i think the problem you're running into has to do with the fact that you're treating languages like it's a representation of situations in reality. and so, a sentence like "john is wearing a hat" could almost be summed up with a drawing of a guy with a hat on his head (and maybe some sort of arrow or something to show the relation between john and the hat, which is that he's the wear-er and the hat is the wear-ee). it seems like such a drawing would make sense because it's got all of the different components, "john", "hat", and "wearing", and it ties them together into a single idea of "john wearing a hat". by contrast, how on earth would you draw a picture of "john /isn't/ wearing a hat"? simply having a hat-less man wouldn't fly, and while it's almost conceivable that you could capture the meaning "isn't wearing" with an arrow that's been X-ed out or something, you'd still have to have a drawing of the hat that he isn't wearing (but then, if would have to be an abstract hat that doesn't even really exist). and so, i can see what you mean.

really though, i (personally) don't see language as a representation of the situations around us. rather, i see /thought/ as a representation of those situations, and then language as a representation of though. -- it might sound like a pedantic distinction, but i think it's really necessary if you want to avoid running into problems like these. i think that the typical way that linguists go about thinking about this stuff is by drawing a sharp line between semantics and pragmatics, and then saying that some aspects of language have a semantic function (which is to say, they attempt to describe real life) and that other aspects have a pragmatic function (which is to say that they try to capture all of the little extra thoughts that speakers inject a sentence regarding information tracking and personal perspective etc). personally, i'm just kinda against this whole analysis (though, if you happen to be into it, then feel free to make an argument as to why it makes good sense), and prefer instead the "reality --> thought --> language" analysis that i described above. (but, maybe then you could say that "not" functions kinda like how other pragmatically oriented words, like "the", do.)

anyhow, if we think of language as only being a representation of thought, then we shouldn't be trying to think in terms of drawing pictures (which,,, maybe no one else beside me ever thought anyway). if a word like "hat" seems like it could code for an actual hat, then that's really only incidental. (really though, how could it? the word "hat" is totally abstract. it has no size or color or anything. but there are no abstract color-free hats in real life.) the word "not" then is just a way to indicate negation (like the examples you showed using logic symbols), and in some cases it actually works backwards from how we might expect (for example, a sentence like "i doubt he's coming" is technically not a negated sentence in any way, but it ends up having the same meaning as a sentence like "i think he's not coming", which /is/ negated). and i'd say that that's really no different from the sort of thing that other logical operators (such as "or") do. and in fact, that's another good one. how would you draw a picture of a sentence containing the word "or"?

really though, i think there might be a whole bunch of examples that have the sorts of problem that "not" does. for example, it would probably be difficult enough to draw a picture of "if that mug falls, it will shatter", but then, what about "if that mug had fallen, it would have shattered"? what's more,, even if you could figure out a way to draw those two sentences, how on earth would you distinguish them? ------ but if these sentences are representations of thoughts, instead of real life, then our only job would be to draw the /thoughts/. now,,, i wouldn't know how to go about doing that, and i'm not quite sure what the "components" of a thought are (although, then again, i'm not really sure what the "components" of real life are either), but i think it's reasonable to assume that the though "he isn't wearing a hat" is every bit as substantial as the thought "he is wearing a hat" (whereas, the event of "him not wearing a hat" is not nearly as real as the event of "him wearing a hat"). --- another question to ponder over is how thoughts relate to real life, but that of course then would be outside of linguistics and i don't know that there's really anyone who has any good ideas on the subject.