Quote:
Originally Posted by IlyaK1986
Okay, this might be fast, but you can read it over and over. Induction is like an infinite set of dominoes.
It works by proving a general case by starting with a base case (the first domino), and then showing that for any such domino, it will make the next one fall over just as easily.
For instance, let's say that I want to prove that the sum of numbers 1 through n was n(n+1)/2
Let's take the base case of 1 and 2. The sum of 1 and 2 is 3. n in this case is 2. So...
2(3)/2=3. It works.
Now, let's do the general case.
n(n-1)/2 (the sum of numbers 1 through n-1) + n = n(n-1)/2+2n/2.
Expanding n(n-1), we have n^2-n, and then we add another 2n. So now we have n^2+n=n(n+1)/2, which is the sum for all numbers 1 through n (as opposed to 1 through n-1), which shows that the general case holds, and thus, our proof is done.
I'm not exactly sure what the hell it's doing in a PHILOSOPHY course though. Mathematics, engineering, and computer science, sure...but PHILOSOPHY? Ahahahah!
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So what you just showed me would be strong induction, yes (or complete induction, I forget the word)? And that second part, where you do the general case, that is the inductive step, right? I think I understand better now- thanks for the reply.
The only thing I got from my lecture was that there is a base case, such as, say with my example, that there is a proof in LFJ of A on hyp Gamma (Gamma being a use-language symbol ranging over formulae). Symbolically it would look like "Gamma |=(LFJ) A". Then there was an inductive step, where it would basically be a linear representation of some structural or non-structural rule of the language such as modus ponens (or cut, or weakening, etc..., depending on the language). Then the last step would be a conclusion that would essentially make the whole thing recursive.
Oh, and yes it's a logic course in the philosophy department (proof theory, foundations of mathematics kind of stuff...in advanced logic the math stuff starts appearing). This is a program with a very strong analytic bent. Unfortunately I find it very difficult to follow the professor in lecture (he's old and mumbles a lot, and doesn't explain everything), and even more difficult to follow when asking him questions face to face.
Part of the problem is that, yeah, you've taken a couple years of symbolic logic and thought you had a handle on the material...(you is figurative)...and then you get to grad school and find out that there is this gap between what you learned and what you're learning now. In other words, they don't make the transition between elementary logic (being able to construct proofs and manipulate formulae IN the languages of first order propositional and predicate logic) and advanced metalogical theory very easy.