"Here we come up against the great question that lies behind all these considerations.--For someone might object against me:'You take the easy way out! You talk about all sorts of language-games, but have nowhere said what the essence of a language-game, and hence of language is: what is common to all these activities, and what makes them into language or parts of language. So you let yourself off the very part of the investigation that once gave you yourself most headache, the part about the general form of propositions and of language.'
And this is true.-- Instead of producing something common to all that we call language, I am saying that these phenomena have no one thing in common which makes us use the same word for all, -- but that they are related to one another in many different ways. And it is because of this relationship, or these relationships, that we call them all 'language'. I will try to explain this."(Wittgenstein, section 65).
Well, he'd better explain all this. Just looking at the above, language is a loose consortium of games that are non-trivially related to one another. This leaves lots of questions like: if language is composed of disparate games, can there be ONE symbolic logic to capture it? It sounds like the answer has to be 'no'. Or, perhaps, their similarity is enough to make our normal logic, like philosophy majors take as undergrads, apply in all circumstances. But, look, the answer just has to be 'yes'.
Listen to what he says about numbers:
"Why do we call something a 'number'? Well, perhaps because it has a -- direct -- relationship with several things that have hitherto been called number; and this may be said to give it an indirect relationship to other things that we call the same name."(Wittgenstein, Section 67)
"How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: 'This and similar things are called 'games'". And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is? -- But this is not ignorance. We do not know the boundaries[between games and non-games] because none have been drawn."(Wittgenstein, Section 69)
"One might say that the concept 'game' is a concept with blurred edges."(Wittgenstein, Section, 71)
So, so far the concept of game is not rigorously defined, and I don't know if it will be. It may be that the philosophy here is showing what games are but not defining what they are. Games are practices of some kind, but not a kind he will define once and for all. Is he on to something? I don't know.