Now some logic

Good morning. Again.

So, dad dug up for me how to get the logic symbols, so I'll talk about it now, since you wanted to know how to write things in maths.
There are more than one kind of logics, but the one I'll be talking about it predicate logic.
I apologize to begin with if I use wrong terms. I have only ever studied logic in Finnish, but I'll try my best to make the correct translations

Basically writing something in predicate logic looks something like this:

∀x (B(x) ∧ F(x) → ∃yL(y,x))

Obviously on it's own this doesn't really tell you much. Let me take that into pieces and explain how you should read that.

First of all, you have the predicates, in this case B(x), F(x) and L(x,y). They tell about a quality something has, like in B(x) and F(x), or if there's more than one variable inside it, then it tells about the relation between what ever is inserted there.
The variables (x, y, z...) only mark the place where you can put some constant, marked with a, b, c... or really with any letter you like that isn't confused with the variables. The constants, unlike the variables have a meaning on their own: they are names given to subjects, and are always used to refer to that specific thing, like a certain chair, or a book, or a specific person.
Also, there are the symbols that you need to know the meaning to, that you might be familiar with from high school math lessons. I'll tell you what they are.

→ implies, if... then
↔ equivalent, if and only if... then
¬ negation, what ever follows isn't true
∧ conjunction, and
∨ disjunction, or
∀ universal quantification, with all
∃ existential quantification, exists

Also, before you can actually translate anything, you have to give meaning to the predicates and the constants in the context of the conversation.

I have given my predicates up there the following meanings:
B(x): x is a book
F(x): x is fantasy/scifi
L(x,y): x likes y
In order to give you more examples, I'll just give meanings to a few more things, so I don't have to do that later.
M(x): x is a movie/series
s: Star Treck
m: me
p: you
i: our little sister
w: Wool (The book. Just finished reading the second of the series. It's brilliant.)

Cause I'm sure this will make for great examples. (Then again, at school our examples are something like "Write "all red balls are red" in logic", so...)

Ok, now let's look at the sentence in the beginning again by putting it into pieces.

∀x (of all things) (B(x) (if a thing is a book) ∧ (and) F(x) (fantasy) → (then) ∃y (there exist) L(y,x) (y that likes the thing))

or, in a more understandable language: For every fantasy book, there is someone who likes it.

Let's try something simpler, and the other way round. Let's translate "you like Star Trek". It's very simple:

L(p,s)

Now let's make that a little more complicated so it's not so boring. Let's say "you like the scifi series Star Trek". We can't actually say it quite like that, but have to take it into pieces, "Star Trek is scifi" being one, "Star Trek is a series" being another, and "you like Star Trek" being the third. Now, to write this, we need p and s, and we need F(x), M(x) and L(x,y).
And then we write:

F(s) ∧ M(s) ∧ L(p,s)
s is fantasy AND s is a series AND you like s.

You getting a hang of this?

I'll write you a few more examples:

You like all scifi series. (If something is a scifi series, you like it)
∀x ((F(x)∧M(x))→L(p,x))

I don't like Star Trek, but our little sister does.
¬L(m,s)∧L(i,s)

There is a fantasy book I like or you do.
∃x(F(x)∧B(x)∧(L(m,x)∨L(p,x)))

Our little sister likes Wool if I do. (If I like Wool, then she does too.)
L(m,w)→L(i,w)

If you like a book, it's also a movie.
∀x((B(x)∧L(p,x))→M(x))

I never said any of those sentences are true. Or that they'd be very good or sensible sentences. (Like I said, we have exercises translating sentences like red balls are red.) They're just examples of how to translate from English to logic.
After weeks of sitting in class, there are still people there who have no idea how any of this works. I trust you're smart enough to get a little bit of a hang on how this works from this little explanation. Tell me if you didn't get something, I can go on in the comments. I for example didn't get to where you should put the brackets and how moving them might change what the sentence means.

Ok, I'm running out of time. But I hope you got something out of that. A couple of small things that I just happen to have in my head, though:
Continuing on the topic of translating stuff did you know scientists are making equipment for translating dolphinian live time? Or what ever you want to call the language dolphins speak.
Also, I watched the last episode of How I Met Your Mother just before starting to write this. I have only one thing to say: Ummm... what? You're kidding me, right?

~matu