handwritten solution please

3. Formalize the following arguments in Li providing a suitable dictionary. Make your formalization as detailed as possible. If you find an ambiguity in the argument, explain it to justify your choice of formalization. Once formalized, say whether the argument is valid in propositional logic. If it is, provide a natural deduction proof. If it is not provide a counterexample. (a) Ali is not happy if he's alone. But he's not alone. Therefore he's happy. (b) If our observations are correct, matter is finitely divisble. For matter is finitely divisible unless there are no atoms. And there are atoms, provided our observations are correct. (c) It's not true that our beliefs about the external world are unjustified. For if there is no knowledge of the external world, our beliefs about it are still justified, or at least they are rational. And if there is knowledge of the external world our beliefs about it are justified