Market monetarist logic fail

Ok, we have total logical garbage coming from Scott Sumner today. I think the best way to show it is a (pedantic) little refresher on basic logic. Sumner is either making an incorrect attempt at a disjunctive syllogism or a circular argument. In either case, it's logically flawed and I'll show you why.
So let's start with some predicates (CB = central bank):

p = CB wants x
q = CB can achieve x
r = CB raises rates
x = inflation target

Here is Robert Hall's logical argument as presented by Ryan Avent:

p ∧q → x premise
¬x → ¬(p ∧q) contrapositive
¬x → ¬p ∨¬q De Morgan's
¬x observation
p assumption
∴¬q

Sumner at first just claims that the conclusion is dubious based on the rather silly "Bernanke says there are things (a) that could be done to achieve inflation (x)"
∃a : q(a) because Bernanke says
∴ q

... but then Sumner makes the incorrect counter-argument:

p ∧q → x premise
¬x → ¬(p ∧q) contrapositive
¬x → ¬p ∨¬q De Morgan's
¬x observation
r → ¬p assumption
∴q (incorrect conclusion/false disjunction)

Here's the the logically correct argument:

p ∧q → x premise
¬x → ¬(p ∧q) contrapositive
¬x → ¬p ∨¬q De Morgan's
¬x observation
r → ¬p assumption
∴q ∨¬q

That is to say the correct conclusion is that raising interest rates tells us nothing about the central bank's ability to achieve its inflation target. In fact, the only way you can achieve Sumner's conclusion is by assuming q -- i.e. assuming the market monetarist model -- making Sumner's argument above circular ...
Sumner's semantically valid, but circular argument:

q assumption
p ∧q → x premise
¬x → ¬(p ∧q) contrapositive
¬x → ¬p ∨¬q De Morgan's
¬x observation
r → ¬p assumption
∴q ∨¬q (correct conclusion)
∴q (circular conclusion)

You can see the illogical mess better if we write it this way:

q assumption
p ∧q → x premise
p → x disjunctive syllogism
¬x → ¬p contrapositive
¬x observation
∴¬p (conclusion that should have been made already)
r → ¬p assumption (we don't need this to assert ¬p)
∴q (circular conclusion)

Now it is possible that the step where Hall assumes p = "CB wants inflation target" is wrong. That could well be a bad assumption. The thing is that the truth value of p doesn't tell you anything about the truth value of q = "CB can achieve inflation target" given the observation ¬x = "the CB isn't achieving its inflation target" [1].
Of course, the entire market monetarist argument could be summarized as:

q assumption
∴q

UPDATE:

Sumner also brings up the exchange rate argument:

c = exchange rates fall

c → q           (market monetarist model assumption)
∴q

However, the ITM gets the direction of exchange rates just right without q being true, so c → q is not true in general -- it's just a model assumption.




Footnotes:

[1] Note that the observation x = "inflation target achieved" is even less conclusive! You'd need to start with an if and only if version of the premise:
p ∧ q ⇔ x

because the converse

x → p ∧ q

isn't always true.