Is binary (Boolean) logic related at all to the two-satisfiability problem?
And is ternary logic related in some way to the three-satisfiability problem?
Would it follow then that if one were to work in a ternary logical system, computations would be much “harder”? (Assuming P does not equal NP?)
https://en.wikipedia.org/wiki/Boolean_satisfiability_problem
https://en.wikipedia.org/wiki/Three-valued_logic
https://en.wikipedia.org/wiki/Many-valued_logic
No, there is no connection of the kind you seem to be imagining.
2-SAT and 3-SAT are both about satisfiability in ordinary two-valued logic. They differ in which shapes of formulas you can ask to have satisfied, but satisfying that formula is a matter of ordinary true/false logic in both cases.
The "canonical" satisfiability problem in computational complexity is 3-SAT, about two-valued logic and formulas with at most three atoms per clause.
2-SAT is a restricted, particularly easy, case of that.