"Philosophy teaches us the logical form of propositions: that is its fundamental task."--LW. Out of those notes grew the only philosophical book he published in his lifetime, the Tractatus. He thought the philosophical journal Mind was filled "with all its impotence and bankruptcy." His notes from classes from 1933-34 circulated in copies and became known as The Blue Book. LW said we are tempted to explain a word like "pain" as being acquired by our own private, incommunicable sensation. A proposition is like a sentence. So we must add that a proposition is a sentence considered with respect to its meaning and not (say) with respect to its sound when spoken or appearance on the page. Propositions express thoughts. He rejected a philosophy of logic as Russell and Frege conceived of it. 'Everything which is possible in logic is also permitted' (TLP 5.473). As LW said, 'In order for a proposition to be capable of being true it must also be capable of being false' (NB 55). Objects combine into states of affairs, in which they stand in a determinate relation to one another 'like the links of a chain' (TLP 2.023). Since facts are the existence and non-existence of states of affairs, it follows that facts too are independent of each other (TLP 1.21). (He must, so to speak, throw away the ladder after he has climbed up it.) He must transcend these propositions and then he will see the world aright' (TLP 6.54). . it is possible to invent words; but I cannot think any thoughts to go with them.' (WWK 68) Wittgenstein believes that the definitive verification of a hypothesis is neither possible nor necessary. (6 and the principle of verification) LW was fond of telling a joke about a French politician who said that it was a characteristic of the French language that in it words occur in the order in which one thinks of them. 3. Doubt presupposes the mastery of a language-game. 'If you are not certain of any fact, you cannot be certain of the meaning of your words either' (OC 114). In other words, universal doubt is impossible. It would be like a student doubting every word that came out of a teacher's mouth. Not calling things in doubt is often a precondition of learning certain games (OC 329). 'Our doubts depend on the fact that some propositions are exempt from doubt, are as if it were like the hinges on which those turn' (OC 341). Hence 'The game of doubting itself presupposes certainty' (OC 115). We can make list of things, as did Wittgenstein and Moore, that cannot be doubted. And saying something cannot be doubted is not the same thing as saying that it can be known. You must look at the practice of language, then you will see it.' (OC 501) He also said: 'Logic must take care of itself.' (NB 2) No proposition is a priori true. LW continued to insist that if a sentence makes sense, its negation must make sense, consequently an real synthetic a priori proposition is impossible. Philosophy is the doctrine of the logical form of scientific propositions (not primitive propositions only).
On the whole Kenny describes Wittgenstein's ideas with remarkable simplicity and clarity, I was even following the bits of his early thought that consisted pretty much entirely of logic notation, which is a blaady miracle.
Kenny's writing style does not really help to make things accesible at first glance either (which, on the other hand, means of course that he does not oversimplify Wittgenstein's concepts): basically what you get is a wall of text, barely any figures, tables or even bullet points.
In particular, there is the "stream" analogy that Wittgenstein uses, where new things come into the stream and things leave, much as some principles come into our thinking about the world, and then leave, as if there are no truly stable principles that must remain with us.