FAAAAARK THIIIIS Right, rage hat on, I'm not buying this. It's an absolute fucking classic on how not to write a maths book. From the PDFs, some crap examples

FAAAAARK THIIIIS Right, rage hat on, I'm not buying this. It's an absolute fucking classic on how not to write a maths book. From the PDFs, some crap examples then. "It is also standard to define two notions of validity. The first is semantic. A valid inference is one that preserves truth, in a certain sense. Specifically, every interpretation (that is, crudely, a way of assign-ing truth values) that makes all the premises true makes the conclu-sion true. We use the metalinguistic symbol ‘|=’ for this. What distin-guishes different logics is the different notions of interpretation they employ." Wut the utter fuck does that mean? What could "preserves truth, in a certain sense" possibly mean? What the fuck is a metalinguistic symbol anyway? He goes on: "The second notion of validity is proof-theoretic. Validity is defined in terms of some purely formal procedure (that is, one that makes reference only to the symbols of the inference). We use the metalinguistic symbol ‘|-’ for this notion of validity" I think he's saying ‘|=’ is the human version of 'therefore' or 'we can deduce', and the formal version is ‘|-’, which corresponds to my understanding but if I didn't have a background in this I'd be fucked. This stuff is not complicated and it's mainly not hard, but it is fucking impossible if you write like this. Later on we have " ⊃ " which is "material conditional" - da fuq? This is logical implication I think, it is standard in classical logic as an arrow '->' but he neither uses that nor relates it to the arrow symbol, nor even gives it the standard name - I have never ever seen it called a Material Conditional. And he doesn't give its definition so I can't tell. Later he gives a tree form of this I've never seen before but can sort-of understand so I guess it is logical implication. And how does material equivalence '≡' differ from equality '='? Explain this shit please. "An interpretation of the language is a function, ν, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as ν(p) = 1 and ν(q) = 0." If I didn't know what this meant I'd be lost. "A branch is closed iff there are formulas of the form A and ¬A on two of its nodes; otherwise it is open." I'm not sure what he's saying although I know it's actually very simple. It's something like 'if you get x on one branch and ¬x on another then you have a contradiction, which means that branch is junk because it's claiming something is true while the other is claiming exactly the same thing is false; they can't both be true. Except that's a bit wordy and more importantly I'd need to sit down and consider the scope; what a branch actually is and how far back up the contradiction goes. This is a lesson on how not to do it. I'm getting lost and the 'branch is closed' example is on page 8! I would love to understand the stuff in this but the medium of conveyance, to wit this book, is a car crash. At best it's lecture notes but you'd better have the lecturer around.