Reddit reviews Metalogic: An Introduction to the Metatheory of Standard First Order Logic
We found 4 Reddit comments about Metalogic: An Introduction to the Metatheory of Standard First Order Logic. Here are the top ones, ranked by their Reddit score.
This is for a general study guide for logic. Very solid.
http://www.logicmatters.net/tyl/
and I think you can give one of these two a try if you find "Computability and Logic" difficult.
https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560
or
https://www.amazon.com/Friendly-Introduction-Mathematical-Logic/dp/1942341075
It's such a good question because it really gets to questioning logic itself. It is a metalogical question, and I think you would find that looking into metalogic will be very interesting for you. I would recommend that you check out Geoffrey Hunter's Metalogic. It is a bit older, but you can find it starts from the most basic concepts and builds up to very advanced concepts in philosophical and mathematical logic. (You can get much of it free on Google Books or even get the physical book for pretty cheap).
In general, the axioms that logicians use to construct their logical systems are chosen by fiat (This fact is sometimes a little shocking to some). There is no special or metaphysical significance to the particular axioms that a logical system has. Usually they are chosen because they have certain qualities that the logician is interested in (e.g. one axiom may make it possible to add conjunction to a line, or another may make it possible to take away a disjunction from a line, etcetera). There are all kinds of different axiomatic systems using differing sets of axioms. Usually the logician will want the system to be complete in some sense (there are different ways a logical system can be complete), consistent, or decidable, or be able to be interpreted in certain ways.
The ultimate goal of all of these systems is to give some account of logical truth and logical consequence. All of logic can be thought of as constructing different models of how logical truth can be expressed, and what expressions can be considered to "follow from" some other expressions.
Your very deep question about how we know the rules we use are correct is a question of epistemology. The goal of logicians is always to follow reason. They use their capacity to reason to analyse whether or not the conclusions "logically follow from" the premises. What the nature of reason is, is still an open question. You can think of a logical truth as being "self-evident" upon reflection. No matter how complex a theorem or logical truth is, it should always be possible to construct it in terms of building blocks, each of which is supremely obvious or "self-evident."
Syntax just refers to the symbols, expressions and rules without regard to any interpretation of them. That means when you are simply moving symbols around according to rules you are using syntax. These rules work just like moving boxes around a room according to rules or moving chess pieces around a board according to rules.
But whether or not you are using chess pieces or boxes is a matter of semantics. Semantics are the interpretations of your syntax. That is when you fill in the values of all the variables like "P"s and "Q"s with meanings. Without semantics you are moving around meaningless symbols. Semantics is when you assign a particular meaning to the syntax. Usually there is a particular intended interpretation of the syntax, but very often the systems will make other interpretations possible too.
I think others ITT have addressed the issue of non-classical logics, so I will leave that issue to them.
The best introductory logic text you will ever find: Logic: Techniques of Formal Reasoning, 2nd Edition Donald Kalish, Richard Montague.
This book is especially good if you have done any programming. The structure of main and sub-proofs corresponds to main program and subroutine calls. You can pick up a used copy for around $23 here: https://www.abebooks.com/book-search/author/kalish-montague-mar/ and you can see the table of contents here: https://www.powells.com/book/logic-techniques-of-formal-reasoning-9780195155044 (but, obviously, don't buy it for $133!)
For meta-theory, take a look at: Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, https://www.amazon.com/Metalogic-Introduction-Metatheory-Standard-First/dp/0520023560. This book explains things in a clear way using ordinary English, before setting out the proofs.
And, if you are interested in model theory, take a look at Model Theory by C.C. Chang and H. Jerome Keisler, https://www.amazon.com/Model-Theory-Third-Dover-Mathematics/dp/0486488217 and you should get a good idea of what additional mathematics you might want to pursue.
I don't know any books off-hand which cover both.
RE: metatheory - I hear Hunter's Metalogic is good, but it's fairly old.
RE: modal logic - I suggest Hughes & Cresswell or Garson. I don't like Mendelsohn.