Answer by sobach'e_pole for Is it possible to obtain a contradiction from...
I think the following is true:$\mathrm{GL}\vdash A$$\;\Leftrightarrow\;$$\mathrm{GL}\vdash \square A\quad$ for any modal formula $A$.The proof is by Solovay's theorem: $\mathrm{GL}\vdash A$ iff for...
View ArticleAnswer by Noah Schweber for Is it possible to obtain a contradiction from...
There's an interesting subtlety here, around the deduction theorem.Certainly accepting each sequent of the form $$\vdash \Box P\rightarrow P$$ is fatal for $\mathsf{GL}$ due to its combination with...
View ArticleAnswer by Dan Doel for Is it possible to obtain a contradiction from...
Using that rule, we presumably have a theorem like:$$\dfrac{ \dfrac{\dfrac{\dfrac{}{\square P ⊢ \square P}}{\square P ⊢ P}}{⊢ \square P → P}}{⊢ \square (\square P → P)}$$Then:$$\dfrac{\dfrac{}{⊢...
View ArticleIs it possible to obtain a contradiction from provability logic GL +...
Is it possible to obtain a contradiction from provability logic GL + inference rule ($\square P$ ==> $P$) ?I suspect that answer is "No".If I am right, then there is a model of such logic.And the...
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