The new riddle of induction

Suppose every blueberry you've observed to date has been blue. You take this to be evidence for:

(H1) All blueberries are blue.

As a consequence, you predict:

(P1) The first blueberry I observe on February 11, 2012 will be blue.

Now, define "bleen" to mean "blue until February 10, 2012--green thereafter". By this definition, every blueberry you've observed to date has been bleen. You take this to be evidence for:

(H2) All blueberries are bleen.

As a consequence, you predict:

(P2) The first blueberry I observe on February 11, 2012 will be bleen.

A blueberry that will be bleen on February 11, 2012, though, will be green--not blue. Thus, (P2) flatly contradicts (P1). Is there any evidence that favors (H1) over (H2)? We might complain that (H2) is couched in terms of a derivative property--"bleen" is defined in terms of "blue" and "green". We might take this observation to favor (H1), and therefore (P1).

But define "grue" to mean "green until February 10, 2012--blue thereafter". Imagine a culture that only understands "bleen" and "grue"--not "green" and "blue". To them, "green" means "grue until February 10, 2012--bleen thereafter", while "blue" means "bleen until February 10, 2012--grue thereafter". Their complaint about (H1) is that it is couched in terms of a derivative property--"blue" is defined in terms of "bleen" and "grue". They take this observation to favor (H2), and therefore (P2). Could we really be right, and they really be wrong?

What else could favor (H1) over (H2)? If nothing, isn't this a problem for every hypothesis of the form "All X are Y"? And don't we (implicitly) make predictions founded upon such hypotheses in our everyday reasoning? Doesn't this problem undermine the very way in which we learn from our observations and experiences?

The problem of the broken clock

Sorry, I'm in a philosophical mood today. Deal with it.

Suppose you have a clock that has never failed you during the many years that you've had it. You look at the clock, which reads 5:00 pm. As a result, you come to believe that it is indeed 5:00 pm. And you're right--it is 5:00 pm.

Now suppose that one hour passes. You look at the clock once more, which still reads 5:00 pm. Of course, it is now in fact 6:00 pm. It seems that your clock is broken.

According to Plato, knowledge is justified, true belief. On the one hand, one hour earlier you believed that it was 5:00 pm. You were justified in believing that it was 5:00 pm, for your hitherto reliable clock said so. And the time really was 5:00 pm. You had a justified, true belief that it was 5:00 pm. On the other hand, if the time had really been, say, 2:30 pm, you would have nevertheless believed it was 5:00 pm. And you would have been wrong.

So... did you know it was 5:00 pm? Or is Plato missing something?

What makes something a cloud?

When you observe a cloud in the sky, it certainly seems like you're observing one thing--namely, the cloud. And yet, if you were to zoom in on it, you might not even notice the cloud. Instead you'd see lots of individual water droplets, some closer to the center of the cloud, some further away. If you look at the edges of the cloud, the water droplets are so spread out that some of them probably are not even part of the cloud in any meaningful sense. And yet there is no hard and fast point at which we'd say this droplet is part of the cloud--that one next to it is not. As a result, there are many groups of water droplets that have just as much claim to being that cloud as any other group of water droplets. It would seem, then, that there are many clouds. And yet it also seems that there is just one. What makes one group of water droplets qualify as that cloud, rather than another? Or is there no cloud at all?

Why isn't mathematics sensitive to experimental scrutiny?

Suppose I have two containers, both full of (what I believe to be) a single fluid. Each container's volume is 500 mL. I then empty the contents of the two containers into a third, the volume of which is 1000 mL. To my surprise, the container is only 75% full--it holds just 750 mL of the fluid. How should I revise my beliefs in light of this discovery?

Among my presumptions was that volume is additive when mixing a single fluid. One conclusion that suggests itself is that the two containers in fact contained different fluids. Another is that volume is not necessarily additive.

A third conclusion, however, does not suggest itself: 500 + 500 = 750. Why not? What is it about 500 + 500 = 1000 that justifies my willingness to concede that the fluids were different, or that volume is not necessarily additive, but not that 500 + 500 = 750? The surprising outcome of my experiment shows that at least one of my presumptions is incorrect, but it does not indicate which. What is it about this experiment that prevents it from lending support to the hypothesis that 500 + 500 = 750?