A course in discrete mathematics that I did recently at University briefly touched on the fundamentals of logic. We covered material like the Laws of Logic, Rules of Inference, quantifiers and existence modifiers. While we were covering logic in one paper we were covering group theory in another course that I was doing. I noticed that primitive logical statements come quite close to being a group but fail when we get to the inverse laws. So i started wondering if there is any use in viewing logic and binary logical operators as a semi-group (I don't know if that is an actual term) of some sorts.
I can't find much information on this simply by googling it, so any information is much appreciated.
You're right to draw an algebraic connection here, but groups and semigroups aren't really the place to look. $\wedge$ and $\vee$ are associative, and so yield semigroup structures; but the class of semigroups is extremely broad, so by itself this doesn't let us say much. And $\implies$, while quite interesting, isn't associative - $\perp\implies(\top\implies \perp)$ is true, but $(\perp\implies\top)\implies\perp$ is false - so it doesn't really fit into this picture.
Instead of trying to make them look individually like groups, the logical operations together give rise to a Boolean algebra (or if we're using intuitionistic logic, a Heyting algebra), which is equivalent in a precise sense to a certain type of ring. The general study of associating algebraic structures to logical systems is called algebraic logic.