## 1. Names and Objects



The "names" spoken of in the *Tractatus* are not
mere signs (i.e., typographically or phonologically identified
inscriptions), but rather signs-together-with-their-meanings --
or "symbols." Being symbols, names are identified and
individuated only in the context of significant sentences. A name is
"semantically simple" in the sense that its meaning does
not depend on the meanings of its orthographic parts, even when those
parts are, in other contexts, independently meaningful. So, for
example, it would not count against the semantic simplicity of the
symbol '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 in
other sentential contexts. For Wittgenstein, however, something else
does 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 names
will be rare and hard to find. Even apparently simple singular terms
such as 'Obama,' 'London,' etc., will not be
counted as "names" by the strict standards of the
*Tractatus* since they will disappear on further analysis.
(Hereafter, 'name' will mean "Tractarian name"
unless otherwise indicated.)



It is a matter of controversy whether the *Tractatus* reserves
the term 'name' for those semantically simple symbols that
refer to particulars, or whether the term comprehends semantically
simple symbols of all kinds. Since objects are just the referents of
names, this issue goes hand in hand with the question whether objects
are one and all particulars or whether they include properties and
relations. The former view is defended by Irving Copi (Copi 1958) and
Elizabeth Anscombe (Anscombe 1959 [1971, 108 ff]), among others. It is
supported by *Tractatus* 2.0231: "[Material properties]
are first presented by propositions -- first formed by the
configuration of objects." This might seem to suggest that
simple properties are not objects but rather arise from the combining
or configuring of objects. The Copi-Anscombe interpretation has been
taken to receive further support from *Tractatus* 3.1432:



> 
> We must not say, "The complex sign '*aRb*'
> says '*a* stands in relation *R* to
> *b*;'" but we must say, "*That*
> '*a*' stands in a certain relation to
> '*b*' says *that aRb*."
> 



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



The opposing view, according to which names include predicates and
relational expressions, has been defended by Erik Stenius and Merrill
and Jaakko Hintikka, among others (Stenius, 1960, 61-69;
Hintikka and Hintikka, 1986, 30-34). It is supported by a
*Notebooks* entry from 1915 in which objects are explicitly
said to include properties and relations (*NB*, 61). It is
further buttressed by Wittgenstein's explanation to Desmond Lee
(in 1930-1) of *Tractatus* 2.01:
"'Objects' also include relations; a proposition is
not 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 propositions
as differing radically from anything we may be familiar with from
ordinary -- or even Fregean -- grammar. It thus respects
Wittgenstein's warning to Waismann in 1929 that "The
logical structure of elementary propositions need not have the
slightest 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 forms
of non-elementary propositions and the forms of elementary
propositions. In "Some Remarks on Logical Form" (1929) he
says: "One is often tempted to ask from an a priori standpoint:
What, after all, can be the only forms of atomic propositions, and to
answer, e.g., subject-predicate and the relational propositions with
two or more terms further, perhaps, propositions relating predicates
and relations with one another, and so on. But this, I believe, is a
mere playing with words" (Klagge and Nordman, 1993, 30). A
similar thought already occurs in a more compressed form in the
*Tractatus* itself: "There cannot be a hierarchy of the
forms of the elementary propositions. Only that which we ourselves
construct can we foresee" (5.556).



It is possible, then, that the options we began with represent a false
dichotomy. Perhaps Wittgenstein simply did not have an antecedent
opinion on the question whether Tractarian names will turn out to be
names of particulars only, particulars and universals, or whatnot. And
perhaps he believed that the final analysis of language would (or
might) reveal the names to defy such classifications altogether. This
broader range of interpretive possibilities has only recently begun to
receive the attention it deserves (See Johnston 2009).


## 2. Linguistic Atomism



By "Linguistic atomism" we shall understand the view that
the analysis of every proposition terminates in a proposition all of
whose genuine components are names. It is a striking fact that the
*Tractatus* contains no explicit argument for linguistic
atomism. This fact has led some commentators -- e.g., Peter
Simons (1992) -- to suppose that Wittgenstein's position
here is motivated less by argument than by brute intuition. And
indeed, Wittgenstein does present some conclusions in this vicinity as
if they required no argument. At 4.221, for example, he says:
"*It is obvious that* in the analysis of propositions we
must come to elementary propositions, which consist of names in
immediate combination" (emphasis added). Nonetheless, some basic
observations about the *Tractatus*'s conception of
analysis will enable us to see why Wittgenstein should have thought it
obvious that analysis must terminate in this way. 


### 2.1 Wittgenstein's Early Conception of Analysis



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



> 
> Formerly, I myself spoke of a 'complete analysis,' and I
> used to believe that philosophy had to give a definitive dissection of
> propositions so as to set out clearly all their connections and remove
> all possibilities of misunderstanding. I spoke as if there was a
> calculus in which such a dissection would be possible. I vaguely had
> in mind something like the definition that Russell had given for the
> definite article (*PG*, 211).
> 



One of the distinctive features of Russell's definition is that
it treats the expression "the *x* such that
*Fx*" as an "incomplete symbol." Such symbols
have no meaning in isolation but are given meaning by contextual
definitions that treat of the sentential contexts in which they occur
(cf. *PM*, 66). Incomplete symbols do, of course, *have*
meaning because they make a contribution to the meanings of the
sentences in which they occur (cf. *Principles*, Introduction,
x). What is special about them is that they make this contribution
without expressing a propositional constituent. (For more on the
nature of incomplete symbols, see Pickel 2013)



Russell's definition is contained in the following clauses (For
the sake of expository transparency, his scope-indicating devices are
omitted.).



> 
> 
> [1] *G*(the *x*: *Fx*) =
> [?]*x*([?]*y*(*Fy*-*y*=*x*)
> & *Gx*)    Df.
>   
> 
> (cf. Russell 1905b; Russell 1990, 173) 
> 
> 
> 
> [2] (the *x*: *Fx*) exists =
> [?]*x*[?]*y*(*Fy*-*y*=*x*)
>    Df.
>   
> 
> (cf. Russell 1990, 174)
> 
> 
> 



The fact that existence is dealt with by a separate definition shows
that Russell means to treat the predicate 'exists' as
itself an incomplete symbol. 



One can understand why Wittgenstein should have taken there to be an
affinity between the theory of descriptions and his own envisioned
"calculus," for one can extract from his remarks in the
*Tractatus* and elsewhere two somewhat parallel proposals for
eliminating what he calls terms for "complexes": 



> 
> [3] *F*[*aRb*] iff *Fa* & *Fb* &
> *aRb*
> 



> 
> [4] [*aRb*] exists iff *aRb*
> 



Clauses [1] to [4] share the feature that any sentence involving
apparent reference to an individual is treated as false rather than as
neither true nor false if that individual should be discovered not to
exist. 



Wittgenstein's first contextual definition -- our [3]
-- occurs in a *Notebooks* entry from 1914 (*NB*,
4), but it is also alluded to in the *Tractatus*: 



> 
> Every statement about complexes can be analysed into a statement about
> their constituent parts, and into those propositions which completely
> describe the complexes (2.0201).
> 



In [3] the statement "about [the complex's] constituent
parts" is "*Fa* & *Fb*," while the
proposition 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 be
necessary: 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 again
to eliminate these terms. If there is going to be a unique final
analysis, each apparent name will have to be *uniquely* paired
with a term for a complex. So the program of analysis at which
Wittgenstein gestures, in addition to committing him to something
analogous to Russell's theory of descriptions, also commits him
to the analogue of Russell's "description theory of
ordinary names" (cf. Russell 1905a). The latter is the idea that
every apparent name not occurring at the end of analysis is equivalent
in meaning to some definite description. 



Wittgenstein's first definition, like Russell's, strictly
speaking, stands in need of a device for indicating scope, for
otherwise it would be unclear how to apply the analysis when we
choose, say, "~*G*" as our instance of
"*F*." In such a case the question would arise
whether the resulting instance of [3] is [5]:
"~*G*[*aRb*] = ~*Ga* & ~*Gb*
& *aRb*," which corresponds to giving the term for a
complex wide scope with respect to the negation operator, or whether
it is: [6] "~*G*[*aRb*] = ~[*Ga* &
*Gb* & *aRb*]," which corresponds to giving
the term for a complex narrow scope. One suspects that
Wittgenstein's intention would most likely have been to follow
Russell's convention of reading the logical operator as having
narrow scope unless the alternative is expressly indicated (cf.
*PM*, 172). 



Definition [3] has obvious flaws. While it may work for such
predicates as "*x* is located in England," it
obviously fails for certain others, e.g., "*x* is greater
than three feet long" and "*x* weighs exactly four
pounds." This problem can hardly have escaped Wittgenstein; so
it seems likely that he would have regarded his proposals merely as
tentative illustrations, open to supplementation and refinement.



Although Wittgenstein's second contextual definition -- our
[4] -- does not occur in the *Tractatus*, it is implied by
a remark from the *Notes on Logic* that seems to anticipate
2.0201: 



> 
> Every proposition which seems to be about a complex can be analysed
> into a proposition about its constituents and ... the proposition
> which describes the complex perfectly; *i.e., that proposition
> which 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 that
*aRb*, Wittgenstein's clarifying addendum amounts to the
claim that the proposition "*aRb*" is equivalent to
the proposition "[*aRb*] exists." And this
equivalence is just our [4].



It turns out, then, that existence is defined only in contexts in
which it is predicated of complexes. Wittgenstein proposal thus
mirrors Russell's in embodying the idea that it makes no sense
to speak of the existence of immediately given (that is, named)
simples (cf. *PM*, 174-5). This is why Wittgenstein was
later to refer to his "objects" as "that for which
there is neither existence nor non-existence" (*PR*, 72).
His view seems to be that when '*a*' is a
Tractarian name, what we try to say by uttering the nonsense string
"*a* exists" will, strictly speaking, be
*shown* by the fact that the final analysis of some proposition
contains '*a*' (cf. 5.535). But of course, the
*Tractatus* does not always speak strictly. Indeed, what is
generally taken to be the ultimate conclusion of the
*Tractatus*'s so-called "Argument for
Substance" (2.021-2.0211) itself tries to say something
that can only be shown, since it asserts the *existence* of
objects. The sharpness of the tension here is only partly disguised by
the oblique manner in which the conclusion is formulated. Instead of
arguing for the existence of objects, the *Tractatus* argues
for the thesis that the world "has substance." However,
because "objects constitute the substance of the world"
(2.021), and because substance is that which *exists*
independently of what is the case (2.024), this is tantamount to
saying that objects exist. So it seems that Wittgenstein's
argument for substance must be regarded as a part of the ladder we are
supposed to throw away (6.54). But having acknowledged this point, we
shall set it aside as peripheral to our main concerns. 



The most obvious similarity between the two sets of definitions is
that each seeks to provide for the elimination of what purport to be
semantically complex referring expressions. The most obvious
difference consists in the fact that Wittgenstein's definitions
are designed to eliminate not definite descriptions, but rather terms
for complexes, for example the expression
"[*aRb*]," which, judging by remarks in the
*Notebooks*, is to be read: "*a in the relation R to
b*" (*NB*, 48) (This gloss seems to derive from
Russell's manner of speaking of complexes in *Principia
Mathematica*, where examples of terms for complexes include, in
addition to "*a* in the relation *R* to
*b*," "*a* having the quality
*q*", and "*a* and *b* and *c*
standing in the relation *S*" (*PM*, 44).). One
might wonder why this difference should exist. That is to say, one
might wonder why Wittgenstein does not treat the peculiar locution
"*a* in the relation *R* to *b*" as a
definite description -- say, "the complex consisting of
*a* and *b*, combined so that *aRb*"? This
description could then be eliminated by applying the
*Tractatus*'s own variant upon the theory of
descriptions: 



> 
> The *F* is *G* - [?]*x*(*Fx*
> & *Gx*) & ~[?]*x*,*y*(*Fx*
> & *Fy*)
>   
> 
> (cf. 5.5321)
> 



Here the distinctness of the variables (the fact that they are
distinct) replaces the sign for distinctness "[?]" (cf.
5.53).



Since Wittgenstein did not adopt this expedient, it seems likely that
he would have regarded the predicate "*x* is a complex
consisting of *a* and *b*, combined so that
*aRb*" as meaningless in virtue of -- among other
things -- its containing ineliminable occurrences of the
pseudo-concepts "complex," "combination," and
"constitution." Only the first of these notions figures on
his list of pseudo-concepts in the *Tractatus* (4.1272), but
there is no indication that that list is supposed to be exhaustive.




There is a further respect in which Wittgenstein's analytical
proposals differ from Russell's. Russell's second
definition -- our [2] -- has the effect of shifting the
burden of indicating ontological commitment from the word
'exists' to the existential quantifier. In
Wittgenstein's definition, by contrast, no single item of
vocabulary takes over the role of indicating ontological commitment.
Instead, that commitment is indicated only after the *final*
application of the definition, by the meaningfulness of the names in
the fully analysed proposition -- or, more precisely, by the fact
that certain symbols are names (cf. 5.535). The somewhat paradoxical
consequence is that one can assert a statement of the form
"[*aRb*] exists" without thereby manifesting any
ontological commitment to the complex [*aRb*] (cf.
*EPB*, 121). What this shows is that the two theories relieve
the assertor of ontological commitments of quite different kinds. In
Russell's case, the analysis -- our [2] -- removes a
commitment to an apparent propositional constituent -- a
"denoting concept"
 [5]
 --expressed by the phrase 'the *F*,' but it
does not remove the commitment to the *F* itself. For
Wittgenstein, by contrast, the analysis shows that the assertor never
was ontologically committed to the complex [*aRb*] by an
utterance of "[*aRb*] exists." 



Russell's conception of analysis at the time of the theory of
descriptions -- c.a. 1905 -- is relatively clear: Analysis
involves pairing up one sentence with another that expresses the very
same Russellian proposition but which does so more perspicuously. The
analysans counts as more perspicuous than the analysandum because the
former is free of some of the latter's merely apparent
ontological commitments. By the time of *Principia
Mathematica*, however, this relatively transparent conception of
analysis is no longer available. Having purged his ontology of
propositions in 1910, Russell can no longer appeal to the idea that
analysans and analysandum express one and the same proposition. He now
adopts "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 relation
between the judging mind and the proposition *Othello loves
Desdemona*, is now a non-dyadic, or, in Russell's
terminology, "multiple," relation whose relata are the
judging mind and those items that were formerly regarded as
constituents of the proposition *Othello loves Desdemona*
(Russell 1910 [1994, 155]). After 1910 Russell can say that a speaker who
sincerely assertively uttered the analysans (in a given context) would
be guaranteed to make the same judgment as one who sincerely
assertively uttered the analysandum (in the same context), but he can
no longer explain this accomplishment by saying that the two sentences
express the same proposition. 



A further departure from the earlier, relatively transparent
conception of analysis is occasioned by Russell's resolution of
the set-theoretic version of his paradox. In this resolution one gives
an analysis of a sentence whose utterance could not be taken to
express *any* judgment. One argues that the sentence
"{*x*: *phx*} e {*x*:
*phx*}" is nonsense because the contextual definitions
providing for the elimination of class terms yield for this case a
sentence that is itself nonsense by the lights of the theory of types
(*PM*, 76). It's (apparent) negation is, accordingly,
also nonsense. In *Principia*, then, there is no very clear
model of what is preserved in analysis. The best we can say is that
Russell's contextual definitions have the feature that a
(sincere, assertive) utterance of the analysans is guaranteed to
express the same judgment as the analysandum, *if* the latter
expresses a judgment at all. 



Some of the unclarity in the conception of analysis introduced by
Russell's rejection of propositions is inherited by
Wittgenstein, who similarly rejects any ontology of shadowy entities
expressed by sentences. In the *Tractatus* a
"proposition" (*Satz*) is a "propositional
sign in its projective relation to the world" (3.12). This makes
it seem as though any difference between propositional signs ought to
suffice for a difference between propositions, in which case analysans
and analysandum could at best be distinct propositions with the same
truth conditions. 



Enough has now been said to make possible a consideration of
Wittgenstein's reasons for describing the position I have been
calling "linguistic atomism" as "obvious."
Since the model for Tractarian analysis is the replacement of apparent
names with (apparently) co-referring "terms for
complexes," 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 expressions
that can be replaced by terms for complexes. 



Wittgenstein, moreover, thinks it obvious that in the case of every
proposition this process of analysis *does* terminate. The
reason he supposes analysis cannot go on forever is that he conceives
an unanalyzed proposition as *deriving* its sense from its
analysis. As *Tractatus* 3.261 puts it: "Every defined
sign signifies via those signs by which it is defined" (Cf.
*NB*, 46; *PT* 3.20102). It follows that no proposition
can have an infinite analysis, on pain of never acquiring a sense. So
the process of analysis must terminate, and when it does so the
product will be propositions devoid of incomplete symbols. 



That much, at least, *is* plausibly obvious, but unfortunately
it does not follow that the final analysis of language will be wholly
devoid of complex symbols. The trouble is that for all we have said so
far, a fully analysed proposition might yet contain one or more
complex symbols *that have meaning in their own right*.
Clearly, Wittgenstein must have been assuming that all genuine
referring expressions must be semantically simple: they must lack
anything like a Fregean sense. But why should that be so? The seeds of
one answer are contained in *Tractatus* 3.3, the proposition in
which Wittgenstein enunciates his own version of Frege's context
principle: "Only the proposition has sense; only in the context
of a proposition has a name meaning" (3.3). Wittgenstein's
juxtaposition of these two claims suggests that the context principle
is supposed to be his ground for rejecting senses for sub-sentential
expressions. But just how it could provide such a ground is far from
clear. Another, more concrete, possibility is that Wittgenstein simply
accepted 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 that
the semantically simple symbols occurring in a proposition's
final analysis refer to simples. The *Tractatus* does not
contain a *distinct* freestanding argument for this thesis,
but, as we will see, the needed argument is plausibly extractable from
the famous "Argument for Substance" of 2.0211-2:




> 
> 2.0211 If the world had no substance, then whether a proposition had
> sense 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 one
needs to appreciate the historical resonances of Wittgenstein's
invocation of the notion of "substance."


### 3.1 Objects as the Substance of the World



The *Tractatus*'s notion of substance is the modal
analogue of Kant's temporal notion. Whereas for Kant, substance
is that which "persists" (in the sense of existing at all
times) (*Critique*, A 182), for Wittgenstein it is that which,
figuratively speaking, "persists" through a
"space" of possible worlds. (Compare the idea of a road
that crosses several U.S. States. Such a road might be said,
metaphorically speaking, to "persist" from one State to
the next: in such a locution, as in Wittgenstein's, a temporal notion
is enlisted to do spatial duty, though in Wittgenstein's case the
space in question is logical rather than physical space and
persistence amounts to reaching through the *whole* of logical
space.) Tractarian substance is the "unchanging" in the
metaphorical sense of that which does not undergo existence change in
the passage (also metaphorical) from world to world. Less
figuratively, Tractarian substance is that which exists with respect
to every possible world. For Kant, to assert that there is substance
(in the schematized sense of the category) is to say that that there
is some stuff such that every existence change (i.e., origination or
annihilation) is necessarily an alteration or reconfiguration of that
stuff. For Wittgenstein, analogously, to say that there is substance
is to say that there are some things such that all "existence
changes" in the metaphorical passage from world to world are
reconfigurations of them. What undergo "existence changes"
are atomic states of affairs (configurations of objects): a state of
affairs exists with respect to one world but fails to exist with
respect to another. Those things that remain in existence through
these 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) are
necessary existents. The *Tractatus* , rather wonderfully,
compresses this whole metaphorical comparison into a single remark:
"The object is the fixed, the existing [*das
Bestehende*]; the configuration is the changing [*das
Wechselnde*]." (2.0271). "*Wechsel*," it
should be noted, is the word that Kant expressly reserves for the
notion of existence change as opposed to alteration
(*Critique*, A 187/B 230). (Unfortunately, however, whether
Wittgenstein had read the *Critique* in time for this
circumstance to have influenced his own phrasing in the
*Tractatus* is unknown.)



Tractarian objects are what any "imagined"--or, more
accurately, *conceivable*--world has in common with the
real 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. (On
Wittgenstein's conception of possibility, the notion of an
"alien" Tractarian object -- one which is
*merely* possible -- is not even intelligible). If the
objects make up the world's form, what makes up its content? The
answer, I think, is the various obtaining atomic states of affairs.
Distinct worlds differ with respect to content because they differ
with respect to which possible states of affairs obtain in them. A
complication arises becasue possible atomic states of affairs also
have both form and content form and content. Their form is the manner
of combination of their components, their content those components
themselves (that is, their contained objects). It follows that
substance -- the totality of objects -- is indeed, as
Wittgenstein says, "both form and content"
(2.024-5). It is at once both the form of the world and the
content of possible states of affairs (These and further details of
this interpretation of Wittgenstein's conception of substance as
the fixed or unchanging are provided in Proops 2004; see also
Zalabardo 2015, Appendix II for more on simples, names, and necessary
existents). 


### 3.2 The Argument for Substance



As we have seen, the immediate goal of the argument for substance is
to establish that there are things that exist necessarily. In the
context of the assumption that anything complex could fail to exist
through decomposition, this conclusion entails that there are simples
(2.021). While the argument is presented as a two-stage *modus
tollens*, it is conveniently reconstructed as a *reductio ad
absurdum* (The following interpretation of the argument is a
compressed version of that provided in Proops 2004. For two recent
alternatives, see Zalabardo 2015, 243-254 and Morris 2008, ch.1,
and 2017; and for criticisms of Morris, see Potter 2009): 



> 
> Suppose, for *reductio*, that
>   
> 
>   [1] There is no substance (that is, nothing exists in
> every possible world).
>   
> 
> Then
>   
> 
>   [2] Everything exists contingently.
>   
> 
> But then
>   
> 
>   [3] Whether a proposition has sense depends on whether
> another proposition is true.
>   
> 
> So
>   
> 
>   [4] We cannot draw up pictures of the world (true or
> false).
>   
> 
> But
>   
> 
>   [5] We *can* draw up such pictures.
>   
> 
> 
> Contradiction
> 
>  So
>   
> 
>   [6] There is substance (that is, some things exist in
> every possible world).
>   
> 
> 



Our [5] is the main suppressed premise. It means, simply, that we can
frame senseful propositions. Let us now consider how we might try to
defend the inference from [2] to [3] on Wittgensteinian principles. As
a preliminary, note that, given Wittgenstein's equation in the
*Notes on Logic* of having sense with having truth-poles
(*NB*, 99), it seems reasonable to suppose that for a sentence
to "have sense" with respect to a given world is for it to
have a truth value with respect to that world. Let us assume that this
is so. Now suppose that everything exists contingently. Then, in
particular, the referents of the semantically simple symbols occurring
in a fully analysed sentence will exist contingently. But then any
such sentence will contain a semantically simple symbol that fails to
refer with respect to some possible world (As we will shortly see,
this step is in fact controversial.). Suppose, as a background
assumption, that there are no contingent simples. (It will be argued
below that this assumption plausibly follows from certain Tractarian
commitments.) Then, if we assume that a sentence containing a
semantically simple term is neither true nor false evaluated with
respect to a world in which its purported referent (namely, a complex
existing 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 truth
valuedness with respect to that world on the truth with respect to
that world of some other sentence, *viz*., the sentence stating
that the constituents of the relevant complex are configured in a
manner necessary and sufficient for its existence. It follows that if
everything exists contingently, then whether a sentence is senseful
with respect to a world will depend on whether another sentence is
true with respect to that world.



The step from [3] to [4] runs as follows. Suppose that whether any
sentence "has sense" (i.e., on our reading, has a
truth-value) depends (in the way just explained) on whether another is
true. Then every sentence will have an "indeterminate
sense" in the sense that it will lack a truth value with respect
to at least one possible world. But an indeterminate sense is no sense
at all, for a proposition by its very nature "reaches through
the whole logical space" (3.42) (i.e., it is truth-valued with
respect 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 a
determinate sense, and so no sentence would have a sense. In which
case 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 is
just to be true or false. How can such a view be attributed to the
Wittgenstein of the *Tractatus* 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 in
a remark from Wittgenstein's lectures at Cambridge during the
year 1934-1935. Looking back on what he'd written in the
*Tractatus*, he says:



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



It is possible, then, that Wittgenstein is thinking of a
*sinnlos* proposition as a proposition that "has
sense" but has it to a zero degree. According to this
conception, a tautology, being true, is, in contrast to a nonsensical
string, in the running for possessing a non-zero quantity of sense,
but is so constructed that, in the end, it doesn't get to have
one. And, importantly, in virtue of being in the running for having a
non-zero quantity of sense its possession of a zero quantity amounts
to 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 the
tradition. Kant, for example, regards rest (motionlessness) as a
species of motion: a zero quantity of it (Bader, Other Internet
Resources, 22-23). If, in a similar fashion,
*Sinnlosigkeit* is a species of *Sinn*, the equation of
having *Sinn* with being true or false will be preserved. To
offer a full defence of this understanding of *Sinnlosigkeit*
would take us too far afield, but I mention it to show that the
current objection is not decisive.



Another apparent difficulty for this reconstruction arises from its
appearing to contradict *Tractatus* 3.24, which clearly
suggests that if the complex entity *A* were not to exist, the
proposition "*F*[*A*]" would be false,
rather than, as the argument requires, without truth value. But the
difficulty is only apparent. It merely shows that 3.24 belongs to a
theory that assumes that the world *does* have substance. On
this assumption Wittgenstein can say that whenever an apparent name
occurs that appears to mention a complex this is only because it is
not, after all, a genuine name -- and this is what he does say.
But on the assumption that the world has no substance, so that
*everything* is complex, Wittgenstein can no longer say this.
For now he must allow that the semantically simple symbols occurring
in a proposition's final analysis do refer to complexes. So in
the context of the assumption that every proposition has a final
analysis, the *reductio* assumption of the argument for
substance entails the falsity of 3.24. But since 3.24 is assumed to be
false only in the context of a *reductio*, it is something that
Wittgenstein can consistently endorse (This solution to the apparent
difficulty 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 Tractarian
commitments extrinsic to the argument for substance rule out
contingent
 simples.[7]
 Suppose *a* is a contingent simple. Then "*a*
exists" must be a contingent proposition. But it cannot be an
elementary proposition because it will be entailed by any elementary
proposition containing '*a*,' and elementary
propositions are logically independent (4.211). So "*a*
exists" must be non-elementary, and so further analyzable. And
yet there would seem to be no satisfactory analysis of this
proposition on the assumption that '*a*' names a
contingent simple -- no analysis, that is to say, that is both
intrinsically plausible and compatible with Tractarian principles.
Wittgenstein cannot analyse "*a* exists" as the
proposition "[?]*x*(*x* = *a*)"
for two reasons. First, he would reject this analysis on the grounds
that it makes an ineliminable use of the identity sign (5.534).
Second, given his analysis of existential quantifications as
disjunctions, the proposition "[?]*x*(*x* =
*a*)" would be further analysed as the
*non-contingent* proposition "*a* = *a*
[?] *a* = *b* [?] *a* =
*c*...". Nor can he analyse "*a*
exists" as "~[ ~*Fa* & ~*Ga* &
~*Ha*...]" -- that is, as the negation of the
conjunction of the negations of every elementary proposition involving
"*a*." To suppose that it could, is to suppose that
the proposition "~*Fa* & ~*Ga* &
~*Ha*..." means "*a* does not
exist," and yet by the lights of the *Tractatus* this
proposition would *show* *a*'s existence --
or, more correctly, it would show something that one tries to put into
words by saying "*a* exists" (cf. 5.535,
*Corr*, 126)). So, pending an unforeseen satisfactory analysis
of "*a* exists," this proposition will have to be
analysed as a complex of propositions not involving *a*. In
other words, '*a*' will have to be treated as an
incomplete symbol and the fact of *a*'s existence will
have to be taken to consist in the fact that objects other than
*a* stand configured thus and so. But that would seem to entail
that *a* 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 that
a name fails to refer with respect to a possible world at which its
actual-world referent does not exist. This amounts to the
controversial assumption that names do not function as what Nathan
Salmon has called "obstinately rigid designators" (Salmon
1981, 34). Secondly, the step leading from [3] to [4] relies on the
assumption that a sentence that is neither true nor false with respect
to some possible world fails to express a sense. As Wittgenstein was
later to realize, the case of intuitively senseful, yet vague
sentences plausibly constitutes a counterexample (cf. *PI*
Section 99). Lastly, one may question the assumption that it makes
sense to speak of a final analysis, given that the procedure for
analysing a sentence of ordinary language has not been made clear (See
*PI*, 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 the
*Tractatus*, and yet it is clear from his retrospective remarks
that during the composition of the *Tractatus* he did think it
possible *in principle* to discover the Tractarian objects (See
*AM*, 11 and *EPB*, 121). So it seems worth asking by
what means he thought such a discovery might be made. 



Sometimes, it can seem as though Wittgenstein just expected to hit
upon the simples by reflecting from the armchair on those items that
struck him as most plausibly lacking in proper parts. This impression
is most strongly suggested in the *Notebooks*, and in
particular in a passage from June 1915 in which Wittgenstein seems to
express confidence that certain objects already within his ken either
count as Tractarian objects or will turn out to do so. He says:
"It seems to me perfectly possible that patches in our visual
field are simple objects, in that we do not perceive any single point
of a patch separately; the visual appearances of stars even seem
certainly to be so" (*NB*, 64). By "patches in our
visual field" in this context Wittgenstein means parts of the
visual field with no noticeable parts. In other words, *points*
in visual space (cf. *KL*, 120). Clearly, then, Wittgenstein at
one stage believed he was in a position to specify some Tractarian
objects. However, the balance of the evidence suggests that this idea
was short-lived. For he was later to say that he and Russell had
pushed the question of examples of simples to one side as a matter to
be settled on a future occasion (*AM*, 11). And when Norman
Malcolm pressed Wittgenstein to say whether when he wrote the
*Tractatus* he had decided on anything as an example of a
"simple object," he had replied -- according to
Malcolm's report -- that "at the time his thought had
been that he was a logician; and that it was not his business as a
logician, to try to decide whether this thing or that was a simple
thing or a complex thing, that being a purely empirical matter"
(Malcolm 1989, 70).



Wittgenstein was not suggesting that the correct way to establish that
something is a Tractarian object is to gather evidence that its
decomposition is *physically* impossible. That reading would
only have a chance of being correct if Wittgenstein had taken
metaphysical possibility to coincide with physical possibility, and
that is evidently not
 so.[8]
 His meaning seems rather to be just that the objects must be
discovered rather than postulated or otherwise specified in advance of
investigation (cf. *AM*, 11). But since Wittgenstein was later
to accuse his Tractarian self of having entertained the concept of a
distinctive kind of *philosophical* discovery (see *WVC*
182, quoted below), we must not jump -- as Malcolm appears to
have done, to the conclusion that he conceived of the discovery in
question as "empirical" in anything like the contemporary
sense of the word. 



We know that Wittgenstein denied categorically that we could
*specify* the possible forms of elementary propositions and the
simples *a priori* (4.221, 5.553-5.5541, 5.5571). But he
did not deny that these forms would be revealed as the result of
logical analysis. In fact, he maintained precisely this view. This
idea is not explicit in the *Tractatus*, but it is spelled out
in a later self-critical remark from G. E. Moore's notes of
Wittgenstein's 1933 lectures at Cambridge: 



> 
> I say in [the] *Tractatus* that you can't say anything
> about [the] structure of atomic prop[osition]s: my idea being the
> wrong 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, this
time from a German version of the *Brown Book*, Wittgenstein
says: "What these [fundamental constituents] of reality are it
seemed difficult to say. I thought it was the job of further logical
analysis to discover them" (*EPB* 121). These remarks
should be taken at face value: it is logical analysis -- the
analysis of propositions -- that is supposed to enable us to
discover the forms of elementary propositions and the objects. The
hope is that when propositions have been put into their final, fully
analysed forms by applying the "calculus" spoken of in the
*Philosophical Grammar* we will finally come to know the names
and thereby the objects. Presumably, we will know the latter *by
acquaintance* in the act of grasping propositions in their final
analysed forms.



Admittedly, Wittgenstein's denial that we can know the objects
*a priori* looks strange given the fact that the analytical
procedure described in Section 2 above seems to presuppose that we
have a priori knowledge both of the correct analyses of ordinary names
and of the contextual definitions by means of which terms for
complexes are to beeliminated. But some tension in
Wittgenstein's position on this point is just what we should
expect in view of his later rather jaundiced assessment of his earlier
reliance on the idea of philosophical discovery:



> 
> I [used to believe that] the elementary propositions could be
> specified at a later date. Only in recent years have I broken away
> from that mistake. At the time I wrote in a manuscript of my book,
> "The answers to philosophical questions must never be
> surprising. In philosophy you cannot discover anything." *I
> myself, however, had not clearly enough understood this and offended
> against it*. (*WVC*, 182, emphasis added)
> 



The remark that Wittgenstein quotes here from "a manuscript of
the *Tractatus*" did not survive into the final version,
but its sentiment is clearly echoed in the related remark that there
can: "never be surprises in logic" (6.1251). Wittgenstein
is clear that in the *Tractatus* he had unwittingly proceeded
as though there could be such a thing as a *philosophical*
surprise or discovery. His idea that the true objects would be
discovered through analysis, but are nonetheless not known *a
priori*, is plausibly one instance of this mistake.



On the conception of the *Tractatus*, objects are to be
discovered by grasping fully analysed propositions, presumably
*with* the awareness that they *are* fully analysed. But
since that is so, we shall not have fully explained how we are
supposed to be able to discover the simples unless we explain how, in
practice, we can know we have arrived at the final analysis of a
proposition. But on this point, unfortunately, Wittgenstein has little
to say. In fact, the only hint he offers is the rather dark one
contained in *Tractatus* 3.24: 



> 
> That a propositional element signifies [*bezeichnet*] a complex
> can be seen from an indeterminateness in the propositions in which it
> occurs. We know that everything is not yet determined by this
> proposition. (The notation for generality contains a prototype).
> (3.24)
> 



It is an indeterminateness in propositions -- whatever this might
amount to -- that is supposed to alert us to the need for further
analysis. In Wittgenstein's view, then, we possess a positive
test for analyzability. However, since the notion of
"indeterminateness" in question is unclear, the test is of
little practical value. The indeterminateness in question is plainly
not the one we considered in section 3: what is in question at the
present juncture is the indeterminateness of propositions, not of
senses. 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 than
one way it can be true. Thus if I say "Barack Obama is in the
United States," I leave open where in particular he might be.
The source of the indeterminacy is the implied generality of this
statement, which is tantamount to: "Obama is *somewhere*
in the United States." This line of interpretation has the merit
of promising to make sense of the closing parenthetical remark of
3.24. But it cannot be correct. The kind of indeterminacy that
Wittgenstein has in mind at 3.24 is supposed to serve as a sign of
further 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 of
elementary propositions.



According to a second line of interpretation, a proposition is
indeterminate in the relevant sense if the result of embedding it in
some context is structurally ambiguous. Consider, for example, the
result of embedding "*F* [*A*]" in the
context "it is not true that," where
'*A*' is temporarily treated as a semantically
simple term designating a complex (Keep in place the assumption that a
sentence containing a non-referring semantically simple term is
neither true nor false). In this case the question would arise whether
the result of this embedding is neither true nor false evaluated with
respect to a world in which *A* does not exist, or simply true.
The first option corresponds to giving the apparent name wide scope
with respect to the logical operator, the second to giving it narrow
scope. Such a scope ambiguity could not exist if
'*A*' were a genuine Tractarian name, so its
presence could reasonably be taken to signal the need for further
analysis.



So far, so good, but where does the business about the generality
notation "containing a prototype" come in? Nothing in the
present explanation has yet done justice to this remark. Nor does the
present explanation really pinpoint what it is that signals the need
for further analysis. That, at bottom, is the fact that we can imagine
circumstances in which the supposed referent of
'*A*' fails to exist. So, again, there is reason to
be dissatisfied with this gloss on indeterminacy. 



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


## 5. The Dismantling of Logical Atomism



Wittgenstein's turn away from logical atomism may be divided
into two main phases. During the first phase (1928-9),
documented in his 1929 article "Some Remarks on Logical
Form" (Klagge and Nordmann, 1993, 29-35), Wittgenstein
exhibits a growing dissatisfaction with certain central details of the
*Tractatus*'s logical atomism, and notably with the
thesis of the independence of elementary propositions. During this
phase, however, he is still working within the broad conception of
analysis presupposed, if not fully developed, in the
*Tractatus*. The second phase (1931-2) involves a
revolutionary break with that very conception.


### 5.1 First phase: The colour-exclusion problem



The so-called "colour-exclusion problem" is a difficulty
that arises for the *Tractatus*'s view that it is
metaphysically possible for each elementary proposition to be true or
false regardless of the truth or falsity of the others (4.211). In
view of its generality, the problem might more accurately be termed
"the problem of the manifest incompatibility of apparently
unanalysable statements." The problem may be illustrated as
follows: Suppose that *a* is a point in the visual field.
Consider the propositions *P*: "*a* is blue at
*t*" and *Q*: "*a* is red at
*t*" (supposing "red" and "blue"
to refer to determinate shades). It is clear that *P* and
*Q* cannot both be true; and yet, on the face of it, it seems
that this incompatibility (or "exclusion" in
Wittgenstein's parlance) is not a *logical*impossibility. In the *Tractatus* Wittgenstein's
response was to treat the problem as merely apparent. He supposed that
in such cases further analysis would reveal the incompatibility to be
logical in nature:



> 
> For two colours, *e.g*., to be at one place in the visual field
> is impossible, and indeed logically impossible, for it is excluded by
> the logical structure of colour. Let us consider how this
> contradiction presents itself in physics. Somewhat as follows: That a
> particle cannot at the same time have two velocities, that is, that at
> the same time it cannot be in two places, that is, that particles in
> different places at the same time cannot be identical (6.3751)
> 



As F. P. Ramsey observes in his review of the *Tractatus*(Ramsey, 1923), the analysis described here actually fails to
reveal a logical incompatibility between the two statements in
question; for, even granting the correctness of the envisaged
reduction of the phenomenology of colour perception to facts about the
velocities of particles, the fact that one and the same particle
cannot be (wholly) in two places at the same time still looks very
much like a synthetic *a priori* truth. It turns out, however,
that Wittgenstein was well aware of this point. He knew that he had
not taken the analysis far enough to bring out a logical
contradiction, but he was confident that he had taken a step in the
right direction. In a *Notebooks* entry from August 1916 he
remarks that: "The fact that a particle cannot be in two places
at the same time does look *more 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 same
time] different, and this in its turn all seems to follow from the
structure of space and particles" (*NB*, 81; emphasis
added). Here Wittgenstein is *conjecturing* that it will turn
out to be a conceptual (hence, for him *logical*) truth about
particles and space (and presumably also time) that particles in two
distinct places (at the same time) are distinct. He does not yet
possess the requisite analyses to demonstrate this conjecture, but he
is optimistic that they will be found. 



The article "Some Remarks on Logical Form" (1929) marks
the end of this optimism. Wittgenstein now arrives at the view that
some incompatibilities cannot, after all, be reduced to logical
impossibilities. His change of heart appears to have been occasioned
by a consideration of incompatibilities involving the attribution of
qualities that admit of gradation -- *e.g*., the pitch of
a tone, the brightness of a shade of colour, etc. Consider, for
example, the statements: "*A* has exactly one degree of
brightness" and "*A* has exactly two degrees of
brightness." The challenge is to provide analyses of these
statements that bring out the logical impossibility of their being
true together. What Wittgenstein takes to be the most plausible
suggestion -- or at least a sympathetic reconstruction of it
-- adapts the standard definitions of the numerically definite
quantifiers to the system described in the *Tractatus*,
analysing these claims as respectively:
"[?]*x*(*Bx* & *A* has *x*)
& ~[?]*x*,*y*(*Bx* & *By*
& *A* has *x* and *A* has *y*)"
("*Bx*" means "*x* is a degree of
brightness") and "[?]*x*,*y*(*Bx*
& *By* & *A* has *x* and *A* has
*y*) & ~[?]*x*,*y*,*z*(*Bx*
& *By* & *Bz* & *A* has *x*
& *A* has *y* & *A* has
*z*)." But the suggestion will not do. The trouble is
that this analysis -- absurdly -- makes it seem as though
when something has just one degree of brightness there could be a
substantive question about which (if any) of the three mentioned in
the analysis of the second claim-- *x* or *y* or
*z* -- it was--as if a degree of brightness were a
kind of corpuscle whose association with a thing made it bright (cf.
Klagge and Nordmann, 33). Wittgenstein concludes that the independence
of elementary propositions must be abandoned and that terms for real
numbers must enter into atomic propositions, so that the impossibility
of something's having both exactly one and exactly two degrees
of brightness emerges as an irreducibly mathematical impossibility.
This, in turn, contradicts the *Tractatus*'s idea that
all necessity is logical necessity (6.37). 


### 5.2 Second phase: Generality and Analysis



Unlike Frege and Russell, the *Tractatus* does not treat the
universal and existential quantifiers as having meaning in isolation.
Instead, it treats them as incomplete symbols to be analysed away
according to the following schemata:



> 
> [?]*x*. *Ph*x - *Pha* &
> *Phb* & *Phc*...
>   
> 
> [?]*x*. *Phx* - *Pha* [?]
> *Phb* [?] *Phc*...
> 



Universal (existential) quantification is treated as equivalent to a
possibly infinite conjunction (disjunction) of propositions.
Wittgenstein's later dissatisfaction with this view is expressed
most clearly in G. E. Moore's notes of Wittgenstein's
lectures 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 the *Tractatus*'s analysis of generality
is not *palpably* absurd, Wittgenstein says: 



> 
> Suppose we say that: Everybody in this room has a hat = Ursell has a
> hat, Richards has a hat etc. This is obviously false, because you have
> to add "& *a*, *b*, *c*,... are
> the 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 or
> colours; and give them names, then there would be no prop[osition]
> analogous to "And *a*, *b*, *c* are the
> only people in the room." (ibid.)
> 



Clearly, in the *Tractatus* Wittgenstein was not making the
simple-minded mistake of forgetting that "Every *F* is
*G*" cannot be analysed as "*Ga* &
*Gb* & *Gc*..." even when *a*,
*b*, *c*, etc. are in fact the only *F*s.
(Unfortunately, his claim that he registered this point in the
*Tractatus* is not borne out by the text). His idea was rather
that the *Tractatus*'s analysis of generality is offered
only for the special case in which *a*, *b*, *c*,
etc, are "individuals" in Russell's sense.
Wittgenstein had supposed that in this case there is no proposition to
express the supplementary clause that is needed in the other cases.
Unfortunately, Wittgenstein does not explain why there should be no
such proposition, but the answer seems likely to be the following:
What we are assumed to be analysing is actually "Everything is
*G*." In this case any allegedly necessary competing
clause -- for example, "*a*, *b*, *c*
etc., are the only *things"* (that is, Tractarian
objects) -- would just be a nonsense-string produced in the
misfired attempt to put into words something that is *shown* by
the fact that when analysis bottoms out it yields as names only such
as figure in the conjunction "*Ga* & *Gb*
& *Gc*..." (cf. *Tractatus* 4.1272).



What led Wittgenstein to abandon the *Tractatus*'s
analysis of generality was his realization that he had failed
adequately to think through the infinite case. He had proceeded as
though the finite case could be used as a way of thinking about the
infinite case, the details of which could be sorted out at a later
date. By 1932 he had come to regard this attitude as mistaken. The
point is made in a passage from the *Cambridge Lectures* whose
meaning can only be appreciated after some preliminary explanation.
The passage in question makes a crucial claim about something
Wittgenstein refers to as "The Proposition". By this
phrase in this context he means the joint denial of all the
propositions that are values of the propositional function
"*x* is in this room". This proposition can be
written: 



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



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

 Here the symbol '[- - - - - T]' symbolizes
the joint-denial operation, and the whole symbol expresses the result
of applying this operation to arbitrarily many values of the
propositional function "*x* is in the room". The
dashes in the symbol for joint denial represent rows in the
truth-table on which one or more of the truth-arguments--that is
values of the propositional function--is true. The result of
applying the operation of joint denial to those truth-arguments is
accordingly false. (In a variant on this notation each of the dashes
could be replaced with 'F'). Wittgenstein is interested in
the fact that while we write down finitely many dashes we intend the
arguments for the joint-denial operation to be arbitrarily many and
possibly infinitely many. His criticism of these conceptions runs as
follows:


> 
> There is a most important mistake in [the] Tract[atus]...I
> pretended that the Proposition was a logical product; but it
> isn't, because "..." don't give you a
> logical product. It is [the] fallacy of thinking 1 + 1 + 1 ... is
> a 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 a
logical product rested on the mistake of confusing "dots of
infinitude" with "dots of laziness.". The upshot
could scarely be more important: if Wittgenstein is right, the
*Tractatus*'s very conception of the general form of the
proposition, because it makes essential appeal to the idea of the
joint denial of arbitrarily many values of a propositional function,
is itself infected with confusion.



Wittgenstein, however, does not think that the confusion of kinds of
dots was the deepest mistake he made in the *Tractatus*. Beyond
this: "There was a deeper mistake -- confusing logical
analysis with chemical analysis. I thought
'([?]*x*)*fx*' *is* a definite
logical sum, only I can't at the moment tell you which"
(November 25, 1932, ibid.; cf. *PG*, 210). Wittgenstein had
supposed that there was a fact of the matter -- unknown, but in
principle knowable -- about which logical sum
"([?]*x*).*fx*" is equivalent to. But
because he had failed to specify the analytical procedure in full
detail, and because he had not adequately explained what analysis is
supposed to preserve, this idea was unwarranted. Indeed, it
exemplified an attitude he was later to characterize as amounting to a
kind of unacceptable "dogmatism" (*WWK*, 182).