Plato and Socrates didn't really unpack the concept of belief. Either you believe it or you don't. That's understandable, since they were just starting to think about what we now call epistemology so they used chunky concepts. In pagan Rome and later Christian Europe people didn't think about it much at all, or if they did it wasn't written down. For Christians, belief was simply not questioned, and people who did question it got tortured to death. Galileo managed to escape that fate, but they did shut him up.
People who have studied philosophy in any depth will no doubt want to take issue with what I am about to say as an oversimplification, but I can't help it, this is just a blog post. In the Enlightenment and post-Enlightenment era, philosophers were so gobsmacked by the discoveries of science that they came to argue -- starting with Comte and leading to the Vienna circle and Wittgenstein -- that only statements that are subject to empirical verification are meaningful, ultimately going so far as to say that the meaning of a sentence is equivalent to the means by which it can be verified, and any statements that cannot be verified are nonsensical. This was called positivism.
Wittgenstein eventually recognized the absurdity of this position. It is mistaken in at least three ways. The first is that many speech acts, in fact most of them, are not even trying to be verifiable assertions. Positivism would render most of what Socrates is purported to have said nonsensical, because he mostly asked questions. But even leaving that aside, if we restrict ourselves to assertions, positivism is mistaken in two more or less orthogonal dimensions.
The first is that there are different kinds of assertions, as Jürgen Habermas notably discussed in The Theory of Communicative Action. Many assertions are about ethics and morality -- what is right and wrong, and socially proper. These are certainly meaningful and criticizeable, but they are not subject to verification in the same way as empirical claims. We also make assertions about our personal feelings and preferences. Others may take issue with these, or believe the person is dissembling, but only the speaker knows if she is being truthful. (Promises can be seen as a special case of this sort of assertion, and we will eventually know if they are broken. I might also come to doubt, for example, that she really loves me. But these sorts of complications are not our concern here.) Many arguments are feckless because people make category errors about assertions.
The second is that even for empirically verifiable claims of the sort that are in the domain of science, belief is not binary -- there are degrees of belief, and even different kinds. Here we get into the philosophically fraught realm of probability, which we'll take up next time.
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