Formally known as the problem of induction (with additional information here), conclusions dependent on empirical evidence cannot be completely proven, only supported. Basically, since we cannot be everywhere all the time to make every relevant observation, we cannot completely verify a claim. For example, the assertion that "all swans are white" could not be proven until all swans everywhere were observed. The claim was strongly evidenced, since hundreds of thousands of swans observed at the time the claim was plausible were indeed white. However, the concept of empirical inconclusiveness is fully illustrated by the later discovery of black swans. So we see that, until we have all the observations, we cannot have a definite conclusion. In this world, we cannot make every observation at every point in time, and so we must satisfy ourselves with approaching certainty.

 

 

A good example of intuitively gained a posteriori knowledge can be found on Wikipedia's "Problem of Induction" page (linked above). It goes as follows:

 

"For instance, from a series of observations that a woman walks her dog by the market at 8am on Monday, it seems valid to infer that next Monday she will do the same, or that, in general, the woman walks her dog by the market every Monday. That next Monday the woman walks by the market merely adds to the series of observations, it does not prove she will walk by the market every Monday. First of all, it is not certain, regardless of the number of observations, that the woman always walks by the market at 8am on Monday. In fact, Hume would even argue that we cannot claim it is "more probable", since this still requires the assumption that the past predicts the future. Second, the observations themselves do not establish the validity of inductive reasoning, except inductively." [3]

 

The key idea for this example is that no matter how much evidence there is supporting a claim, it's always possible that it is wrong, since we don't know the whole story. Another way to look at it is through this, more extreme, example:

 

Person A has lived in Phoenix, Arizona for his entire life. He was born in Phoenix, and has never left the state. He has been told about other places (let's say Europe) through schooling and conversation. He has also read books on the subject, watched documentaries filmed there, and seen photographs taken there. Furthermore, he has met people who were European. He feels certain that Europe exists. However, it is still possible that his schooling was false, that the conversations were tall tales fabricated by the tellers, that the books were wrong, that the documentaries were fictional and were filmed elsewhere, that the photographs were fakes, and that the 'Europeans' were posing, all for any number of reasons.

 

From the perspective of everyday life, it seems ridiculous not to believe in the existence of Europe. But as we see here, the existence of Europe is not truly proven to Person A, it is only evidenced. Of course, the reason why it seems so ridiculous to reject the idea of Europe in everyday life is because we have so much evidence supporting it, and although it seems like we accept facts of that nature based on proof, this example illustrates how we actually accept them based on how much partial evidence they have behind them, and therefore, how likely they are to be true. Additionally, when we are presented with multiple, conflicting pictures of  reality, we pick the one that has the most backing, and is most likely to be true.

 

This is, after all, the most logical and reasonable approach to take in an uncertain world, and most of the time, it is the approach we choose. However, we sometimes abandon this logical way of thinking when we want a less-likely option to be the truth. The key is to remain objective and to take notice when you, or others around you, reject a conclusion that is well-supported, and makes intuitive sense therein.