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

6.1 The propositions of logic are tautologies.
6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions.
6.3 The exploration of logic means the exploration of everything that is subject to law. And outside logic everything is accidental.
6.4 All propositions are of equal value.
6.5 When the answer cannot be put into words, neither can the question be put into words. The riddle does not exist. If a question can be framed at all, it is also possible to answer it.

A logical picture of facts is a thought.

The logical picture of the facts is the thought. (3)

The proposition constructs a world with the help of a logical scaffolding, and therefore one can actually see in the proposition all the logical features possessed by reality if it is true. One can draw conclusions from a false proposition

When I have asked thinking men what reason they had to believe that every fact in the universe is precisely determined by law, the first answer has usually been that the proposition is a "presupposition " or postulate of scientific reasoning. Well, if that is the best that can be said for it, the belief is doomed. Suppose it be " postulated " : that does not make it true, nor so much as afford the slightest rational motive for yielding it any credence. It is as if a man should come to borrow money, and when asked for his security, should reply he "postulated " the loan. To "postulate" a proposition is no more than to hope it is true. There are, indeed, practical emergencies in which we act upon assumptions of certain propositions as true, because if they are not so, it can make no difference how we act. But all such propositions I take to be hypotheses of individual facts. For it is manifest that no universal principle can in its universality be compromised in a special case or can be requisite for the validity of any ordinary inference.

The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.

"The gramophone record, the musical thought, the score, the waves of sound, all stand to one another in that pictorial internal relation which holds between language and the world. To all of them the logical structure is common. (Like the two youths, their two horses and their lilies in the story. They are all in a certain sense one) " (4.014). The possibility of a proposition representing a fact rests upon the fact that in it objects are represented by signs.

Believing a given proposition is a matter of believing that it faithfully represents some state of the world, and this fact yields some immediate insights into the standards by which our beliefs should function. In particular, it reveals why we cannot help but value evidence and demand that propositions about the world logically cohere.

There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible.

The thought is the significant proposition. (4)

6.4 All propositions are of equal value.
6.41 The sense of the world must lie outside the world. In the world everything is as it is, and everything happens as it does happen: in it no value exists — and if it did exist, it would have no value. If there is any value that does have value, it must lie outside the whole sphere of what happens and is the case. For all that happens and is the case is accidental. What makes it non-accidental cannot lie within the world, since if it did it would itself be accidental. It must lie outside the world.
6.42 So too it is impossible for there to be propositions of ethics. Propositions can express nothing that is higher.
6.43 If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts — not what can be expressed by means of language. In short the effect must be that it becomes an altogether different world. It must, so to speak, wax and wane as a whole. The world of the happy man is a different one from that of the unhappy man.
6.44 It is not how things are in the world that is mystical, but that it exists.
6.45 To view the world sub specie aeterni is to view it as a whole — a limited whole. Feeling the world as a limited whole — it is this that is mystical.

A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective.

The thought is the significant proposition