I would assume that the vaccine couldn't be semi-effective against cervical cancer caused by all strains without it being effective against all strains, since your first link says 95%+ of cervical cancer is caused by HPV and there's no known mechanism for the vaccine to act directly against cervical cancer without first acting against the HPV virus.

I apologize for the ambiguity of my prior comment. To illustrate what I'm thinking: Suppose cervical cancer is caused by two strains, 1 and 2, which are equally present in the population. A vaccine works 100% against strain 1 and 0% against strain 2. That vaccine is 50% effective against all strains causing cancer, despite having 0 power to eradicate strain 2.

Therefore, it's not clear to me that being 'semi-effective against all strains' implies 'semi-effective against each strain.'

I am now looking for sources that say Gardasil is semi-effective against each strain of HPV.

Edit: Reading this source[1] I see that Gardasil protects against strains 31/33/45/51 (grouped) with 33%-51% effectiveness. Still can't find conclusive evidence that Gardasil protects against each strain.

[1] http://cvi.asm.org/content/early/2015/01/29/CVI.00591-14.ful...

I would assume that the context ("already provide some protection against additional strains of HPV") implies that it provides additional protection against additional, individual strains, but will grant that the wording doesn't conclusively say that and alternate interpretations are possible.

Agreed. From the evidence I was able to find, certainly Gardasil seems to prevent at least some closely related strains. I am wondering if it prevents all cancer-causing strains though, which would be needed for eradication. Probably, given the title of the article, but I wish I could find a primary or secondary source.

I'm not clear on the distinction that you're drawing - cervical cancer is almost always caused by HPV. What were you alluding to?