Municipal elections in Spain have just been run with interesting results. In Barcelona, the nationalist ruling party CiU, noted by its recent shift towards unreserved support for the independence of Catalonia, has lost to Barcelona en Comú, a political alliance loosely associated to left-wing Podemos, ambivalent on these matters. In Madrid, the party in office and undisputed winner for more than twenty years, right-wing PP, has lost a great deal of support, almost yielding first place to Ahora Madrid, also linked to Podemos, which is now in a better position to form government. Commenting on the elections, CiU's Artur Mas, president of the Catalonian regional government, says (translation mine):
Mas has discarded the possibility that CiU's defeat in Barcelona has been influenced by the party's pro-independence discourse. "Madrid might also have a mayor from Podemos, and there is no independentist movement there", he said.
Is Mr. Mas's argument sound? Let us analyze it formally.
Logical analysis
Write
i(x) = party in power in county x supports independence of the region,
p(x) = Podemos (or associated alliances) gains support in x over the party in power,
p(x) = Podemos (or associated alliances) gains support in x over the party in power,
where x ranges over all counties in Spain. Then, Mas's reasoning can be formulated as
It is not the case that ∀x i(x) → p(x) because ¬i(Madrid) ∧ p(Madrid).
Put this way, the argument is blatantly false: a refutation of ∀x i(x) → p(x) would ask for a city with a pro-independence governing party where Podemos did not succeed, which is not the case of Madrid. Let us try another approach.
Probabilistic analysis
If Mas was thinking in statistical terms, a plausible formulation could be
It is not the case that P(p(x) | i(x)) > P(p(x)) because ¬i(Madrid) ∧ p(Madrid),
that is, the probability that Podemos gains power in a pro-independence context is not higher that in general, as supported by the results in Madrid. If we write
PI = number of counties with pro-independence councils where Podemos wins,
I = number of counties with pro-independence councils where Podemos runs,
P = number of counties where Podemos wins,
N = total number of counties where Podemos runs,
I = number of counties with pro-independence councils where Podemos runs,
P = number of counties where Podemos wins,
N = total number of counties where Podemos runs,
then Mas afirms that
PI / I ≤ P / N
or, equivalently,
D = (P / N) − (PI / I) ≥ 0.
Is the fact that ¬i(Madrid) ∧ p(Madrid) giving any support to this contention? Let
D' = (P' / (N − 1)) − (PI' / I')
be the value resulting from considering every county in Spain where Podemos ran except Madrid. When taking Madrid into account, we have
D = ((P' + 1) / N) − (PI' / I')
and, consequently,
D − D' = ((P' + 1) / N) − (P' / (N − 1)),
which is always non-negative (since P' is never greater than N − 1), and furthermore strictly positive if P' < N − 1, that is, if Podemos has lost somewhere (which is certainly the case). So, knowing about the existence of counties without pro-independence councils where Podemos won increases the chances that, in effect, D ≥ 0 and, consequently, independentism not play a beneficial role in the success of this party. But if this knowledge is of just one county out of the hundreds where Podemos ran, the degree of extra confidence we gain is extremely small. This is the very weak sense in which Mas's argument can be held valid.
