I understand why consistency and soundness are important. It would undermine the purpose of reasoning if we can make conclusions that contradict our assumptions. At some level, when we desire to formalize our arguments, we are looking to make sure that we are treating the same forms of reasoning the same way.
With soundness, we just want to ensure that, based on the model, that every conclusion that we may derive according to the system is true. I don't really understand what it means for a formal system to be unsound though: Either it means that it is the wrong model for the proof calculus, or it means that there can be no consistent model for the proof calculus. Any clarification here would be appreciated.
But I don't really understand the value, at least for philosophical logic, of completeness and decidability. An incomplete formal system would have truths that can't be proven. This seems sensible to me, for the same reason that there are truths about the world that we will never be able to prove, like the existence of a flower in a forest fire.
Similarly for decidability. What I've been able to pull up online seems concerned about decidability in the context of theorem proving software and artificial intelligence. Yet, first order predicate logic seems to be used very heavily, even in philosophy, yet it is known to be undecidable. I would think that second-order logic would be even more useful for philosophy, as many propositions, like the identity of indiscernibles and the principle of sufficient reason, can really only be expressed in second order logic.
So why is it so important that formal logic systems be complete and decidable?