这个定语已经不是定语从句了，而是带关系代词的不定式短语。其结构为：介词+which+to do. 这种结构为定语从句的简化，可以还原为定语从句：

Without facts,we can't form a correct opinion, for we need to have actual knowledge on which to base our thinking.

= Without facts,we can't form a correct opinion, for we need to have actual knowledge on which we will base our thinking.

我们知道情态动词后接不带to的不定式。介词+which引导的定语从句都属于正式语体（非正式语体介词留在句末）。你会发现，定语从句省去主语we和情态动词，并将不定式的to还原，就得到了on which to base our thinkin这个带关系词的不定式短语。这种结构只用于正式语体。

你再注意非正式语体的演变：

Without facts,we can't form a correct opinion, for we need to have actual knowledge which/that/( ) we will base our thinking on. 介词on位于句末，此时关系代词可以用which，that或省略关系代词。

=Without facts,we can't form a correct opinion, for we need to have actual knowledge we will base our thinking on. 进一步省略主语和情态动词，并将不定式的to还原，就得到：

Without facts,we can't form a correct opinion, for we need to have actual knowledge to base our thinking on. 红色部分就是你熟悉的不定式短语作定语了。这是非正式语体。如果把介词on移到不定式前，就要加which了。on which to base our thinking.