Wittgenstein's Tractatus Logico-Philosophicus is based around demonstrating the truth of the following ideas:

(i) "Every proposition has a unique final analysis which reveals it to be a truth-function of elementary propositions ((Tractatus 3.25, 4.221, 4.51, 5)"(Proops, 2013):

3.25 - There is one and only one complete analysis of the proposition

4.221 - It is obvious that in the analysis of propositions we must come
to elementary propositions, which consist of names in immediate
combination. The question arises here, how the propositional connexion
comes to be

4.51 - Suppose all elementary propositions were given me: then we
can simply ask: what propositions I can build out of them. And
these are all propositions and so are they limited.

5 - Propositions are truth-functions of elementary propositions.
(An elementary proposition is a truth-function of itself.)

(ii) "These elementary propositions assert the existence of atomic states of affairs (3.25, 4.21)"(Proops, 2013):

3.25 - There is one and only one complete analysis of the proposition

4.21 - The simplest proposition, the elementary proposition, asserts
the existence of an atomic fact.

(iii) "Elementary propositions are mutually independent — each one can be true or false independently of the truth or falsity of the others (4.211, 5.134)"(Proops, 2013)

4.211 - It is a sign of an elementary proposition, that no elementary
[The importance of the law of non-contradiction, itself essential the truth-functional propositional calculus upon which the Tractatus is founded, is evident here. It is impossible for an elementary proposition to be both true and false. We will also see the importance of the law of excluded middle, the other of the two pillars of truth-functional propositional calculus, for Wittgenstein's Tractatus; namely, that all propositions are either true or false and that it is an, if not the, essential function of language to affirm the truth or falsity of propositions].

5.134 - From an elementary proposition no other can be inferred.

(iv) "Elementary propositions are immediate combinations of semantically simple symbols or “names” (4.221)"(Proops, 2013):

4.221 - It is obvious that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination. The question arises here, how the propositional connexion comes to be

(v) "Names refer to items wholly devoid of complexity, so-called “objects” (2.02 & 3.22)"(Proops, 2013):

2.02 - The object is simple.

3.22 - In the proposition the name represents the object.

(vi) "Atomic states of affairs are combinations of these simple objects (2.01)"(Proops, 2013)

2.01 - An atomic fact is a combination of objects (entities, things).

"Names" in the Tractatus:

A name is not a mere "sign" but a symbol(Proops, 2013). That is, it is the lexical aspect and the semantic aspect, the signifier and the meaning given to the signifier. A name is semantically simple in that it does not depend for its meaning on any of its orthographic parts even when an orthographic chunk of a word may have meaning wrested from its original context(Proops, 2013). For example, it does not count against the semantic simplicity of the name "Daniel" that "an" is an indefinite article when taken from its context and placed elsewhere.

There are two related questions we can ask about the function of a name(Proops, 2013):

1) Does a "name" refer only to semantically simple symbols referring to particulars or does it comprehend semantically simple symbols of all kinds(Proops, 2013)?

2) Are objects solely particulars or do they also refer to the properties of that object and the relations they bear to other objects(Proops, 2013)?

Irving Copi and Elizabeth Anscombe favor the latter because of proposition 2.0231(Proops, 2013):

"2.0231 - The substance of the world can only determine a form and not any material properties. For these are ﬁrst presented by the propositions—ﬁrst formed by the conﬁguration of the objects."

That is:

“[Material properties] are first presented by propositions — first formed by the configuration of objects"(Proops, 2013). Some suggest that for the Tractarian Wittgenstein, properties are not objects but arise from the combination or configuration of objects(Proops, 2013).

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”

So Wittgenstein seems to be saying here that we should not say that what we have is a set of discrete particular standing in relation to one another but that the very relation borne among the particulars is itself an object such that Tractarian objects are not mere particulars but also relations borne by particulars to one another(Proops, 2013).

Hintikka defends the idea that "names" are used to denote not simply particulars but also predicates and relational expressions. Wittgenstein explicitly says in a Notebooks entry from 1915 that objects include properties and relations(NB, 61) such that names are not confined to particulars but include relations and properties and predicates(Proops, 2013).

He also says in an explanation of Tractatus 2.01 to Desmond Lee that "Objects' also include relations(Proops, 2013); a proposition is not two things connected by a relation(Proops, 2013). 'Thing' and 'relation' are on the same level." Therefore, the relation is itself an object(Proops, 2013). This is what he means when he says that they are on the same level. He is explaining the following to Desmond Lee:

2.01 - An atomic fact is a combination of objects (entities, things).

“In an atomic state of affairs objects hang one in another, like the links of a chain.” (2.03).

We even have Wittgenstein's own commentary on this remark in an explanation to C.K. Ogden as meaning that "There isn't anything third that connects the links but the links themselves make that connection with one another." So there is therefore a sense in which the link or relation has some sort of objective autonomy or dignity in its own right such that it itself constitutes a kind of object(Proops, 2013). Wittgenstein in other words seems to see the link in the chain as itself having a kind of being as being on the same level(Proops, 2013).

Each link in a chain is linked both by and to its neighbor. It links to it but is itself also linked to it(Proops, 2013). No link is exclusively a saturator or saturatee, but instead plays both roles by virtue of its mutual relation with other particulars(Proops, 2013).

A state of affairs is given unity by each of its objects, be these objects particulars nor relations (between particulars). When we utter propositions concerning these states of affairs, the names by which these propositions are constituted are themselves mutually saturated by and in and through other names as well(Proops, 2013). When we utter an atomic proposition concerning a state of affairs concerning the world, this atomic proposition is itself saturated by another atomic proposition, as well as saturating another, such that each contributes to the unity of the complex proposition it makes up(Proops, 2013).

Proops, Ian, "Wittgenstein's Logical Atomism", The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.), URL = <plato.stanford.edu/archives/sum2013/entries/wittgenstein-atomism/>.

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