An epistemological theory of the way beliefs are justified. Under foundationalism a belief is justified if it is based on sound basic beliefs. These are the 'foundations' of knowledge from which justification can flow.

When making a statement, one can always be questioned on 'why do you think that?'. And the reasons given can in turn be questioned. And so on. This could go on forever. It could also turn out that you really don't have any good reason to believe x after all. Or there could be something that you can be sure about. These 'certain things', if they can be found, would be your basic beliefs.

Strong Foundationalism says that you need to have basic beliefs that are absolutely infallible, incorrigible, or indubitable. But since it is very hard to find beliefs that are that strong, we also have Moderate Foundationalism, in which the basic beliefs only have to be 'most likely' true. Under this system you are justified in believing something only if it has a greater than 50% chance of being so. But even this can be hard to get, so we also have Weak Foundationalism--If you can get any justification at all, go ahead and use it.

Needless to say, people are having a hard time figuring out what we are justified in believing. Foundationalism is only one theory. You might also want to check out Coherentism, Reliabilism and Scepticism.

If you are looking for more information (and there is a lot more to foundationalism than I have here) Here is a good philosophy search engine. And any introduction to epistemology book should have a section on foundationalism.

One of the most important things to remember about Foundationalism, the belief that there is some basic knowledge that other things can be deduced from, is that it is an implicit belief of many different philosophical, religious or scientific schools. Behind the often noisy debates amongst differing positions, they do have some common ground, which is the belief that a set of objective, undeniable beliefs are possible. In some ways, this makes all debates about what belief system is true to be sectarian, and like all sectarian conflicts, they tend to be the bloodiest.

For example, rationalists and (some) empiricists are foundationalists, and debate, sometimes bitterly, whether pure reason or experience is the foundation of knowledge. Both of them are based on the individual's judgment about whether something is true, and thus both of these systems can be equally bitter in their debates with systems that consider some form of group knowledge, whether it be religious or social to be foundational. But all of these groups, and many other schools, believe that there is an objective reality, and that people can analyze it using some basic form of concepts, to form further knowledge, that is often judged by its utility.

Questioning the validity of foundationalism is a good way to cut the Gordian Knot of some arguments, or at least stop people's yammering. Richard Dawkins and Ted Haggard ended up at each other's throats, but their debate was non-productive, as most sectarian debates are, because neither was questioned about the justification for foundationalist belief in the first place. A non-foundationalist, such as a post-modernist or Discordian would have looked at the whole thing with some skewed eyebrows, and the debates that would have emerged between non-foundationalists and foundationalists would have been much less emotional.

Although there are some non-foundationlist philosophical schools, most established schools of philosophy or other beliefs do have some sort of foundational belief, or at least the possibility of one. It seems that in most cases, people are naturally somewhat foundationalist, believing that there are some things that are basic, even if these are only believed from a pragmatic point of view.

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