Because we can write down syntactical rules for parts of our language, and because we use these to demonstrate certain things, we forget that a demonstration of the validity of our arguments in general cannot be based on rules. Similarly with establishing semantic consequences - we can write down a few of these in a rule based way, but we can't completely eliminate semantic imports into the rules that we use. We can't give a general account of how language comes to have semantic content.
When I say that I can say 'The cat is hungry' - that this is a legitimate move in the language game I am playing (i.e. that it is 'true' for that game) - because of the way the world is, I am not saying that I can say 'The cat is hungry' because the cat is hungry. This is a kind of category mistake - or a mistake about the meta-level we are dealing with.
Of course it is true that if I say (seriously) 'the cat is hungry' I must also believe that the cat is hungry (or I'm not playing the game properly). For the purposes of that game, I must believe that this is the way the world is. But that is only because I am using this language to describe the world, and not some other language - e.g. the language I would use to explain how this language (the one in which the cat is hungry is a fact) works.
If I try to explain how this language works, I end up saying silly things like "'The cat is hungry' is true because the cat is hungry" (or their equivalent). Or, I say: "'The cat is hungry' is true in this language game because the world allows this language game to work". And the second statement doesn't say anything specific about the world except that we can talk about it the way we do, in fact, talk about it ...
Maybe this is 'the fact' of the slingshot - I would need to look at this again. It's where we go when we try to make general statements about what must, ultimately, be true in order for anything to be true. We end up with just this one feature of the world: that something can be true about it.
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