Test discussion, and propositional variables

I think the test went well? It's difficult to say. It was one of those tests where at no point was answering anything problematic in some way, and at no point did I run into something that looked like it could have been an error ... but at the same time I'd feel genuinely queasy claiming I aced it. Maybe I'm just being superstitious. At any rate I think I'll have to see what happens when it gets returned.

Not finding structural induction on the test made me a bit sad, which might be because I'd focussed much of my studying on structural induction because a part of it still kind of eludes me. I'd done a bit of studying on this because I feel like it's one of those things I "understand" but don't fully understand. Superficially, it's pretty simple to just say that structural induction is just a little more abstract a version of the other induction flavours we've seen. But I think that explanation is - in addition to being a little too vague - eliminating the question of what is a propositional variable anyway.


What is a propositional variable anyway

It appears to be some variable that can be either true or false. So we've sort of seen these before in that we can consider P(n) a propositional item. The exact P(n) for some given number n is something that can be either true or false. If we recall the problem about making any cost of postage using only, say, 3- and 5- cent stamps, then we have the following:

P(n) - "Postage totalling n cents may be made of a linear combination of 3- and 5- cent stamps."

and we can compute

P(0) - F
P(1) - F
P(2) - F
P(3) - T
P(4) - F
P(5) - T
P(6) - T
P(7) - F
P(8) - T
P(9) - T
P(10) - T
P(11) - T
And it looks like it's true from here on in. 

In this case it really looks like P is more a propositional formula which has a variable n. n isn't a propositional variable though, because n can't be true or false; n is a number (n /in N).

According to the few minutes I spent looking at what the internet brought up:
- a propositional variable is like the 'atom' of propositional logic

But that doesn't explain much, for me anyway. Next time I think I'll tackle what our course notes have to say about this, since judging from the course outline we're not going to have the time to cover this in class.