Question 1: Why propositional logic has no axioms?
There are different proof systems for propositional calculus; some - called Hilbert-style - have axioms and rules; some, like e.g. Natural Deduction rules only.
When we speak of propositional logic, we usually speak of the language and the calculus: thus, we say that propositional logic is consistent because we cannot derive $\bot$ in the calculus.
Question 2: Why propositional logic is consistent?
We associate to the logic a semantics: for classical logic, the usual semantics is defined via the truth tables for the connectives.
We define the concepts of tautology and tautological consequence as well as the related properties of soundness and completeness.
Soundness implies that the calculus derives only tautologies (while completeness means that the calculus derives all the tautologies).
The key points of soundness are:
Having said that, we immediately have the consistency of the claculus:
$\bot$ is not a tautology.
For an introduction, we can see: S.Simpson, Mathematical Logic (2013).
You can define anything you want. However, the point of defining something is to make it easier to refer to, which means that the most useful definitions are for things that are:
(a) frequently referred to;
(b) not trivial; and often
(c) similar to something else
So, for example, we define $\wedge$ because it allows for a lot of shortcuts in writing the propositional logic, and it happens to align with the general understanding of the word "and". The "=" in the definition isn't really part of the logic, it's a part of the language surrounding it, and we know that there's a level at which we have to resort to shared understanding since you can only abstract things so far.
On the other hand, I probably wouldn't bother coming up with a definition for "the set of all even prime numbers in $\mathbb{N}$", because it's simple enough to just say $\{2\}$. Or if I did define it, it would only be for a very limited context (for example, one where I actually needed to prove that 2 is the only element in the set), so I could get away with a generic definition like $A$.
Best Answer
You can see a full exposition of the Completeness Theorem for propositional logic in every good math log textbook, like :
The proof system used is Natural Deduction; here is a sketch of the proof.
The proof of it needs the rules of the proof system.
For (a) : by assumption we have (see def of consistency) $\Gamma, \lnot \varphi \vdash \bot$. Now apply the (RAA) rule [i.e. if we have a derivation of $\bot$ from $\lnot \varphi$, we can infer $\varphi$, "discharging" the assumption $\lnot \varphi$] to conclude with : $\Gamma \vdash \varphi$.
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