I'd like to ask you a question about logic.
I study philosophy in a Spanish Christian university. In the first year, we study logic but it's the classical one, following Aristotle's Organon, the Scholastic authors and so on (I think it's called propositional logic or zeroth-order logic?).
Now, I'm a bit confused, because as far as I know the XIX and XX centuries have created new and different kinds of logics based on mathematics.
The thing is, I'm really lost in these sort of logics. I have no idea about them and I would like to learn them from books.
My thing is continental philosophy, but it's important to learn this stuff. And here is my question: each time that I read an entry in The Stanford Encyclopedia of Philosophy I find a lot of this stuff that I really don't understand. For instance, let's say I want to find info about "Realism", so I start reading the entry and suddenly I find this weird symbols:
Suppose, first of all, that one wished to deny the existence claim which is a component of platonic realism about arithmetic. One way to do this would be to propose an analytic reduction of talk seemingly involving abstract entities to talk concerning only concrete entities. This can be illustrated by considering a language the truth of whose sentences seemingly entails the existence of a type of abstract object, directions. Suppose there is a first order language L, containing a range of proper names ‘a’, ‘b’, ‘c’, and so on, where these denote straight lines conceived as concrete inscriptions. There are also predicates and relations defined on straight lines, including ‘ … is parallel to …’. ‘D( )’ is a singular term forming operator on lines, so that inserting the name of a concrete line, as in ‘D(a)’, produces a singular term standing for an abstract object, the direction of a. A number of contextual definitions are now introduced:
(A) ‘D(a) = D(b)’ is true if and only if a is parallel to b.
(B) ‘ΠD(x)’ is true if and only if ‘Fx’ is true, where ‘… is parallel to …’ is a congruence for ‘F( )’.
(To say that ‘… is parallel to …’ is a congruence for ‘F( )’ is to say that if a is parallel to b and Fa, then it follows that Fb).
(C) ‘(∃x)Πx’ is true if and only if ‘(∃x)Fx’ is true, where ‘Π’ and ‘F’ are as in (B).
What's this? First-order logic ("∃x means there exists a... ") or what? Which books should I read in order to understand these entries and this logic from scratch and be able to write reasoned arguments like this?
Thanks for your help.