What are some real-life examples using reason (deductive or inductive) in a sound and valid manner and coming up with a false statement of reality? In other words, I'm trying to prove that reason is not always a reliable way of knowing.

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It might help to start with some definitions. As philosophers and logicians use the term "valid," a piece of reasoning is valid, roughly, if it's impossible for the premises to be true unless the conclusion is also true. That means that any argument with true premises and a false conclusion is automatically invalid. And as philosophers and logicians use the word "sound," a sound piece of reasoning is valid and has true premises. That means that any sound argument automatically has a true conclusion.

Of course, valid arguments can lead us to bad conclusions. That happens when they start with false premises. The following argument is valid, but the conclusion is false:

Some whales are fish. All fish have gills. Therefore, some whales have gills.

The problem, of course, is the first premise. But the reasoning isn't at fault.

So far, we've talked about deductive reasoning, and we can say that there are principles of deductive reasoning that are reliable in this sense: when applied to true premises, they produce true conclusions. But reasoning is often inductive, which means among other things that even if the premises are true, it isn't guaranteed that the conclusion will true as well, no matter how meticulous the induction. Inductive reasoning aims to show that its conclusions are probable given the premises, but this means that a person could be an impeccable inductive reasoner, could start from true premises, and still end up believing false things. That's the risk of trying to extend what you know beyond whatever follows strictly from what you already believe.

Then there's the question of where our premises come from in the first place, and one thing's for sure: they don't all come from reasoning. My belief that I had yogurt for breakfast is a matter of memory; my belief that I'm in the Philosophy Department lounge as I type is a matter of what I see. And on it goes. My memory can fail me; my perceptions can be illusory; my information-processing can be addled.

So there we are. If we reason from valid deductive principles, we'll end up with truths if we start with truths. That's a kind of reliability, but no guarantee of truth. If we're good inductive reasoners, and we start with true premises, then our conclusions are probably true, which is arguably a kind of reliability, as well, but not guaranteed to be true. And there are lots of ways for the beliefs we reason from to be off the mark. In short, there are no guarantees, but this isn't really surprising. And it's perfectly compatible with our being overall reliable about a good many things, even if we can fully expect that we'll sometimes go astray.

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