In Propositional Logic you can find out about the basic rules governing most current logical systems1. But as useful as A → ¬B is for translating 'If it rains then I will not go to the party', we have problems if we want to translate an argument as simple as
All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.
It's all very well translating 'All men are mortal' as 'A' and 'Socrates is a man' as 'B', but then all you can come up with is an argument that says 'A, B therefore C - Socrates is mortal', which isn't much use at all.
What we need, therefore, is a new system that enables us to translates ideas such as 'All As are F' and 'Socrates is F'. The most commonly used system of this sort is called 'Predicate'.
Subjects / Predicate Analysis
If you've ever learnt a foreign language in school, you'll probably remember write 'S', 'V' and 'O' over various parts of the sentence - marking out which word is the subject, which is the verb, and which is the object. In English, however, people don't tend to do this (though I imagine they might if it's taught as a foreign language) - instead they're more interested into subjects and predicates.
Perhaps it's a sign of the post-Freudian world we live in, but when analysing sentences a useful way of separating things out are into people or objects, and the things that happen to them. The person or object is the 'subject' of the sentence, and the thing that happens to them, which can be expressed by a verb / verbal phrase 'eats the doughnut' or an adjective / adjectival phrase 'is mortal' is the 'predicate'.
Upper and Lowercase Letters
In Predicate, the language, we denote a subject with a lowercase letter, frequently plucked from the middle of the alphabet, using m, n and o first, though any letters will do, though you should really avoid x, y and z for reasons that will become apparent in just a minute. Frequently, however, people choose obvious letters to make translation back and forwards easier - so Socrates becomes 's'. This can be of differing degrees of usefulness, since it makes the initial task easier, but since half the point of logic is to simplify things and then apply rigorous rules to reach a conclusion, you shouldn't need to be able to quickly translate back and forth along the way, and it can actually be quite distracting if there's an unobvious sentence someone in the middle of the working, even though you got to it by correct methods - the beauty of logic is that it doesn't matter how wacky the lines you go through, so long as the steps between them are valid.
Predicates are denoted by uppercase letters, again with the option of choosing useful letters for particular predicates ('M' for 'is mortal'), and almost invariably starting at 'F' if this option is not taken.
Subjects and predicates always go together in predicate, so you shouldn't see any 'a's or 'F's on their own2. So if 's' denotes 'Socrates' and 'M' means ...is a man' then 'Ms' means 'Socrates is a man'. Likewise 'Socrates is mortal', if we take 'N' to mean '...is mortal' (we've already used capital 'M' to mean '...is a man', so we'll have to go for another letter for '...is mortal' - remember what I said about using obvious letters not always being terribly useful?) then 'Ns' means 'Socrates is mortal'.
So now we can say 'Socrates is a man' and 'Socrates is mortal', but what about 'All men are mortal'? To form a quantified sentence you need two additional weapons in our armoury - variable subjects and the quantifiers themselves. A variable subject, which can stand in for any subject are written just as you would write any other subject, but we tend to use the letters x, y and z first (and then begin again at 'a' - which is why we avoid these letters for names (what we call subjects that point to a particular thing) and why we tend to start naming at the letter 'm'. Any letter, however, can be a variable, and we can only really identify them with certainty when they are coupled with a quantifier.
The Universal Quantifier
The 'Universal Quantifier' is written '∀' and means 'For all...', so '∀x' means 'For all x'. This isn't much use on its own, but when coupled with a sentence involving 'x' is very useful indeed. '(∀x)(Mx)' means 'For all x, x is a man', for example. We can now translate the final line of our problematic argument, since 'All men are mortal' means just about the same as '(∀x)(Mx → Nx)' or 'For all x, if x is a man then x is a mortal' in English - think about it.
The Existential Quantifier
The 'Existential Quantifier' is so-named because it denotes that there exists something with a certain property. '∃x' therefore means 'There is an x'. Please be sure to note that this quantifier does not say that there is only one thing with a condition, but that there is at least one - the English is ambiguous, but the predicate is not. '(∃x)(Mx)' therefore means 'There is an x such that x is a man', and happens to be true because there is indeed at least one man in the world3.
It happens to be the case that we can translate almost all sentences with just these two quantifiers, though we can invent as many quantifiers as we like - it just so happens that these are particularly easy ones to use4. In fact, even these two are overdoing it - since we can express each in terms of the other. If '(∀x)(Mx)' then '¬(∃x)(¬Mx)' (think about it!) and if '(∃x)(Mx)' then '¬(∀x)(¬Mx)'.
Stuff to add
Show that Socrates argument works
Discuss arguments involving there exists