis of some propositioncontains ‘a’ (cf. 5.535). But of course, theTractatus does not always speak strictly. Indeed, what isgenerally taken to be the ultimate conclusion of theTractatus’s so-called “Argument forSubstance” (2.021–2.0211) itself tries to say somethingthat can only be shown, since it asserts theexistence ofobjects. The sharpness of the tension here is only partly disguised bythe oblique manner in which the conclusion is formulated. Instead ofarguing for the existence of objects, theTractatus arguesfor the thesis that the world “has substance.” However,because “objects constitute the substance of the world”(2.021), and because substance is that whichexistsindependently of what is the case (2.024), this is tantamount tosaying that objects exist. So it seems that Wittgenstein’sargument for substance must be regarded as a part of the ladder we aresupposed to throw away (6.54). But having acknowledged this point, weshall set it aside as peripheral to our main concerns.

The most obvious similarity between the two sets of definitions isthat each seeks to provide for the elimination of what purport to besemantically complex referring expressions. The most obviousdifference consists in the fact that Wittgenstein’s definitionsare designed to eliminate not definite descriptions, but rather termsfor complexes, for example the expression“[aRb],” which, judging by remarks in theNotebooks, is to be read: “a in the relation R tob” (NB, 48) (This gloss seems to derive fromRussell’s manner of speaking of complexes inPrincipiaMathematica, where examples of terms for complexes include, inaddition to “a in the relationR tob,” “a having the qualityq”, and “a andb andcstanding in the relationS” (PM, 44).). Onemight wonder why this difference should exist. That is to say, onemight wonder why Wittgenstein does not treat the peculiar locution“a in the relationR tob” as adefinite description — say, “the complex consisting ofa andb, combined so thataRb”? Thisdescription could then be eliminated by applying theTractatus’s own variant upon the theory ofdescriptions:

TheF isG ↔ ∃x(Fx&Gx) & ~∃x,y(Fx&Fy)
(cf. 5.5321)

Here the distinctness of the variables (the fact that they aredistinct) replaces the sign for distinctness “≠” (cf.5.53).

Since Wittgenstein did not adopt this expedient, it seems likely thathe would have regarded the predicate “x is a complexconsisting ofa andb, combined so thataRb” as meaningless in virtue of — among otherthings — its containing ineliminable occurrences of thepseudo-concepts “complex,” “combination,” and“constitution.” Only the first of these notions figures onhis list of pseudo-concepts in theTractatus (4.1272), butthere is no indication that that list is supposed to be exhaustive.

There is a further respect in which Wittgenstein’s analyticalproposals differ from Russell’s. Russell’s seconddefinition — our [2] — has the effect of shifting theburden of indicating ontological commitment from the word‘exists’ to the existential quantifier. InWittgenstein’s definition, by contrast, no single item ofvocabulary takes over the role of indicating ontological commitment.Instead, that commitment is indicated only after thefinalapplication of the definition, by the meaningfulness of the names inthe fully analysed proposition — or, more precisely, by the factthat certain symbols are names (cf. 5.535). The somewhat paradoxicalconsequence is that one can assert a statement of the form“[aRb] exists” without thereby manifesting anyontological commitment to the complex [aRb] (cf.EPB, 121). What this shows is that the two theories relievethe assertor of ontological commitments of quite different kinds. InRussell’s case, the analysis — our [2] — removes acommitment to an apparent propositional constituent — a“denoting concept”^[5] —expressed by the phrase ‘theF,’ but itdoes not remove the commitment to theF itself. ForWittgenstein, by contrast, the analysis shows that the assertor neverwas ontologically committed to the complex [aRb] by anutterance of “[aRb] exists.”

Russell’s conception of analysis at the time of the theory ofdescriptions — c.a. 1905 — is relatively clear: Analysisinvolves pairing up one sentence with another that expresses the verysame Russellian proposition but which does so more perspicuously. Theanalysans counts as more perspicuous than the analysandum because theformer is free of some of the latter’s merely apparentontological commitments. By the time ofPrincipiaMathematica, however, this relatively transparent conception ofanalysis is no longer available. Having purged his ontology ofpropositions in 1910, Russell can no longer appeal to the idea thatanalysans and analysandum express one and the same proposition. He nowadopts “the multiple relation theory of judgment,”according to which the judgment (say) that Othello loves Desdemona,instead of being, as Russell had formerly supposed, a dyadic relationbetween the judging mind and the propositionOthello lovesDesdemona, is now a non-dyadic, or, in Russell’sterminology, “multiple,” relation whose relata are thejudging mind and those items that were formerly regarded asconstituents of the propositionOthello loves Desdemona(Russell 1910 [1994, 155]). After 1910 Russell can say that a speaker whosincerely assertively uttered the analysans (in a given context) wouldbe guaranteed to make the same judgment as one who sincerelyassertively uttered the analysandum (in the same context), but he canno longer explain this accomplishment by saying that the two sentencesexpress the same proposition.

A further departure from the earlier, relatively transparentconception of analysis is occasioned by Russell’s resolution ofthe set-theoretic version of his paradox. In this resolution one givesan analysis of a sentence whose utterance could not be taken toexpressany judgment. One argues that the sentence“{x:φx} ε {x:φx}” is nonsense because the contextual definitionsproviding for the elimination of class terms yield for this case asentence that is itself nonsense by the lights of the theory of types(PM, 76). It’s (apparent) negation is, accordingly,also nonsense. InPrincipia, then, there is no very clearmodel of what is preserved in analysis. The best we can say is thatRussell’s contextual definitions have the feature that a(sincere, assertive) utterance of the analysans is guaranteed toexpress the same judgment as the analysandum,if the latterexpresses a judgment at all.

Some of the unclarity in the conception of analysis introduced byRussell’s rejection of propositions is inherited byWittgenstein, who similarly rejects any ontology of shadowy entitiesexpressed by sentences. In theTractatus a“proposition” (Satz) is a “propositionalsign in its projective relation to the world” (3.12). This makesit seem as though any difference between propositional signs ought tosuffice for a difference between propositions, in which case analysansand analysandum could at best be distinct propositions with the sametruth conditions.

Enough has now been said to make possible a consideration ofWittgenstein’s reasons for describing the position I have beencalling “linguistic atomism” as “obvious.”Since the model for Tractarian analysis is the replacement of apparentnames with (apparently) co-referring “terms forcomplexes,” together with eliminative paraphrase of the latter,it follows trivially that the endpoint of analysis, if there is one,will contain neither “terms for complexes” nor expressionsthat can be replaced by terms for complexes.

Wittgenstein, moreover, thinks it obvious that in the case of everyproposition this process of analysisdoes terminate. Thereason he supposes analysis cannot go on forever is that he conceivesan unanalyzed proposition asderiving its sense from itsanalysis. AsTractatus 3.261 puts it: “Every definedsign signifies via those signs by which it is defined” (Cf.NB, 46;PT 3.20102). It follows that no propositioncan have an infinite analysis, on pain of never acquiring a sense. Sothe process of analysis must terminate, and when it does so theproduct will be propositions devoid of incomplete symbols.

That much, at least,is plausibly obvious, but unfortunatelyit does not follow that the final analysis of language will be whollydevoid of complex symbols. The trouble is that for all we have said sofar, a fully analysed proposition might yet contain one or morecomplex symbolsthat have meaning in their own right.Clearly, Wittgenstein must have been assuming that all genuinereferring expressions must be semantically simple: they must lackanything like a Fregean sense. But why should that be so? The seeds ofone answer are contained inTractatus 3.3, the proposition inwhich Wittgenstein enunciates his own version of Frege’s contextprinciple: “Only the proposition has sense; only in the contextof a proposition has a name meaning” (3.3). Wittgenstein’sjuxtaposition of these two claims suggests that the context principleis supposed to be his ground for rejecting senses for sub-sententialexpressions. But just how it could provide such a ground is far fromclear. Another, more concrete, possibility is that Wittgenstein simplyaccepted the arguments Russell had given in “On Denoting”for rejecting senses for sub-sentential expressions.

3. Metaphysical Atomism

By “Metaphysical atomism” we will understand the view thatthe semantically simple symbols occurring in a proposition’sfinal analysis refer to simples. TheTractatus does notcontain adistinct freestanding argument for this thesis,but, as we will see, the needed argument is plausibly extractable fromthe famous “Argument for Substance” of 2.0211–2:

2.0211 If the world had no substance, then whether a proposition hadsense would depend on whether another proposition was true.

2.0212 It would then be impossible to draw up a picture of the world(true or false).

To see what precisely is being contended for in this argument oneneeds to appreciate the historical resonances of Wittgenstein’sinvocation of the notion of “substance.”

3.1 Objects as the Substance of the World

TheTractatus’s notion of substance is the modalanalogue of Kant’s temporal notion. Whereas for Kant, substanceis that which “persists” (in the sense of existing at alltimes) (Critique, A 182), for Wittgenstein it is that which,figuratively speaking, “persists” through a“space” of possible worlds. (Compare the idea of a roadthat crosses several U.S. States. Such a road might be said,metaphorically speaking, to “persist” from one State tothe next: in such a locution, as in Wittgenstein's, a temporal notionis enlisted to do spatial duty, though in Wittgenstein's case thespace in question is logical rather than physical space andpersistence amounts to reaching through thewhole of logicalspace.) Tractarian substance is the “unchanging” in themetaphorical sense of that which does not undergo existence change inthe passage (also metaphorical) from world to world. Lessfiguratively, Tractarian substance is that which exists with respectto every possible world. For Kant, to assert that there is substance(in the schematized sense of the category) is to say that that thereis some stuff such that every existence change (i.e., origination orannihilation) is necessarily an alteration or reconfiguration of thatstuff. For Wittgenstein, analogously, to say that there is substanceis to say that there are some things such that all “existencechanges” in the metaphorical passage from world to world arereconfigurations of them. What undergo “existence changes”are atomic states of affairs (configurations of objects): a state ofaffairs exists with respect to one world but fails to exist withrespect to another. Those things that remain in existence throughthese existence changes, and which are reconfigured in the process,are Tractarian objects. It follows that the objects that“constitute the substance of the world” (2.021) arenecessary existents. TheTractatus, rather wonderfully,compresses this whole metaphorical comparison into a single remark:“The object is the fixed, the existing [dasBestehende]; the configuration is the changing [dasWechselnde].” (2.0271). “Wechsel,” itshould be noted, is the word that Kant expressly reserves for thenotion of existence change as opposed to alteration(Critique, A 187/B 230). (Unfortunately, however, whetherWittgenstein had read theCritique in time for thiscircumstance to have influenced his own phrasing in theTractatus is unknown.)

Tractarian objects are what any “imagined”—or, moreaccurately,conceivable—world has in common with thereal world (2.022). Accordingly, they constitute the world’s“fixed form” (2.022–3). ‘Fixed’ because,unlike the world’s content, objects (existentially speaking)hold fast in the transition from world to world. ‘Form’because they constitute what is shared by all the worlds. (OnWittgenstein’s conception of possibility, the notion of an“alien” Tractarian object — one which ismerely possible — is not even intelligible). If theobjects make up the world’s form, what makes up its content? Theanswer, I think, is the various obtaining atomic states of affairs.Distinct worlds differ with respect to content because they differwith respect to which possible states of affairs obtain in them. Acomplication arises becasue possible atomic states of affairs alsohave both form and content form and content. Their form is the mannerof combination of their components, their content those componentsthemselves (that is, their contained objects). It follows thatsubstance — the totality of objects — is indeed, asWittgenstein says, “both form and content”(2.024–5). It is at once both the form of the world and thecontent of possible states of affairs (These and further details ofthis interpretation of Wittgenstein’s conception of substance asthe fixed or unchanging are provided in Proops 2004; see alsoZalabardo 2015, Appendix II for more on simples, names, and necessaryexistents).

3.2 The Argument for Substance

As we have seen, the immediate goal of the argument for substance isto establish that there are things that exist necessarily. In thecontext of the assumption that anything complex could fail to existthrough decomposition, this conclusion entails that there are simples(2.021). While the argument is presented as a two-stagemodustollens, it is conveniently reconstructed as areductio adabsurdum (The following interpretation of the argument is acompressed version of that provided in Proops 2004. For two recentalternatives, see Zalabardo 2015, 243–254 and Morris 2008, ch.1,and 2017; and for criticisms of Morris, see Potter 2009):

Suppose, forreductio, that
  [1] There is no substance (that is, nothing exists inevery possible world).
Then
  [2] Everything exists contingently.
But then
  [3] Whether a proposition has sense depends on whetheranother proposition is true.
So
  [4] We cannot draw up pictures of the world (true orfalse).
But
  [5] Wecandraw up such pictures.

Contradiction

So
  [6] There is substance (that is, some things exist inevery possible world).

Our [5] is the main suppressed premise. It means, simply, that we canframe senseful propositions. Let us now consider how we might try todefend the inference from [2] to [3] on Wittgensteinian principles. Asa preliminary, note that, given Wittgenstein’s equation in theNotes on Logic of having sense with having truth-poles(NB, 99), it seems reasonable to suppose that for a sentenceto “have sense” with respect to a given world is for it tohave a truth value with respect to that world. Let us assume that thisis so. Now suppose that everything exists contingently. Then, inparticular, the referents of the semantically simple symbols occurringin a fully analysed sentence will exist contingently. But then anysuch sentence will contain a semantically simple symbol that fails torefer with respect to some possible world (As we will shortly see,this step is in fact controversial.). Suppose, as a backgroundassumption, that there are no contingent simples. (It will be arguedbelow that this assumption plausibly follows from certain Tractariancommitments.) Then, if we assume that a sentence containing asemantically simple term is neither true nor false evaluated withrespect to a world in which its purported referent (namely, a complexexisting contingently at the actual world) fails to exist — and,for now, we do — then, for any such fully analysed sentence,there will be some world such that the sentence depends for its truthvaluedness with respect to that world on the truth with respect tothat world of some other sentence,viz., the sentence statingthat the constituents of the relevant complex are configured in amanner necessary and sufficient for its existence. It follows that ifeverything exists contingently, then whether a sentence is sensefulwith respect to a world will depend on whether another sentence istrue with respect to that world.

The step from [3] to [4] runs as follows. Suppose that whether anysentence “has sense” (i.e., on our reading, has atruth-value) depends (in the way just explained) on whether another istrue. Then every sentence will have an “indeterminatesense” in the sense that it will lack a truth value with respectto at least one possible world. But an indeterminate sense is no senseat all, for a proposition by its very nature “reaches throughthe whole logical space” (3.42) (i.e., it is truth-valued withrespect to every possible world).^[6] So if every sentence depended for its “sense” (i.e.,truth-valuedness) on the truth of another, no sentence would have adeterminate sense, and so no sentence would have a sense. In whichcase we would be unable to frame senseful propositions (i.e., to“draw up pictures of the world true or false”).

One apparent difficulty concerns the assumption that to have sense isjust to be true or false. How can such a view be attributed to theWittgenstein of theTractatus given his view that tautology,which is true, and contradiction, which is false, are without sense(sinnlos) (4.461)? The seeds of an answer may be contained ina remark from Wittgenstein’s lectures at Cambridge during theyear 1934–1935. Looking back on what he’d written in theTractatus, he says:

When I called tautologies senseless I meant to stress a connectionwith a quantity of sense, namely 0. ([AM]), 137)

It is possible, then, that Wittgenstein is thinking of asinnlos proposition as a proposition that “hassense” but has it to a zero degree. According to thisconception, a tautology, being true, is, in contrast to a nonsensicalstring, in the running for possessing a non-zero quantity of sense,but is so constructed that, in the end, it doesn’t get to haveone. And, importantly, in virtue of being in the running for having anon-zero quantity of sense its possession of a zero quantity amountsto its, broadly speaking, ‘having sense’. Such a view,according to which, for some non-count noun N, an N-less entity has N,but has a zero quantity of it, is not without precedent in thetradition. Kant, for example, regards rest (motionlessness) as aspecies of motion: a zero quantity of it (Bader, Other InternetResources, 22–23). If, in a similar fashion,Sinnlosigkeit is a species ofSinn, the equation ofhavingSinn with being true or false will be preserved. Tooffer a full defence of this understanding ofSinnlosigkeitwould take us too far afield, but I mention it to show that thecurrent objection is not decisive.

Another apparent difficulty for this reconstruction arises from itsappearing to contradictTractatus 3.24, which clearlysuggests that if the complex entityA were not to exist, theproposition “F[A]” would be false,rather than, as the argument requires, without truth value. But thedifficulty is only apparent. It merely shows that 3.24 belongs to atheory that assumes that the worlddoes have substance. Onthis assumption Wittgenstein can say that whenever an apparent nameoccurs that appears to mention a complex this is only because it isnot, after all, a genuine name — and this is what he does say.But on the assumption that the world has no substance, so thateverything is complex, Wittgenstein can no longer say this.For now he must allow that the semantically simple symbols occurringin a proposition’s final analysis do refer to complexes. So inthe context of the assumption that every proposition has a finalanalysis, thereductio assumption of the argument forsubstance entails the falsity of 3.24. But since 3.24 is assumed to befalse only in the context of areductio, it is something thatWittgenstein can consistently endorse (This solution to the apparentdifficulty for the present reconstruction is owed, in its essentials,to David Pears (see Pears 1987 [1989, 78]).

To complete the argument it only remains to show that Tractariancommitments extrinsic to the argument for substance rule outcontingent simples.^[7] Supposea is a contingent simple. Then “aexists” must be a contingent proposition. But it cannot be anelementary proposition because it will be entailed by any elementaryproposition containing ‘a,’ and elementarypropositions are logically independent (4.211). So “aexists” must be non-elementary, and so further analyzable. Andyet there would seem to be no satisfactory analysis of thisproposition on the assumption that ‘a’ names acontingent simple — no analysis, that is to say, that is bothintrinsically plausible and compatible with Tractarian principles.Wittgenstein cannot analyse “a exists” as theproposition “∃x(x =a)”for two reasons. First, he would reject this analysis on the groundsthat it makes an ineliminable use of the identity sign (5.534).Second, given his analysis of existential quantifications asdisjunctions, the proposition “∃x(x =a)” would be further analysed as thenon-contingent proposition “a =a∨a =b ∨a =c…”. Nor can he analyse “aexists” as “~[ ~Fa & ~Ga &~Ha…]” — that is, as the negation of theconjunction of the negations of every elementary proposition involving“a.” To suppose that it could, is to suppose thatthe proposition “~Fa & ~Ga &~Ha…” means “a does notexist,” and yet by the lights of theTractatus thisproposition wouldshowa’s existence —or, more correctly, it would show something that one tries to put intowords by saying “a exists” (cf. 5.535,Corr, 126)). So, pending an unforeseen satisfactory analysisof “a exists,” this proposition will have to beanalysed as a complex of propositions not involvinga. Inother words, ‘a’ will have to be treated as anincomplete symbol and the fact ofa’s existence willhave to be taken to consist in the fact that objects other thana stand configured thus and so. But that would seem to entailthata is not simple.

The argument for substance may be criticized on several grounds.First, the step leading from [2] to [3] relies on the assumption thata name fails to refer with respect to a possible world at which itsactual-world referent does not exist. This amounts to thecontroversial assumption that names do not function as what NathanSalmon has called “obstinately rigid designators” (Salmon1981, 34). Secondly, the step leading from [3] to [4] relies on theassumption that a sentence that is neither true nor false with respectto some possible world fails to express a sense. As Wittgenstein waslater to realize, the case of intuitively senseful, yet vaguesentences plausibly constitutes a counterexample (cf.PISection 99). Lastly, one may question the assumption that it makessense to speak of a final analysis, given that the procedure foranalysing a sentence of ordinary language has not been made clear (SeePI, Sections 60, 63–4, and Section 91).

4. The Epistemology of Logical Atomism

How could we possibly know that something is a Tractarian object?Wittgenstein has little or nothing to say on this topic in theTractatus, and yet it is clear from his retrospective remarksthat during the composition of theTractatus he did think itpossiblein principle to discover the Tractarian objects (SeeAM, 11 andEPB, 121). So it seems worth asking bywhat means he thought such a discovery might be made.

Sometimes, it can seem as though Wittgenstein just expected to hitupon the simples by reflecting from the armchair on those items thatstruck him as most plausibly lacking in proper parts. This impressionis most strongly suggested in theNotebooks, and inparticular in a passage from June 1915 in which Wittgenstein seems toexpress confidence that certain objects already within his ken eithercount as Tractarian objects or will turn out to do so. He says:“It seems to me perfectly possible that patches in our visualfield are simple objects, in that we do not perceive any single pointof a patch separately; the visual appearances of stars even seemcertainly to be so” (NB, 64). By “patches in ourvisual field” in this context Wittgenstein means parts of thevisual field with no noticeable parts. In other words,pointsin visual space (cf.KL, 120). Clearly, then, Wittgenstein atone stage believed he was in a position to specify some Tractarianobjects. However, the balance of the evidence suggests that this ideawas short-lived. For he was later to say that he and Russell hadpushed the question of examples of simples to one side as a matter tobe settled on a future occasion (AM, 11). And when NormanMalcolm pressed Wittgenstein to say whether when he wrote theTractatus he had decided on anything as an example of a“simple object,” he had replied — according toMalcolm’s report — that “at the time his thought hadbeen that he was a logician; and that it was not his business as alogician, to try to decide whether this thing or that was a simplething or a complex thing, that being a purely empirical matter”(Malcolm 1989, 70).

Wittgenstein was not suggesting that the correct way to establish thatsomething is a Tractarian object is to gather evidence that itsdecomposition isphysically impossible. That reading wouldonly have a chance of being correct if Wittgenstein had takenmetaphysical possibility to coincide with physical possibility, andthat is evidently not so.^[8] His meaning seems rather to be just that the objects must bediscovered rather than postulated or otherwise specified in advance ofinvestigation (cf.AM, 11). But since Wittgenstein was laterto accuse his Tractarian self of having entertained the concept of adistinctive kind ofphilosophical discovery (seeWVC182, quoted below), we must not jump — as Malcolm appears tohave done, to the conclusion that he conceived of the discovery inquestion as “empirical” in anything like the contemporarysense of the word.

We know that Wittgenstein denied categorically that we couldspecify the possible forms of elementary propositions and thesimplesa priori (4.221, 5.553–5.5541, 5.5571). But hedid not deny that these forms would be revealed as the result oflogical analysis. In fact, he maintained precisely this view. Thisidea is not explicit in theTractatus, but it is spelled outin a later self-critical remark from G. E. Moore’s notes ofWittgenstein’s 1933 lectures at Cambridge:

I say in [the]Tractatus that you can’t say anythingabout [the] structure of atomic prop[osition]s: my idea being thewrong one, that logical analysis would reveal what it would reveal(entry for 6 February, 1933, Stern et. al., 2016, 252)

Speaking of Tractarian objects in another retrospective remark, thistime from a German version of theBrown Book, Wittgensteinsays: “What these [fundamental constituents] of reality are itseemed difficult to say. I thought it was the job of further logicalanalysis to discover them” (EPB 121). These remarksshould be taken at face value: it is logical analysis — theanalysis of propositions — that is supposed to enable us todiscover the forms of elementary propositions and the objects. Thehope is that when propositions have been put into their final, fullyanalysed forms by applying the “calculus” spoken of in thePhilosophical Grammar we will finally come to know the namesand thereby the objects. Presumably, we will know the latterbyacquaintance in the act of grasping propositions in their finalanalysed forms.

Admittedly, Wittgenstein’s denial that we can know the objectsa priori looks strange given the fact that the analyticalprocedure described in Section 2 above seems to presuppose that wehave a priori knowledge both of the correct analyses of ordinary namesand of the contextual definitions by means of which terms forcomplexes are to beeliminated. But some tension inWittgenstein’s position on this point is just what we shouldexpect in view of his later rather jaundiced assessment of his earlierreliance on the idea of philosophical discovery:

I [used to believe that] the elementary propositions could bespecified at a later date. Only in recent years have I broken awayfrom that mistake. At the time I wrote in a manuscript of my book,“The answers to philosophical questions must never besurprising. In philosophy you cannot discover anything.”Imyself, however, had not clearly enough understood this and offendedagainst it. (WVC, 182, emphasis added)

The remark that Wittgenstein quotes here from “a manuscript oftheTractatus” did not survive into the final version,but its sentiment is clearly echoed in the related remark that therecan: “never be surprises in logic” (6.1251). Wittgensteinis clear that in theTractatus he had unwittingly proceededas though there could be such a thing as aphilosophicalsurprise or discovery. His idea that the true objects would bediscovered through analysis, but are nonetheless not knownapriori, is plausibly one instance of this mistake.

On the conception of theTractatus, objects are to bediscovered by grasping fully analysed propositions, presumablywith the awareness that theyare fully analysed. Butsince that is so, we shall not have fully explained how we aresupposed to be able to discover the simples unless we explain how, inpractice, we can know we have arrived at the final analysis of aproposition. But on this point, unfortunately, Wittgenstein has littleto say. In fact, the only hint he offers is the rather dark onecontained inTractatus 3.24:

That a propositional element signifies [bezeichnet] a complexcan be seen from an indeterminateness in the propositions in which itoccurs. We know that everything is not yet determined by thisproposition. (The notation for generality contains a prototype).(3.24)

It is an indeterminateness in propositions — whatever this mightamount to — that is supposed to alert us to the need for furtheranalysis. In Wittgenstein’s view, then, we possess a positivetest for analyzability. However, since the notion of“indeterminateness” in question is unclear, the test is oflittle practical value. The indeterminateness in question is plainlynot the one we considered in section 3: what is in question at thepresent juncture is the indeterminateness of propositions, not ofsenses. But what does that amount to?

According to one line of interpretation, due originally to W. D. Hart(Hart 1971), a proposition is indeterminate when there is more thanone way it can be true. Thus if I say “Barack Obama is in theUnited States,” I leave open where in particular he might be.The source of the indeterminacy is the implied generality of thisstatement, which is tantamount to: “Obama issomewherein the United States.” This line of interpretation has the meritof promising to make sense of the closing parenthetical remark of3.24. But it cannot be correct. The kind of indeterminacy thatWittgenstein has in mind at 3.24 is supposed to serve as a sign offurther analysability. But Hart’s notion cannot play this role,since any disjunctive proposition would be indeterminate in his sense,even a fully analysed proposition consisting of a disjunction ofelementary propositions.

According to a second line of interpretation, a proposition isindeterminate in the relevant sense if the result of embedding it insome context is structurally ambiguous. Consider, for example, theresult of embedding “F [A]” in thecontext “it is not true that,” where‘A’ is temporarily treated as a semanticallysimple term designating a complex (Keep in place the assumption that asentence containing a non-referring semantically simple term isneither true nor false). In this case the question would arise whetherthe result of this embedding is neither true nor false evaluated withrespect to a world in whichA does not exist, or simply true.The first option corresponds to giving the apparent name wide scopewith respect to the logical operator, the second to giving it narrowscope. Such a scope ambiguity could not exist if‘A’ were a genuine Tractarian name, so itspresence could reasonably be taken to signal the need for furtheranalysis.

So far, so good, but where does the business about the generalitynotation “containing a prototype” come in? Nothing in thepresent explanation has yet done justice to this remark. Nor does thepresent explanation really pinpoint what it is that signals the needfor further analysis. That, at bottom, is the fact that we can imaginecircumstances in which the supposed referent of‘A’ fails to exist. So, again, there is reason tobe dissatisfied with this gloss on indeterminacy.

It is hard to resist the conclusion that Wittgenstein never suppliedan adequate way of recognizing when a proposition is fully analysed,and consequently that he failed to specify a means for recognizingsomething as a Tractarian object.

5. The Dismantling of Logical Atomism

Wittgenstein’s turn away from logical atomism may be dividedinto two main phases. During the first phase (1928–9),documented in his 1929 article “Some Remarks on LogicalForm” (Klagge and Nordmann, 1993, 29–35), Wittgensteinexhibits a growing dissatisfaction with certain central details of theTractatus’s logical atomism, and notably with thethesis of the independence of elementary propositions. During thisphase, however, he is still working within the broad conception ofanalysis presupposed, if not fully developed, in theTractatus. The second phase (1931–2) involves arevolutionary break with that very conception.

5.1 First phase: The colour-exclusion problem

The so-called “colour-exclusion problem” is a difficultythat arises for theTractatus’s view that it ismetaphysically possible for each elementary proposition to be true orfalse regardless of the truth or falsity of the others (4.211). Inview of its generality, the problem might more accurately be termed“the problem of the manifest incompatibility of apparentlyunanalysable statements.” The problem may be illustrated asfollows: Suppose thata is a point in the visual field.Consider the propositionsP: “a is blue att” andQ: “a is red att” (supposing “red” and “blue”to refer to determinate shades). It is clear thatP andQ cannot both be true; and yet, on the face of it, it seemsthat this incompatibility (or “exclusion” inWittgenstein’s parlance) is not alogicalimpossibility. In theTractatus Wittgenstein’sresponse was to treat the problem as merely apparent. He supposed thatin such cases further analysis would reveal the incompatibility to belogical in nature:

For two colours,e.g., to be at one place in the visual fieldis impossible, and indeed logically impossible, for it is excluded bythe logical structure of colour. Let us consider how thiscontradiction presents itself in physics. Somewhat as follows: That aparticle cannot at the same time have two velocities, that is, that atthe same time it cannot be in two places, that is, that particles indifferent places at the same time cannot be identical (6.3751)

As F. P. Ramsey observes in his review of theTractatus(Ramsey, 1923), the analysis described here actually fails toreveal a logical incompatibility between the two statements inquestion; for, even granting the correctness of the envisagedreduction of the phenomenology of colour perception to facts about thevelocities of particles, the fact that one and the same particlecannot be (wholly) in two places at the same time still looks verymuch like a synthetica priori truth. It turns out, however,that Wittgenstein was well aware of this point. He knew that he hadnot taken the analysis far enough to bring out a logicalcontradiction, but he was confident that he had taken a step in theright direction. In aNotebooks entry from August 1916 heremarks that: “The fact that a particle cannot be in two placesat the same time does lookmore like a logical impossibility[than the fact that a point cannot be red and green at the same time].If we ask why, for example, then straight away comes the thought:Well, we should call particles that were in two places [at the sametime] different, and this in its turn all seems to follow from thestructure of space and particles” (NB, 81; emphasisadded). Here Wittgenstein isconjecturing that it will turnout to be a conceptual (hence, for himlogical) truth aboutparticles and space (and presumably also time) that particles in twodistinct places (at the same time) are distinct. He does not yetpossess the requisite analyses to demonstrate this conjecture, but heis optimistic that they will be found.

The article “Some Remarks on Logical Form” (1929) marksthe end of this optimism. Wittgenstein now arrives at the view thatsome incompatibilities cannot, after all, be reduced to logicalimpossibilities. His change of heart appears to have been occasionedby a consideration of incompatibilities involving the attribution ofqualities that admit of gradation —e.g., the pitch ofa tone, the brightness of a shade of colour, etc. Consider, forexample, the statements: “A has exactly one degree ofbrightness” and “A has exactly two degrees ofbrightness.” The challenge is to provide analyses of thesestatements that bring out the logical impossibility of their beingtrue together. What Wittgenstein takes to be the most plausiblesuggestion — or at least a sympathetic reconstruction of it— adapts the standard definitions of the numerically definitequantifiers to the system described in theTractatus,analysing these claims as respectively:“∃x(Bx &A hasx)& ~∃x,y(Bx &By&A hasx andA hasy)”(“Bx” means “x is a degree ofbrightness”) and “∃x,y(Bx&By &A hasx andA hasy) & ~∃x,y,z(Bx&By &Bz &A hasx&A hasy &A hasz).” But the suggestion will not do. The trouble isthat this analysis — absurdly — makes it seem as thoughwhen something has just one degree of brightness there could be asubstantive question about which (if any) of the three mentioned inthe analysis of the second claim—x ory orz — it was—as if a degree of brightness were akind of corpuscle whose association with a thing made it bright (cf.Klagge and Nordmann, 33). Wittgenstein concludes that the independenceof elementary propositions must be abandoned and that terms for realnumbers must enter into atomic propositions, so that the impossibilityof something’s having both exactly one and exactly two degreesof brightness emerges as an irreducibly mathematical impossibility.This, in turn, contradicts theTractatus’s idea thatall necessity is logical necessity (6.37).

5.2 Second phase: Generality and Analysis

Unlike Frege and Russell, theTractatus does not treat theuniversal and existential quantifiers as having meaning in isolation.Instead, it treats them as incomplete symbols to be analysed awayaccording to the following schemata:

∀x.Φx ↔Φa &Φb &Φc…
∃x.Φx ↔Φa ∨Φb ∨Φc…

Universal (existential) quantification is treated as equivalent to apossibly infinite conjunction (disjunction) of propositions.Wittgenstein’s later dissatisfaction with this view is expressedmost clearly in G. E. Moore’s notes of Wittgenstein’slectures from Michaelmas term 1932.

Now there is a temptation to which I yielded in [the]Tractatus, to say that

(x).fx = logical product,^[9]fa .fb .fc…

(∃x).fx = [logical] sum,fa ∨fb ∨fc…

This is wrong, but not as absurd as it looks. (entry for 25 November,1932, Stern et. al., 2016, 215)^[10].

Explaining why theTractatus’s analysis of generalityis notpalpably absurd, Wittgenstein says:

Suppose we say that: Everybody in this room has a hat = Ursell has ahat, Richards has a hat etc. This is obviously false, because you haveto add “&a,b,c,… arethe only people in the room.” This I knew and said in [the]Tractatus. But now, suppose we talk of“individuals” in R[ussell]’s sense, e.g., atoms orcolours; and give them names, then there would be no prop[osition]analogous to “Anda,b,c are theonly people in the room.” (ibid.)

Clearly, in theTractatus Wittgenstein was not making thesimple-minded mistake of forgetting that “EveryF isG” cannot be analysed as “Ga &Gb &Gc…” even whena,b,c, etc. are in fact the onlyFs.(Unfortunately, his claim that he registered this point in theTractatus is not borne out by the text). His idea was ratherthat theTractatus’s analysis of generality is offeredonly for the special case in whicha,b,c,etc, are “individuals” in Russell’s sense.Wittgenstein had supposed that in this case there is no proposition toexpress the supplementary clause that is needed in the other cases.Unfortunately, Wittgenstein does not explain why there should be nosuch proposition, but the answer seems likely to be the following:What we are assumed to be analysing is actually “Everything isG.” In this case any allegedly necessary competingclause — for example, “a,b,cetc., are the onlythings” (that is, Tractarianobjects) — would just be a nonsense-string produced in themisfired attempt to put into words something that isshown bythe fact that when analysis bottoms out it yields as names only suchas figure in the conjunction “Ga &Gb&Gc…” (cf.Tractatus 4.1272).

What led Wittgenstein to abandon theTractatus’sanalysis of generality was his realization that he had failedadequately to think through the infinite case. He had proceeded asthough the finite case could be used as a way of thinking about theinfinite case, the details of which could be sorted out at a laterdate. By 1932 he had come to regard this attitude as mistaken. Thepoint is made in a passage from theCambridge Lectures whosemeaning can only be appreciated after some preliminary explanation.The passage in question makes a crucial claim about somethingWittgenstein refers to as “The Proposition”. By thisphrase in this context he means the joint denial of all thepropositions that are values of the propositional function“x is in this room”. This proposition can bewritten:

(x is in this room) [- - - - - T]

(Entry for November 25th, 1932, Compare Stern et. al, 217)

There is a most important mistake in [the] Tract[atus]…Ipretended that the Proposition was a logical product; but itisn’t, because “…” don’t give you alogical product. It is [the] fallacy of thinking 1 + 1 + 1 … isa sum. It is muddling up a sum with the limit of a sum (ibid.)

His point is that the Proposition does not, despite appearances,express a logical product. It rather, he now seems to be saying,expresses something like an indefinitely extensible process.Wittgenstein came to see his earlier hope that it did express alogical product rested on the mistake of confusing “dots ofinfinitude” with “dots of laziness.”. The upshotcould scarely be more important: if Wittgenstein is right, theTractatus’s very conception of the general form of theproposition, because it makes essential appeal to the idea of thejoint denial of arbitrarily many values of a propositional function,is itself infected with confusion.

Wittgenstein, however, does not think that the confusion of kinds ofdots was the deepest mistake he made in theTractatus. Beyondthis: “There was a deeper mistake — confusing logicalanalysis with chemical analysis. I thought‘(∃x)fx’is a definitelogical sum, only I can’t at the moment tell you which”(November 25, 1932, ibid.; cf.PG, 210). Wittgenstein hadsupposed that there was a fact of the matter — unknown, but inprinciple knowable — about which logical sum“(∃x).fx” is equivalent to. Butbecause he had failed to specify the analytical procedure in fulldetail, and because he had not adequately explained what analysis issupposed to preserve, this idea was unwarranted. Indeed, itexemplified an attitude he was later to characterize as amounting to akind of unacceptable “dogmatism” (WWK, 182).

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• Soames, S., 1983, “Generality, Truth Functions, andExpressive Capacity in theTractatus,”ThePhilosophical Review, XCII(4): 573–89.
• Stenius, E., 1960,Wittgenstein’s Tractatus: A CriticalExposition of its Main Lines of Thought, Ithaca, New York:Cornell University Press.
• Zalabardo, J., 2015,Representation and Reality inWittgenstein’s Tractatus, Oxford: Oxford UniversityPress.

Suggestions for further reading

• Carruthers, P., 1990,The Metaphysics of the Tractatus,Cambridge: Cambridge University Press.
• Dreben, B. and Floyd, J., 1991, “Tautology: How Not to Use aWord,”Synthese, 87(1): 23–49.
• Griffin, J., 1964.,Wittgenstein’s Logical Atomism,Oxford: Oxford University Press.
• Livingston, P.M., 2001, “Russellian and WittgensteinianAtomism,”Philosophical Investigations, 24 (1):30–54.
• Moss, S., 2012, “Solving the Color IncompatibilityProblem,”Journal of Philosophical Logic, 41(5):841–51.
• Rogers, B. and Wehmeier, Kai F., 2012, “TractarianFirst-Order Logic: Identity and the N–Operator,”TheReview of Symbolic Logic, 5(4): 538–573.
• Potter, M., 2020, “The Rise of Analytic Philosophy1879–1930: From Frege to Ramsey”, Routledge, ch. 52.
• Sullivan, P.M.S., 2003, “Simplicity and Analysis in earlyWittgenstein,”European Journal of Philosophy, 11:72–88.
• Tejedor, C., 2003, “Sense and Simplicity:Wittgenstein’s Argument for Simple Objects,”Ratio, 16(3): 272–89.

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Acknowledgments

My thanks to Juliet Floyd, Dick Schmitt, and Bryan Rogers forcorrections.