Just for the sake of reminding, the paradox is "The barber shaves all and only those who do not shave themselves". I found a Youtube channel by some professor named Frederic Schuller that contains a few set theory lectures (I recommend them by the way).
In the first one he says at some point that this famous statement is a paradox and claims that whether or not it is true is undecidable.
I had little background in propositional logic theory but having watched the video up until that point I felt like I could use what I had learnt from him to easily prove that the statement was false assuming every proposition to either be true or false (i.e. assuming from the start that undecidable propositions do not exist). I'm sure Mr Schuller must be right of course, but here's my work.

Let $A$ be the proposition "The barber shaves himself". Then the initial statement merely says $(A \implies \neg A) \wedge (\neg A \implies A)$.
If we agree that a proposition is always either true or false, then one of the statements on either side of the wedge is false (whichever one carries $\text{true} \implies \text{false}$, according to the binary truth table). Then the "and" operator represented by $\wedge$ yields false. And here I cannot see how I failed to prove that the Barber's "paradox" is false.