Fallacy of the Week #2

Whenever you turn on the TV or pick up a newspaper, you're likely to see someone shouting about one thing leading to another, leading to another, leading to something horrible. Whenever you see this, you're seeing an example of the slippery slope fallacy. The curious thing about this fallacy is that while it isn't a valid form of argument, it nevertheless doesn't invalidate the conclusions. As we discussed last week, the following is more or less a true statement (and is, regardless, a reasonable thing to say):

This mixture of cyanide and arsenic is poisonous, because cyanide is poisonous and arsenic is poisonous.

We can restate this as a conditional, just to make things a little easier later:

If cyanide is poisonous and arsenic is poisonous, then the mixture of cyanide and arsenic is poisonous.

But that's not quite right. In our daily discourse, we generally don't bother to distinguish between the claims we make and the meanings they express (or the conclusions they bring us to). As a general rule, our arguments consist of three types of elements:

1. logical connectors (of which conditionals are one subtype)
   
2. premises (information that we're arguing from, and which we assume to be true)
   
3. conclusions
   

In order for an argument to be effective, all three must "work." Typically, when we go to the trouble of making a claim like this:

It's a good idea to study a lot, because knowledge is power,

the structure of our argument is something like this:

1. if you study, then you will gain knowledge
2. power is desirable; knowledge really is power
   
3. it's a good idea to study a lot

Or, more generally:

1. If A, then B
2. A
3. Therefore B

To illustrate this, let's consider an example where the conditional holds but the premise and conclusion are false:

If I were a teapot, I could be used to boil water.

I am (fortunately) not a teapot and I (tragically) cannot be used to boil water, but the conditional nevertheless holds. On the other hand, we can come up with statements where the premise and conclusion are true but the conditional fails:

If the Earth is roughly spherical, its core must be very hot.

While the Earth is roughly spherical, and while its core is very hot, the conditional itself fails. This obviously doesn't mean that the Earth's core isn't hot, but it does mean that we cannot conclude from the Earth's roundness that its core is hot. 

We can come up with lots of other examples of arguments that fail because one or more of those three elements fails. The point, though, is to really get a feel for the difference between the assertion that "If A, then B" and the conclusion that "B is the case."

To get back to slippery slopes, we can reconstruct such an argument more or less like this:

1. If A, then B
2. If B, then C
3. If C, then D
4. A
5. Therefore D

So why is this problematic? Well, it's really not, assuming that all of those conditionals work. The problem comes in because each of those added steps might fail, causing the whole chain to fail. That is, increasingly complex systems face increasing margins of doubt. Because most of the reasoning we do is inductive (arguing from particular cases to general principles) and therefore not truth-preserving (we are not guaranteed true conclusions, even with true premises and working conditionals), each additional step increases the chances of failure.

This is why it's generally reasonable to make claims like,

If we all stop buying stocks, the stock market will suffer and stock prices will drop,

but not claims like,

If we all stop buying stocks, the stock market will suffer and stock prices will drop, and the government will privatize banks, and we'll start wars across the globe to stimulate industry, and everyone will hate us, and then we'll all nuke each other and the human race will die out,

even if we can accept that every step along that chain is a pretty reasonable one.