Unfortunately, it seems that these students do not trust you as a subject matter expert. This is a difficult place to be. If they are convinced that your proofs are invalid, that your understanding of logic is faulty, and that your understanding of mathematical induction is circular, it will be very difficult to regain any sort of healthy classroom atmosphere.

Put yourself in their shoes: if you were taking a class and the instructor was clearly spouting nonsense (for instance, a biology class where the instructor clearly had a Lamarckian rather than Darwinian understanding of evolution), you too might act disruptively. I certainly would!

This is an extremely serious problem and it calls for serious solutions. Here are my suggestions:

1. Hold a meeting with the disruptive students and see if I am right: do they consider you to be mathematically or logically incompetent?
2. If so, you need to be prepared with a plan to address their concerns. Since the students themselves (clearly) do not have the logical prowess necessary to judge you, all you are left with is an appeal to authority. See if there are any instructors at your institution who they trust as mathematical authorities.
3. Invite these students, and their chosen authority, to a meeting together. Discuss these issues together (vacuous implications, mathematical induction, the standards you are holding them to in their logical arguments, etc.). When their authority figure confirms that they are wrong, and that you are right, this might have some impact. Hopefully, once they realize that they are wrong, they can begin really listening to your instruction and trying to learn from it.
4. Make it very clear to them that you will not tolerate disruptive conversations and behaviors during class. Make a clear distinction between genuine questions (in which both the asker and the receiver carry a curiosity about each other, and a humbleness in their responses), to questions which are designed only to inflame or to dismantle authority. Let the students know that the later sort of question will be identified as inappropriate and that you will ask a student who persists in these attacks to leave the classroom. Same comments apply to derailing groupwork: let them know that they are derailing their group, and that if they cannot stay on task they will be asked to leave.

EDIT: I would like to clarify that my response below is not intended to be definitive. This is an extremely difficult problem to have. It is perhaps the most difficult problem one can have as a teacher: a complete breakdown of the trust in each other which is needed to make communication possible. The idea below is only a stab in the dark, which has many obvious potential drawbacks. Very open to hearing other approaches which are less problematic.

Unfortunately, it seems that these students do not trust you as a subject matter expert. This is a difficult place to be. If they are convinced that your proofs are invalid, that your understanding of logic is faulty, and that your understanding of mathematical induction is circular, it will be very difficult to regain any sort of healthy classroom atmosphere.

Put yourself in their shoes: if you were taking a class and the instructor was clearly spouting nonsense (for instance, a biology class where the instructor clearly had a Lamarckian rather than Darwinian understanding of evolution), you too might act disruptively. I certainly would!

This is an extremely serious problem and it calls for serious solutions. Here are my suggestions:

1. Hold a meeting with the disruptive students and see if I am right: do they consider you to be mathematically or logically incompetent?
2. If so, you need to be prepared with a plan to address their concerns. Since the students themselves (clearly) do not have the logical prowess necessary to judge you, all you are left with is an appeal to authority. See if there are any instructors at your institution who they trust as mathematical authorities.
3. Invite these students, and their chosen authority, to a meeting together. Discuss these issues together (vacuous implications, mathematical induction, the standards you are holding them to in their logical arguments, etc.). When their authority figure confirms that they are wrong, and that you are right, this might have some impact. Hopefully, once they realize that they are wrong, they can begin really listening to your instruction and trying to learn from it.
4. Make it very clear to them that you will not tolerate disruptive conversations and behaviors during class. Make a clear distinction between genuine questions (in which both the asker and the receiver carry a curiosity about each other, and a humbleness in their responses), to questions which are designed only to inflame or to dismantle authority. Let the students know that the later sort of question will be identified as inappropriate and that you will ask a student who persists in these attacks to leave the classroom. Same comments apply to derailing groupwork: let them know that they are derailing their group, and that if they cannot stay on task they will be asked to leave.