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Long comment Regarding inconsistency, we will use the "standard" definition : $\Gamma$ is inconsistent iff for some $\phi : \ \Gamma \vdash \phi$ and $\Gamma \vdash \lnot \phi$. I assume that we can use also the definition of $\lnot \phi$ as : $\phi \to \bot$. Thus, the above definition is equivalent to : $\Gamma$ is inconsistent iff $\Gamma ... 3 The "paradoxical" configuration is : Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. We have here Bob's assumption : "Ann believes that Bob’s assumption is wrong". Now consider Ann's belief attitude towards Bob's assumption; we can say that if$p\$ is any statement, a "reasonable" principle will be : ...