The Choice of Basic Beliefs is Arbitrary
According to foundationalism, there are some beliefs that do not need external justification. These beliefs are justified, even without any further reason for believing that they are true.
However, if we accept the idea that some beliefs are basic, needing no external justification in order to be justified, then there is a danger that anyone seeking to defend a contentious belief will be able do so by claiming that that belief is basic: Alvin Plantinga, a Christian philosopher, has suggested that belief in God is basic, denying that there is a need for rational argument to justify religious belief; discussions of ethics can often come down to appeals to ethical axioms, fundamental principles that are taken to be beyond criticism.
Foundationalism holds that all non-basic beliefs are ultimately justified by being inferred from basic beliefs. If we cannot criticise someone’s choice of basic beliefs, therefore, then any world-view carefully inferred from them will appear to be justified. This could potentially let many world-views that we tend to think of as irrational qualify as epistemically justified, which would be an unwelcome conclusion.
We cannot avoid this conclusion, however, by offering reasons for preferring one choice of basic beliefs to another (e.g. for the principles of logic and mathematics in preference to the principles of astrology). The whole point of basic beliefs is that they are intrinsically justified, that we don’t need to offer reasons for holding them. If we attempt to justify our basic beliefs, then we demonstrate that we don’t really think that they are basic after all.
Foundationalists must accept, then, that our choice of basic beliefs is entirely arbitrary. As all of our beliefs are ultimately derived from our basic beliefs, though, this means not only that the foundation of our beliefs is arbitrary, but also that all of our beliefs are arbitrary.