In general, a shallow broad understanding of a subject is useless for doing anything. Including learning. In fact, back when I took calculus 1 & 2 in my first year at university, students who had gained a "shallow" understanding of calculus from high school had an extremely hard time adjusting...
Well, I'll take your word for it then. I am not a logician, it's entirely possible that I may be talking out of my ass without even knowing it. Fortunately, this is not my day job.
That's a very good point, of course. I'm not too familar with the middle ground between first-order logic and second-order logic, or else I would've tried to pick something between the two.
Hurkyl's definition of a truth valuation looked an awful lot like "assign P a value of true if P is derivable from the axioms of the theory". In that case, I don't see how "P is true in this theory" and "P is provable from the axioms in this theory" are any different.
How does this...
But does that only work for first-order logic? I'm would expect that you can do that sort of thing for FOL given how "nice" it is. I'm not so sure about what happens when you move to something like second-order logic.
Hmm, I should have been able to think of that. Still, this does assume that you can algorithmically determine if a string of symbols is a proof of P; is it true that you can always do that?
I can't think of any useful mathematical meaning of the word "truth" other than "provable from the axioms". That said, I would be more than happy to never again have to use the word "truth" in mathematics. If only everybody else would stop using it...
If I can algorithmically determine...
Completeness of FOL basically says that if something is provable in every model of a first-order theory, then it is provable from the axioms of the theory.
Meanwhile, completeness of a FO theory says that every statement in the theory can be either proved or disproved.
So you can see that...
Of course, we're the ones colouring the balls, and we've done so after the fact. Pulling a ball out of a jar of red balls, painting the ball blue and then exclaiming "Wow, what are the chances we would pull out a blue ball?" is basically what we are doing when we label the current state of the...
scanf discards whitespace. If you enter a blank line it skips over it and keeps waiting for you to enter data.
To copy buffer into strings[strings_index] you can use strcpy, as in strcpy(strings[string_index], buffer). But that won't work unless you allocate memory for the string first...
There's quite a few problems with this code.
First of all, you can't do things like buffer != "" in C. To compare strings you'll need to use the strcmp function. Doing buffer != "" just compares two pointers, which is not at all what you want.
Second, when you do strings[string_index] =...
Destructors are called deterministically. When a stack-based object goes out of scope, its destructor is called immediately.
I think what you're doing is undefined behaviour. Your operator wrapper& is fine, but I don't think it's valid to return a reference to a temporary; I think that the...
Alright, let f(x)=\lvert x\rvert. So, given any \varepsilon>0 take \delta=\varepsilon. Then if \lvert x-0\vert<\delta then \lvert f(x)-f(0)\rvert=\lvert f(x)\rvert<\varepsilon and so \lim_{x\rightarrow 0}f(x)=0=f(0) and therefore f(x) is continuous at x=0.