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Is there any software around that using the Fitch format (used in Language, Proof and Logic), allows one to put a specific set of premises and goals and have it show us the full list of steps needed to solve the problem?
Short answer: No.
Medium Answer: Can’t really be done, though one could write a program to check the validity of a given proof fairly easily. In the case of propositional logic, the problem of automatically finding a proof is NP-complete (though it is decidable!), and in first order logic there are true theorems for which the prover would never stop. (undecidable) (via Gödel’s incompleteness proof)
If you’re interested in writing such a thing, you can take a stab at it, and maybe get it to work for some smaller cases, but it’s not doable in general.
If you’re looking for such a thing to get answers for your homework, quit trying. (a) you won’t find it and (b) the problems from that book are pretty easy, and can be fun! Just give them a try, and seek out help if needed. and, of course, (c) you won’t learn anything if you cheat.