Regarding Protagoras and how does his attempt to demonstrate that truth

The test case normally used is temperature. If Ms. X. says “it is hot,” then the statement (unless she is lying) is true for her. Another person, Ms. Y, may simultaneously claim “it is cold.” This statement could also be true for her. If Ms. X normally lives in Alaska and Ms. Y in Florida, the same temperature (e. g. 25 Celsius) may seem hot to one and cool to the other. The measure of hotness or coldness is fairly obviously the individual person. One cannot legitimately tell Ms. X she does not feel hot — she is the only person who can accurately report her own perceptions or sensations. In this case, it is indeed impossible to contradict as Protagoras is held to have said (DK80a19). But what if Ms. Y, in claiming it feels cold, suggests that unless the heat is turned on the pipes will freeze? One might suspect that she has a fever and her judgment is unreliable; the measure may still be the individual person, but it is an unreliable one, like a broken ruler or unbalanced scale. In a modern scientific culture, with a predilection for scientific solutions, we would think of consulting a thermometer to determine the objective truth. The Greek response was to look at the more profound philosophical implications.

Even if the case of whether the pipes will freeze can be solved trivially, the problem of it being simultaneously hot and cold to two women remains interesting. If this cannot be resolved by determining that one has a fever, we are presented with evidence that judgments about qualities are subjective. If this is the case though, it has alarming consequences. Abstractions like truth, beauty, justice, and virtue are also qualities and it would seem that Protagoras’ dictum would lead us to conclude that they too are relative to the individual observer, a conclusion which many conservative Athenians found alarming because of its potential social consequences. If good and bad are merely what seem good and bad to the individual observer, then how can one claim that stealing or adultery or impiety or murder are somehow wrong? Moreover, if something can seem both hot and cold (or good and bad) then both claims, that the thing is hot and that the thing is cold, can be argued for equally well. If adultery is both good and bad (good for one person and bad for another), then one can construct equally valid arguments for and against adultery in general or an individual adulterer. What will make a case triumph in court is not some inherent worth of one side, but the persuasive artistry of the orator. And so, Protagoras claims he is able to “make the worse case the better” (DK80b6). The oratorical skills Protagoras taught thus had potential for promoting what most Athenians considered injustice or immorality.