Consider an example 2-SAT instance with the constraint (x1 ∨ x2). This CNF has these two implications:
¬x1→x2 and ¬x2→x1.
"They actually mean, if x1 is false then x2 must be true, and if x2 is false then x1 must be true, respectively. Any other case would make the 2-sat problem unsatisfiable."
My doubt is regarding this statement. I can understand the above written two implications, but I don't understand how only those two values of x1 and x2 make this 2-SAT instance satisfiable. Clearly setting both x1 and x2 to true can also satisfy this constraint. I don't see that case being captured in the two implications written above.
Can someone explain this to me? Apologies if this is a stupid question, I haven't formally studied logic.