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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.

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    Please use [MathJax](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference). – 2017-02-16
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    I actually find it clear as is. However MathJax would work as well. – 2017-02-16
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    See example page 123 bottom: $Fa$ is $^*$ in $k$ while $Fb$ is $^*$ in $n$. Thus $\square Fa$ and $\square Fb$ are both $0$ in $n$ and $k$. The intuitive semantics for $\square \varphi$ is : $\varphi$ is "necessarily true", i.e. must be *true* in every world. – 2017-02-16

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