Everything in the wedding scene in Much Ado goes to shit real fast, but even before I was playing Leonato, it struck me as a bit odd that he would be so quick to publicly denounce his own daughter as a slut. Now that I'm playing the role, I am finding that the answer to that question lies in the foundations of scientific thought as we know it.
First, a little bit of background. While the first English-language work of experimental science was Robert Boyle's Experiments Physio-Mechanical Touching on the Spring of the Air in 1660 (remember Boyle's Law from high school chemistry? That's the guy), the first work of experimental science printed in England was William Gilbert's De Magnete (that's Latin for "On the Loadstone"), printed in 1600. For those of you keeping score at home, that was the same year that Much Ado About Nothing was first printed in quarto form.
The term "science" in this context is a bit anachronistic. Scientists of this period, and indeed for centuries afterward, considered themselves "philosophers." They tended to be gentlemen with university educations, and even when they were experimentalists (like Gilbert and Boyle), they very rarely got their hands dirty, hiring skilled craftsmen to build, use, and sometimes design the apparatuses that enabled their program of discovery. This is why scientists get a "Doctor of Philosophy" degree, and is part of the reason why, to this day, there tends to be some condescension from scientists proper toward the engineers who actually do the work. Even among the sciences, those that are more purely theoretical tend to be viewed as less "pure." My comp sci peeps who spend any amount of time talking with traditional math students know what I mean.
In the medieval period, there used to be a lot more talking and a lot less experimenting in science. Part of the reason for this, of course, is that best minds in the world just didn't have access to the sort of basic instruments that you can get in a children's science set for under $100 today: they were limited to understanding the universe based on what they could see with the naked eye, and then using reason to figure out the rest. Rhetoric was as important to these early scientists as it was to lawyers and poets: if you couldn't write and speak effectively, you were less able to argue your case for what made the universe work the way it did.
If you're interested in seeing this in action, you just have to turn to the works of Thomas Hobbes (you probably know him as the political philosopher behind Leviathan, but you can find links to his collected works on his Wikipedia page.). Hobbes was a famous opponent of experimental science (and the royal society) based on the principle that the instruments of experimental science could be faulty, and since there really was no way to tell if the instrument was faulty, there was no way to confirm that it was helping you see something that was actually there. This is why no new radical finding is accepted by the scientific community today until it has been independently verified by lots and lots of scientists working from the same basic methodology.
In the early 1600s, of course, the broader scientific community was still a new thing, and they didn't have peer-reviewed journals and an established postal system (to say nothing of email). If you had a new device/ida/observation, and you wanted to share it with the world, you had to publish a book about it, try disseminating it to the most respected minds in your country and others, and then wait for the commentary to come back to you. That took a lot of time, and the result was that a lot of things that were just plain wrong got handed down from teacher to student for... well... millennia. This is not to vilify those early philosophers, of course, they were doing the best they new how without the resources that every elementary school child takes for granted, but when Shakespeare was writing Much Ado About Nothing, the way we thought about evidence and proof was changing rapidly.
So here we have Leonato, who believes his daughter, hitherto the apple of his eye, quickly turning his opinion of her to one that has her "fallen in a pit of ink, the wide sea hath drops to few to wash her clean again," seemingly on nothing more than the accusations of other men. I like to think that, if my would-be son in law started calling my daughter a whore at the alter, I would bust his head open with the claw end of a hammer and dump his body in a ditch. Leonato, on the other hand goes along with it, but if you look carefully, it sort of starts to make sense why.
Right after he offers that he doesn't think Don Pedro or Claudio would lie, which still keeps things on the level of their word vs. Hero's, Leonato begins citing some proof: "would Claudio lie, who loved her so that, speaking of her foulness, washed it with tears?" Claudio is able to prove the veracity of his claim that he saw Hero engaging in some lewd behavior by crying while he tells it. We know that Leonato has a lot of stock in that because one of his earliest lines in the play describes tears as "a kind overflow of kindness: there are no faces truer than those that are so washed."
And why can Leonato forgive Claudio in the end? Well, for starters, he gets to hear the entire plot confessed, and so knows that Don Pedro and Claudio were themselves deceived. But also, when Leonato offers Claudio the chance to make up for killing Hero with slander by marrying her cousin, Claudio responds "Your over-kindness doth wring tears from me." The crying clinches it.
In this world crying is an outward sign of an inner state of feeling, Leonato fully believes, and events seem to confirm, that no one can lie while crying, and really no one does. Don Pedro and Claudio, perhaps overly cruel by our standards, are proceeding from what they believe is the truth. This truth, however, comes from faulty observation: they have witnessed an event (which the audience has not, as that scene is not in the play), but the event is, of course, a charade engineered by the bastard Don Jon and his men.
In Shakespeare's Messina, observations of character are signified by indelible outward signs, while observations of actions and events are faulty. This is very much in keeping with the medieval philosopher's view of science: we know what must be based on argument. Where there is a division between reason and observation, we must re-conceive the significance of our observation in keeping with what we know through logical proof.
Shakespeare was a product of a grammar school education that probably included a lot of Aristotelian natural philosophy, and there is little indication that he took an interest in natural philosophy beyond that. Tom Berger, the Shakespeare and Performance program's scholar in residence, is fond of cautioning us not to apply post-Enlightenment reasoning to pre-Enlightenment, and this has been a brief explication of why, and the dangers even a well meaning actor can fall into by doing so.
If you're interested in learning a little more about the rise of experimental science after the reformation, I heartily recommend Steve Shapin's The Leviathan and the Air Pump, and if you're interested in learning more about the rise of modern science in early modern London, Deborah Harkness' The Jewel House is definitely worth a read. Obligatory links to Amazon follow: