if 2+2=5, then bertrand russel is the pope!

[Entropia]

Inconsistant logical systems are not useful in getting at "the truth". The following is a cute "proof" by Bertrand Russel:

If 2+2=5...
that means 4=5...
so, lets subtract 2 from each side...
that gives us...
2=3....
transposing, we have 3=2...
now, lets subtract 1 from each side...
2=1...
Now, since the Pope and Russel are two different people, and 2=1...
Therefore, the Pope and Russel are one.

[HallsofIvy]

It's not necessary to be that complicated.
In general, If p is a false statement then
p=> q
is true for any statement q:

The direct statement: "if (2+2= 5) then Bertrand Russel is the pope" is a true statement.

[metacristi]

There is an anecdote about this demonstration (I do not know if this really happened).It is said that during a public lecture Russell was making the remark that from a contradictory set of axioms practically everything can be deduced.At this a man from the public interrupted the lecture and challenged him to prove that from 4=5 results that he is the Pope.Which he brilliantly did...

In another sequence of ideas [my comments are general,possible you know very well,I do not intend to sell flowers to the gardener] the necessity to avoid internal incoherences is what make epistemology so important:its task in mathematics and applied sciences is to establish whether the scientific hypotheses (or the axioms of mathematics) are coherent therefore acceptable logically:all sets of premises (including axioms) must have internal consistence.Indeed in order to be experimentally adequate all scientific hypotheses must pass first the logical test.

Is logic a feature of reality?Well,there is no necessary connection between ontological reality and logic (since logic is a feature of human reasoning).From the fact that we understood so many facts about nature using conceptualizations based on logic does not follow that this will always be the case;in fact there are enough many scientists who are skeptical about the prospects of finding a 'theory of everything' exactly due to this problem.
Anyway,as Einstein once remarked,it is a real miracle that we have been able to understand so many things so far...