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In such solvents, could there be chemicals which could donate/accept protons, and if so could their solutions display acidic/basic properties?

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Certainly! There are various definitions for acid and base, such as Brønsted–Lowry, that describe acid/base reactions as the transfer of protons ($\ce{H+}$), without requiring hydroxyl ions ($\ce{OH-}$), and Lewis, which goes beyond $\ce{H+}$ and deals with electron exchange.

An example of a Brønsted–Lowry acid/base system is ammonia as the solvent. The ammonium ion ($\ce{NH4+}$) corresponds to the hydronium ion, and the amide ion ($\ce{NH2-}$) to the hydroxide ion.

An example of a Lewis acid is $\ce{BF3}$, an electron acceptor rather than a proton donor.

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