I am reading Rod Girle's Modal Logics Philosophy section 8.6 on Free Logic.
And I have a problem understanding how truth values of modal logic formulas are determined in free logic.
In Rod Girle's example, the counterexample for validity of ((x)□Fx ⊃ □(x)Fx) would be;
in n, Fa =*, (x)Fx = 1 □Fa = ∗, (x)□Fx = 1, □(x)Fx=0, * means truth gap;
in k, Fa = 0, (x)Fx = 0, □Fa = 0, (x)□Fx = 0, □(x)Fx = 0
Q(n) is empty, and Q(k) has a as its member.
So a does not exist in n, Fa = *, and □Fa = *.
Yet in another example of Rod Girle, the counterexample for (□(x)Fx ⊃ (x)□Fx) is,
Fa 1 * (x)Fx 1 1 □Fa 0 0 (x)□Fx 0 0 □(x)Fx 1 1 (in n, k respectively)
This time Q(k) is empty, and Q(n) has a as its member.
What I don't get is that □Fa is false in k even though there is no such thing as a in k, making Fa truth-value gap in k.
How should I grasp the relation between Fa and □Fa in free logic?
Please help.