1. Wittgenstein observes:
4.02 ...we understand the sense of a propositional sign without its having been explained to us.
4.03 ...A proposition must use old expressions to communicate a new sense...
What conclusions does Wittgenstein draw from these observations? Explain his arguments, beginning with logically simple propositions. Can the arguments be extended to complex propositions? Evaluate the arguments
One of the things to remember with Wittgenstein's Tractatus is that he asserts that things that make sense are logical and therefore expressible; we can think them, say them, etc. Thus, a propositional sign expresses a fact (atomic or complex) precisely because it is logical: it makes sense. Anything that doesn't make sense, that is not logical, is nonsense and is therefore, inexpressible. Such an inexpressible proposition could not (or should not, according to Wittgenstein) be expressed. The only way to express something illogical is to say it is senseless. So, a propositional sign is understood because it is logical - because we can logically think it, because it makes sense.
The Picture Theory of Meaning:
When Wittgenstein talks of a proposition asserting something by virtue of being a logical picture, he is saying that it is logically possible. It is thinkable. The relationship of parts of a proposition make sense just as the juxtaposition of objects in a picture make sense by the logic of space.
The idea is that there is a connection between the structural relations in language with the structural relations in reality. So, he used the concept of the visual picture to serve as a metaphor showing how objects in a picture represent something in reality spatially - just as words in a sentence are arranged in such a way to represent states of affairs. The arrangement (relations) of the parts is what constructs meaning: in the sentence, in reality, or in the picture. The arrangement (proposition) makes sense if it is logical. And the atomic proposition is the most basic propositional sign; and it is understood because it logically makes sense.
Basically speaking, there are simple or atomic propositions, things that have meanings that can not be dissected further. (Note that these atomic propositions do have parts, but their parts are not meaningful in the way atomic or complex propositions are). From atomic propositions, we can form complex propositions. Wittgenstein concludes that if we knew all the atomic propositions (all logical, singular propositions), and all negations of those propositions, we could know or formulate all complex propositions.
Wittgenstein was trying to show the structural similarity between language and reality. Complex and atomic propositions would become more clear if we understood their structures (in language and in reality). Part of his project was to show this in a mathematical kind of way. Thus "Socrates was wise" and "Plato was wise" state two different things but with a similar or constant structure, both shown as Arb (for example). A complex proposition simply has two or more atomic parts. Knowing the logical structures, we could infer the atomic propositions of a complex, and vice versa.
All of this provides an interesting way to look at the structure of language and its relation to reality, but one problem is that it is too limiting. Wittgenstein would later refute some of his ideas with Philosophical Investigations, saying that a formal structure can not adequately describe the function of language.