We were introduced to these rules during class and not sure how to grasp it. And what does this symbol mean ⊢?
UPDATE: Added rules 1a and 1b
Rule 1a: if the goal list has a proposition that is also in the hypothesis list, the conjecture is true.
Rule 1b: if both the goal list and hypothesis list contain propositions only, and no common propositions, the conjecture is disproven.
These are the rules for Sequent calculus.
It formalizes the rules regarding the derivability relation: $\vdash$ (the symbol is called "the turnstile").
Each rule is expressed in terms of sequents: $A_1,\ldots, A_n \vdash B_1, \ldots B_k$.
The formulas to the left of the turnstile are called the antecedent, and the formulas on the right are called the succedent or consequent.