Epistemic closure is the idea that if you know something, then you also know all of its logical implications. For example, if you know that all dogs are mammals, then you also know that all dogs are animals.

The Gettier problem is the problem of how to justify knowledge. It arises from the fact that there are cases where someone seems to know something, but they do not have a justification for their belief.

A Gettier case is a case where someone seems to know something, but they do not have a justification for their belief. For example, someone might believe that it is going to rain tomorrow because they saw a dark cloud in the sky. However, this belief is not justified because there are other possible explanations for the dark cloud, such as a dust storm or a forest fire.

There are a number of different responses to the Gettier problem. Some philosophers argue that knowledge is defeasible, meaning that it can be overturned by new evidence. Others argue that knowledge is invariant, meaning that it cannot be overturned by new evidence. Still others argue that knowledge is contextual, meaning that it depends on the context in which it is acquired.

The belief that the sun will rise tomorrow is a defeasible belief because it is possible that the sun will not rise tomorrow, for example, if the Earth were to be destroyed by an asteroid.

The belief that 2 + 2 = 4 is an invariant belief because it is not possible for 2 + 2 to equal anything other than 4.

The belief that I am in pain is a contextual belief because it depends on the context in which it is acquired. For example, if I am dreaming, then I might believe that I am in pain, but this belief would not be justified because it is not based on any real evidence.

The main objection to the defeasibility theory of knowledge is that it is too skeptical. It allows for the possibility that all of our beliefs could be overturned by new evidence, which would mean that we could never really know anything.

The main objection to the invariantism theory of knowledge is that it is too dogmatic. It claims that there are some beliefs that cannot be overturned by new evidence, which seems to be too strong a claim.

The main objection to the contextualism theory of knowledge is that it is too complicated. It requires us to take into account all of the different contexts in which a belief is acquired, which can be very difficult to do.

There are a number of criticisms of the Gettier problem. Some philosophers argue that it is too abstract and that it does not reflect the way that we actually acquire knowledge. Others argue that it is not a real problem because it is based on a misunderstanding of knowledge.

The main conclusion of the Gettier problem is that knowledge is defeasible. This means that it is possible for us to have beliefs that we are justified in holding, but which are nevertheless false.

The Gettier problem has a number of implications for our understanding of knowledge. It suggests that we should be more skeptical about our beliefs, more careful about how we acquire knowledge, and more open-minded to new evidence.

The Gettier problem has had a significant influence on contemporary epistemology. It has led to a renewed interest in the nature of knowledge, the development of new theories of knowledge, and a more critical approach to knowledge claims.

There are a number of open questions about epistemic closure and the Gettier problem. These include questions about the best way to define knowledge, how we can justify our beliefs, and how we can acquire knowledge.