Positivist Fallacy: the assumption, often implicit, that historical sources and archaeological remains document significant events of the past. The expression was coined by archaeologist Anthony Snodgrass.
Like the Everest Fallacy, the Positivist Fallacy can best be introduced with an example. There are four sources for the battle in the Teutoburg Forest (Velleius Paterculus, Tacitus, Florus, and Cassius Dio). Generations of scholars have written about the clash and have, considering the battle to have been decisive, argued that (a) the Romans were forced to accept the Rhine as their frontier, (b) the limes was created, (c) Germany remained unoccupied, and (d) this caused an antagonism between Romans/Franks/French and Germans that would continue to influence European history for centuries to come.
We now know that this was exaggerated. Not only did the Romans conduct several campaigns on the east bank, where they continued to find allies, but they also continued to have access to the resources of Germany (e.g., lead and gold). The Rhine limes was not created until the reign of Claudius; it was only then that the Romans accepted rivers as permanent frontiers and started to develop a defensive strategy. This change in perspective is the consequence of more accurate dendrochronological dates and improved pottery seriations. Only with these results, archaeologists came to realize that some forty years passed between the defeat of Varus and the creation of the limes. We now know that the Claudian army reforms were what really mattered.
But even without archaeological advances, the error could have been avoided. Ancient historians have allowed themselves to be misled by the fact that they had four sources on the battle. But they ought not to have been fooled. When we have a great number of sources, that does not mean that an event was significant. Nor does a small number of sources mean that nothing happened. We have no written sources about the Claudian army reforms, but they were important.
To state it differently: scholars have implicitly assumed that the sources we have, refer to the most significant events in the past. Of course the massacre was, in terms of human suffering, important. But decisive it was not. What scholars did wrong, is that they forgot that there are many historical facts for which we have no evidence. Instead they focused on the facts for which positive evidence exists (hence the name "Positive Fallacy").
Another example, again from military history, can be found in numerous books about the Persian Wars, which often end after accounts of the battle of Mycale or Plataea. These are the last events mentioned by Herodotus and are the last facts about which details are known. But the war continued: we know that a Greek navy attacked Cyprus, we know that the Spartans invaded Thessaly, we know that a coalition army was active in the Bosphorus, and we know that the Persian fortress at Eïon was captured. It was only then, when the Persians were expelled from Europe, that hostilities ceased. This final stage of the war is poorly documented, but a historian cannot make his account dependent on the randomness of the tradition.
The problem is also important in archaeology. Pottery can break, but is hard to destroy. Wood, on the other hand, can disappear in a number of ways. Silver and gold are too valuable to leave behind when a house has been destroyed. As a consequence, precious metals and objects made of wood will always be underrepresented in an excavation, while pottery will be overrepresented.
Archeologists have developed a "reconstruction theory", which attempts to balance this problem. I am unaware of a similar attempt by historians. Yet, the Positivist Fallacy is not inevitable, and at least some awareness is possible. A type of thought experiment that I find valuable is to think about a hypothetical archaeological discovery (e.g., a second-century fortress of VIIII Hispana in the north of the Netherlands), and start to think which known facts support and contradict this hypothesis. This exercise allows us to see at least some flaws in our arguments. Still, historians ought to do more to improve the quality of their arguments; this is the main, paradoxical task today's historians are facing.