If I say, I understood everything Wittgenstein said in this book then that will be lying. I might re-read this book again. Then I'll be able to review this review again and see where my thoughts differ.

This book is about philosophy, and logic in particular (any one can deduce that from the name, meh!). However, I think philosophers haven't been agreed upon what is a part of philosophy and what is not. Therefore, we'll be sticking to what Wittgenstein thinks philosophy is:

Philosophy is not one of the natural sciences.

(The word “philosophy” must mean something which stands above or below, but not beside the natural sciences.)

The object of philosophy is the logical clarification of thoughts.

Philosophy is not a theory but an activity.

A philosophical work consists essentially of elucidations.

The result of philosophy is not a number of “philosophical propositions”, but to make propositions clear.

Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred.

Now, that is both narrowing and liberating relative to the way we normally see the philosophy. It is liberating because now it's an action, therefore, opens up new possibilities. Narrowing in the sense that now it works only with the clarification of thoughts and have no apparent common ground with natural science.

The main section of the book starts with a German quote which can roughly be translated to "Everything you have to say, can be said in three words". This, actually covers a lot of the book. Wittgenstein expounded on some core elements of logic but he expounded on proposition most. A proposition is, as I understood reading him, a statement that express a fact. It comes with its own mental picture. And, if that picture is clear, or unthinkable, then the proposition itself is not valid. Now, this validity has nothing to do with if the statement is correct in real life or not. This is an inherent quality of a proposition. Once, a proposition is logically correct, we can proceed to observe the real world and check if the proposition is right or wrong.

Simple independent propositions can be joined with operators to create a complex one. In this way, a consistent logical picture can be generated.

Now, this thing is not simple, and of course, has a lot of consequences. For example, here's Wittgenstein's view on probability:

Probability is a generalization.

It involves a general description of a propositional form. Only in default of certainty do we need probability.

If we are not completely acquainted with a fact, but know something about its form.

Or, consider his view on time:

We cannot compare any process with the “passage of time”—there is no such thing—but only with another process (say, with the movement of the chronometer).

Hence the description of the temporal sequence of events is only possible if we support ourselves on another process.

It is exactly analogous for space. When, for example, we say that neither of two events (which mutually exclude one another) can occur, because there is no cause why the one should occur rather than the other, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry. And if there is such an asymmetry, we can regard this as the cause of the occurrence of the one and of the non-occurrence of the other.

Now, this is a small book, so I don't want to exhaust anyone with a long review. He can be right, he can be wrong. My trust in natural science says sometimes that he is wrong in many things. However, I can't deny his genius. His way of thinking is simply mind-blowing.