Unless I've badly misunderstood it, which is entirely possible, your argument is designed to show that the claim that there can be irresolvable paradoxes is itself a paradox.(source)
Your premise (i) states your opponents' position, for the second of argument. Your (ii) then apparently tries to deduce some further proposition from (i) (since you say "Given (i)..."). What you deduce from (i) is that the negation of (i) (i.e., that there cannot be an irresolvable paradox) would be "either a paradox or a real contradiction".
But as I've pointed out, this is just a non sequitur. You've given no good reason to think this follows from (i).
Perhaps the idea is that, if irresolvable paradoxes are possible, then for just any proposition p we affirm, we must also be prepared to affirm non-p. But again, this is simply a non sequitur. It doesn't follow from (i) at all. Why think that it does?
One might as well argue that, if irresolvable paradoxes are possible, and we believe that the Earth orbits the Sun, then we should also be prepared to believe that the Earth doesn't orbit the Sun. The problem, of course, is that we have plenty of good reason to affirm the former and no good reason to affirm the latter. So your suggestion (if I read you correctly) that if the irresolvable-paradox view is true then it must (by its own lights) be on a par with the no-irresolvable-paradox view, begs the question entirely.
Mr. Anderson is no lightweight when it comes to critical thinking, so I've taken a good bit of time to mull over his comments. Nevertheless, I see a few problems with his critique, or at least a few weaknesses. Let's see if I can explain.
Restatement of the Main Argument
The main argument against irresolvable paradoxes is this:
Suppose for the sake of the argument, a first premise
(P1) The situation that both proposition P is true and P is false (at the same time and in the same way) is a possible situation for any given P.
P1 actually combines two ideas: (1) irresolvable paradox is possible, where irresolvable paradox is defined by a given statement being both true and false in the same way and at the same time; and (2) paradoxes are not limited to only certain categories of propositions.
If P1 is accepted, and if we further add a second premise
(P2) P1 is a proposition, i.e. a member of the set of "any given P"
then we may conclude
(C1) It is possible that (P1) is also false.
Or in other words, if we accept the existence of unlimitable paradoxes, we must also be prepared to accept at least the possibility of the nonexistence of unlimitable paradoxes.
Enhancement to the Main Argument
The main argument may be enhanced, however, through simplification. One enhancement is as follows:
(P3) Reasoned thought is present IFF (i.e. if and only if) the law of non-contradictions is not violated;
(P4) Paradoxes violate the law of non-contradiction; and
(C2) Therefore, reasoned thought is not present when paradoxes are present.
Responses to Objections
Mr. Anderson's main objection seems to be to the boundless aspect of P1. Mr. Anderson, if I have understood him correctly, believes in the existence of irresolvable paradoxes, but only within certain bounds. I'm not sure what objection Mr. Anderson would be able to give to the enhancement argument.
Mr. Anderson's main objection does not appear to be sustainable. It is, of course, handy to say that paradox only exists within special, contained boundaries. And if that were strictly true that would seem to address the problem. Unfortunately, we cannot be assured (within a system that accomodates paradox) that the boundaries themselves are strictly true as opposed to merely paradoxically true.
I don't see any good reason to accept the existence of irreconcilable paradoxes. Such things, were they to exist, would seem to be outside the realm of rational discussion. Accordingly, it would be odd to call any basis for accepting them a "reason." Furthermore, I have seen no reason to reject the strongly intuitive position of the universality of the laws of logic and particularly the law of non-contradiction. I also would see no valid reason for setting boundaries on irreconcilable paradoxes if I were to accept them at all. I'm willing to hear arguments for why I should deny the universality of the laws of logic, but so far I haven't seen any that are logical ... and I'm willing to hear reasons to set boundaries on irreconcilable paradoxes but so far, again, I haven't seen anything beyond simple fiat to support the idea that irreconcilable paradoxes only exist within specific boundaries.