# Classical logic, symbolic logic, higher-order logic, First-order logic? Learning from scratch

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?

• Yes; a lot of things happened in the field of logic since e.g. Kant. See The Emergence of First-Order Logic. – Mauro ALLEGRANZA Jan 15 '19 at 9:45
• And yes; the quoted passage above use the Language of First-order logic with quantifiers : ∃x is the existential quantifier and reads "some (thing) ...", while ∀x is the universal quantifier and reads "every (thing)...". See also Quantifiers and Quantification. – Mauro ALLEGRANZA Jan 15 '19 at 9:48
• Thanks! So I supppose the thing is to learn about First-order logic and formalizacion of natural languages in FOL. Do you know good books about this? Thanks in advace. P.S.: Also: If I learn First-order logic, would I be able to understand this formalized language used in Plato Standford? (taking apart entries that deal specifically with logical contents of course). – Jasso Jan 15 '19 at 9:52
• @Mauro But quantifiers still apply only to defined domain. There is no domain of "Everything". So, it should be read as "Every(thing) in ..." – rus9384 Jan 15 '19 at 10:14
• @Jasso I guess that depends. Aside from FOL there are higher order logics, type theory, category theory and so on. Whether SEP contains it, I dunno, but they are different. – rus9384 Jan 15 '19 at 10:15

The OP has this question:

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?

To learn to use truth-functional logic and first order logic from a natural deduction perspective you might try forallx. The text is available on-line without cost and there exists a proof checker that is also available on-line for practice. This would be one way to learn these. The links are below.

Additional resources would be this stack exchange. Search under tags such as "fitch" and "symbolic logic". You may even find posts associated with the forallx text by searching for "forallx".

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

• I've just downloaded forallx, it looks fantastic! Let me ask you a question: I've been browsing Peter Smith's An Introduction to Formal Logic, which looks like a (non so) friendly introduction, plus his Teach yourself logic which is little bit more advanced in content. Do you recommend these books for a beginner? – Jasso Jan 17 '19 at 9:46
• @Jasso I haven't read either of those texts, but it is good to have multiple approaches to logic that one can consult for comparison. One text I do like is Harry Gensler's Introduction to Logic because it covers modal logic as well from a natural deduction perspective. It also has an introduction to Aristotle's term logic, but you are already familiar with that. – Frank Hubeny Jan 17 '19 at 9:52