Replacing equivalent formulas and Rule N in epistemic logic

Question

I am studying Epistemic Logic on my own from different books and articles. I have some knowledge of Modal Logic. In one of the basic steps in epistemic logic, I am having some difficulties. Please help me to resolve the problem.

In this article a very interesting motivation is given for epistemic logic. It is written on page $6$ of the article that :

“Given that Lake Superior is the largest fresh-water lake, we can infer that Lake Superior is blue if and only if the largest fresh-water lake is blue. The operator “know” is referentially opaque; substituting an equal term inside a knowledge operator may change the truth of a sentence. It is possible that Andrea knows that Lake Superior lies partly in the US, but does not know that the largest fresh-water lake lies partly in the US.”

What I understand from above that in propositional calculus if two wff $\alpha \text{ and } \beta$ are equivalent (i.e $\alpha \to \beta \text{ and } \beta\to \alpha$ both true) then in any formula we can replace $\alpha$ by $\beta$. But from the above statement I think that in the case when we are using modal operator $K_a$ then this kind of substitution is perhaps not possible.

Now I have read some modal logic and I know that if we have inferred $\alpha \to \beta$ then by necessitation rule (Rule N) we can infer $\square \alpha \to \square \beta $ and thus when $\alpha \to \beta \text{ and } \beta\to \alpha$ are both inferred i.e. $\alpha \text{ and } \beta$ are then we can infer both $\square \alpha \to \square \beta $ and $\square \beta \to \square \alpha $ which gives $\square \alpha \text{ and } \square \beta $ are equivalent.

Now since $K_a$ is a box modal operator in epistemic logic, hence it should happen that when we are able to replace $\alpha$ by $\beta$ then we must be able to replace $\square \alpha$ by $\square \beta$ in any formula. But this somehow contradicts my thought from the example I mentioned at the very beginning.

So I am confused about my understandings. Perhaps I am missing something or inferring something wrong. But I cannot find where I am wrong.

Please help me to solve this problem. It will be very helpful for me. Thnx in advance.

Answer 1

I think that the problem you're having here is similar to the following apparent "paradox" we often see in more general modal logics: if we can infer $\phi$, then by necessitation we can infer $\square \phi$, yet the sentence $\phi \rightarrow \square \phi$ is not in general valid. The point is that if we can really infer $\phi$, starting from no axioms, then $\phi$ is a logical tautology, and this is a much stronger statement than $\phi$ happening to be true in some model. It may be true that I wore a red shirt today, but we cannot on that basis conclude it was necessarily true.

In this particular case, if $\alpha$ denotes "Lake Superior is blue" and $\beta$ denotes "the largest freshwater lake is blue", then $\alpha \rightarrow \beta$ is certainly true in this world, but it is not a logical tautology -- it depends on the fact that in this particular world, the largest freshwater lake is in fact Lake Superior. That is, it is true in this model, but need not be true of every model of the relevant logic. For this reason, we don't get to infer $\square(\alpha \rightarrow \beta)$, which appears to be what you're assuming.