Proving that a sentence is inconsistent

Question

I'm trying to understand if the sentence $\square\bot\land \phi$ is consistent in KD. I don't think it is true because it looks like no serial model where this sentence is satisfiable exists. As I understand it, to prove it is not consistent, I must provide a formal proof of $\neg(\square\bot\land \phi)$ in KD. But I failed to do that. Are there other ways of proving inconsistency? (I don't want to appeal to completeness.)

Answer 1

There are essentially two ways to prove that a sentence $\varphi$ is inconsistent: Provide a proof of $\lnot \varphi$, or (assuming you know a completeness theorem for your logic) show that $\varphi$ does not hold in any models. But since you specified in the question that you don't want to appeal to completeness, providing a proof is really your only option.

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As Noah points out in the comments, it suffices to prove $\lnot \square \bot$, because $\lnot p$ entails $\lnot (p\land q)$ by propositional logic. But $\lnot \square \bot$ is equivalent to $\lozenge \top$, and this is easy to prove!

Start with $\top$. By necessitation, $\square \top$. By D (which is $\square p\rightarrow \lozenge p$), $\lozenge \top$.