Hi,

Please do not scream about semantics and everything. This imaginary website have a good markup for it's content. But there is no W3C endorsed doctype on the top. Other than that it is a perfectly fine HTML 4.1 document.

Would search engines care about the doctype not being there?

I'm not sure, but I do know that the average browser usually flips from standards mode to quirks mode, if the markup at the beginning of a page does not start with the doctype - this can cause the page to be displayed wrongly.

In relation to SEO, I'm not sure, but Google uses hundreds of methods to rank it's sites and I wouldn't be surprised if one of these hundreds was having a valid code, or at least one that uses web standards - after all, Google seeks to provide it's visitors with related quality sites, and ones that conform to web standards can improve the experience of visiting a site. While this is a generalization, site's that conform to web standards, TEND to be better built, and may have a slightly higher chance of containing quality content, than sites which have been built by spammers/scammers who have no concern with web standards etc.

As long as it didn't lead to any major problems in spiders crawling the page I doubt it would make any difference. I think there are plenty of other good reasons to use a doctype and make sure your pages validate, but at the moment I don't think it's a big deal to search engines.

They might care in the future though and if your page ends up with serious coding issues then it probably will have a negative impact.

Would search engines care about the doctype not being there

Absolutely not!

doctypes are for rendering engines so they "know" what type engine to use and how to display (or not) the various elements and what level of script code should be used.

SE crawlers and indexers do not render the code so have no requirement of a doctype.