TITLE: History of the predicate calculus
QUESTION [6 upvotes]: My situation. Recently I began to study logic. In particular, I want to understand the "classical results": completeness theorem, compactness theorem, Löwenheim-Skolem, and maybe the incompleteness theorem. In order to understand these theorems, one first needs to grasp some concepts. For example, one needs to know what a model for a given language is, what a first-order formula is, and so on.
Right now I am dealing with the concept of a formal deduction in a Hilbert-style calculus. More precisely, I try to understand the logical axioms of this calculus. These are the following (as given in the wikipedia article I linked to):
1. Propositional axioms (axiom schemes)

P1. ${\displaystyle \phi \to \phi }$
P2. $\displaystyle \phi \to \left(\psi \to \phi \right)$
P3. $\displaystyle \left(\phi \to \left(\psi \rightarrow \xi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \xi \right)\right)$
P4. $\displaystyle \left(\lnot \phi \to \lnot \psi \right)\to \left(\psi \to \phi \right)$

2. Quantifier axioms (axiom schemes)

Q5. $\displaystyle \forall x\left(\phi \right)\to \phi [x:=t]$ where $t$ may be substituted for $x$ in $\displaystyle \,\!\phi$
Q6. $\displaystyle \forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\phi \right)\to \forall x\left(\psi \right)\right)$
Q7. $\displaystyle \phi \to \forall x\left(\phi \right)$ where $x$ is not a free variable of $\displaystyle \,\!\phi$

3. Equality axioms (I9 is also an axiom scheme)

I8. $\displaystyle x=x$ for every variable $x$.
I9. $\displaystyle \left(x=y\right)\to \left(\phi [z:=x]\to \phi [z:=y]\right)$

Modus-ponens can be used as the rule of inference (deduction rule).
For me, as a beginner, the rules and axioms of this calculus seem very unnatural and artificial. I wonder how somebody can invent such a calculus. I mean, how did somebody get the idea to use exactly these axioms? I am curious about the development process of this calculus. Could somebody of you give a brief introduction in the history of how it came that the predicate calculus has the above axioms and rules?

REPLY [3 votes]: Your question has a somewhat false premise. The particular set of rules you have mentioned is not the only possible set, and there are very different alternative deductive systems for first-order logic, including Fitch-style natural deduction, tree-style natural deduction, sequent calculus, and finally Hilbert-style systems (including the one you mentioned).
The real reason for each deductive system is that they suffice. What does this mean? Well, we want to be able to prove every logically necessary sentence, given some set of axioms. This means that if every model satisfies some sentence, we want our deductive system to be able to prove it. It turns out that each of these systems can do so! This is known as the completeness theorem for first-order logic. Now you may complain that this means that the completeness theorem is tied to the choice of deductive system. In a way, yes, but it in fact provides a way for us to figure out what deductive rules we need, because at every step of the completeness theorem we just need to include enough sound deductive rules that let the proof of the completeness theorem go through. If you look at the proof, you will realize that for any reasonable kind of deductive system that we wish to design, we would always be able to add such rules. In that sense the completeness theorem is a foregone fact!
There is another way to arrive at any one of these deductive systems. We can start with any other, such as a natural deduction system, which we find most intuitive, and then we can see that we need certain kind of rules to be able to 'implement' the natural deduction rules in the new system. For example:

P1: restatement.
P2: restatement under an assumption.
P3: modus ponens under an assumption.
P4: somewhat like proof by contradiction.
and so on...

There may not be direct parallels, but as long as the new system has only sound rules and can implement each rule of natural deduction (suitably translated), we automatically have that the new system is complete too.
Hilbert-style systems have a major advantage in that its only inference rule is modus ponens, and so it is easier to prove meta-theorems about it. But they have a major disadvantage in that they are totally unusable in actual mathematical practice. But the translation between Hilbert-style systems and other more intuitive systems means that all the meta-theorems for Hilbert-style systems that are not sensitive to proof length also hold for the more intuitive systems! (Actually it turns out that Hilbert-style systems tend to be terribly inefficient because of the repetition.)