Since there are so many different possible systems for modal logic, it is important to determine which systems are equivalent, and which ones distinct from others. Figure 11.1 lays out these relationships for some of the best-known modal logics. It names systems by listing their axioms. So, for example, M4B is the system that results from adding (M), (4), and (B) to K. In boldface, we have also indicated traditional names of some systems, namely, S4, B, and S5. When system S appears below and/or to the left of S′ connected by a line, then S′ is an extension of S. This means that every argument provable in S is provable in S′, but S is weaker than S′, that is, not all arguments provable in S′ are provable in S.

Showing Systems Are Equivalent

One striking fact shown in Figure 11.1 is the large number of alternative ways of formulating S5. It is possible to prove these formulations are equivalent by proving the derivability of the official axioms of S5 (namely, (M) and (5)) in each of these systems and vice versa. However, there is an easier way. By the adequacy results given in Chapter 8 (or Chapter 9), we know that for each collection of axioms, there is a corresponding concept of validity. Adequacy guarantees that these notions of provability and validity correspond. So if we can show that two forms of validity are equivalent, then it will follow that the corresponding systems are equivalent.