I am trying to learn math "from scratch" and started by reading an introductory logic textbook (A Concise introduction to logic) that did a great job explaining predicate logic. Among other things, proofs, that is, syntactic string manipulation where from a set of premises P, via a number of inference rules, a new statement, aka conclusion C, can be derived.
Slide 42 from here
But how does this connect to predicate logic? The predicate logic proofs begin with premises and here the theorem to be proven looks like the conclusion but no premises are given. Which I can understand: in propositional logic, a theorem is "a sentence that can be derived without premises" (ctrl + f for "theorem" here).
But where do I go from here?
But how can I go about proving it? The techniques in the textbook call for a sequence of conditional derivations and I would guess some universal instantiation or generalizations.
What am I missing to connect these 2 proofs? What is the link between a predicate logic proof and a math proof written in English?
I see that there is a square function involved that can be thought of as a relation (a form of predicate) and Even(x) is also a predicate, but aside from that i see no clues.
The math proof almost feels like a different underlying language and the predicate logic is like its skeleton that can not really be used here. Is that a valid notion?
Proving in math almost feels like using rules of arithmetic to get a matching definition while proving in predicate logic is more about producing a new string from the existing ones.
I see how this is a lot of thoughts pointing in potentially different directions but I am hoping that someone more experienced could notice/relate to the struggle and point me in the right direction.
Thank you everyone!
I believe my question is similar to the mathematical proof vs. first-order logic deductions but I did not find anything to answer my particular example.