(1') $\Diamond (\mathit{sofabed} \vee \mathit{guestbed})$ (2') $\Diamond \mathit{sofabed} \wedge \Diamond \mathit{guestbed}$

Producing a valid proof in this problem will enable you to understand The Free Choice Permission Paradox (FCPP), discovered in 1941 by Ross (“Imperatives and Logic,”: 53–71). Given that the proof in question yields an absurdity, FCPP can be taken to show that(Standard Deontic Logic) =leads to inconsistency when applied; or, put in AI terms, you wouldn’t want a robot to base its ethical decision-making on! Fortunately, the RAIR Lab ’s modern cognitive calculus $\mathcal{DCEC}^\ast$ allows FCPP to be avoided. (A recent paper explaining the use by an ethically correct AI of this calculus is available here .) Here's the paradox. Suppose that you travel to visit a friend, arrive late at night, and are weary. Your friend says hospitably: “You may either sleep on the sofa-bed or sleep on the guest-room bed.” (1) From this statement it follows that you are permitted to sleep on the sofa-bed, and you are permitted to sleep on the guest-room bed. (2) In, this pair gets symbolized like this:But (2') doesn't follow deductively from (1') in, as a call to the provability oracle forin the HyperSlatefile for this problem confirms. A suggested repair is to add tothe schema $$\Diamond (\phi \vee \psi) \rightarrow (\Diamond \phi \wedge \Diamond \psi),$$ but as your proof will (hopefully) show, this addition allows a proof of the absurd theorem that if anything is morally perimssible, everything is! Your finished proof is allowed to make use of the PC provabiity oracle, but of no other oracle.