# Compound Probabilities and the Hypothetical Syllogism: The Slippery Slope Fallacy

March 13, 2012

While working on a larger treatise regarding reasoning and argumentation, I found it worthwhile and applicable to extract this part for the contemporary state of politics and pandering. The slippery slope is a rather familiar term, but I often wonder how much people understand it. In fact, the slippery slope by itself doesn’t make it a fallacy. The fallacy is based on the premises of the argument including the basis and conclusion. For example, there is validity to the hypothetical syllogism if the premises that make them up are valid and true:

If A, then B.

If B, then C.

Therefore, If not A, then not C.

However, this can become difficult if the probabilities are not 1 (or 100%) or of greater, or likely, probability. This is where the probability of events and the validity of premises challenge the conclusion and the slippery slope can become fallacious. This is where you get to the cumulative concept of probability. So, B depends on A and C depends on B ( B½A and C½B to understand AÇBÇC). To make it simpler to understand we can apply different relevant arguments to see this fallacy in action. For example, we can use same-sex marriage, of course, I don’t want to indict any individual or any particular groups opinion to this, so I have just arranged a generic argument.

Suppose:

If we allow same-sex marriage [A], then it will soon become legal for people to practice bigamy or polygamy, legal incest, and the legal ability to marry other species for zoophilia or bestiality purposes [B]. Therefore, Society will erode into a morally destitute and chaotic state without regard for what is right [C].

In order to understand this we need to not only examine the initial premise of same-sex marriage but the probability or likelihood of the following arguments. So, as stated B depends on A and C depends on B. What is the probability of A happening? At this time there are seven states and the district of Columbia that have legalized same-sex marriage and the trend appears to be heading toward more and more states over time. So what probability do we give this premise? You can put whatever number you want, but for this example I will use two: 100% and 51% which we‘ll call A1 and A2. As for the second premise, B, we have to consider a few more things. First, each of the arguments are separate aspects with separate pathways to being legal. Whether, or not, they are done is different versus being legalized which requires a legislative process. It could be safely said that it would more likely be probable, if probable at all, that polygamy would occur before the ability to marry your pet or commit incest legally. So what probability do we give to this premise? This is more difficult but for the sake of completing the example we’ll use two numbers again: 51% and 10% which we’ll call B1 and B2 (the 10% because all of the arguments in the premise occurring would seem very unlikely). Now we can utilize these numbers to understand the probability that moral structure in society will decay to become non-existent or conclusion C.

So, let’s do the math.

A1 then, B1 is 1(0.51) = C is 0.51 or 51%

A1 then, B2 is 1(0.1) = C is 0.1 or 10%

A2 then, B1 is 0.51(0.51) = C is .26 or 26%

A2 then, B2 is 0.51(0.1) = C is 0.051 or 5.1%

Thus, the ability to get to C, indefinitely, requires that each premise be 1 or 100%. So even when both A and B have a majority likelihood the ultimate outcome of C is rather unlikely percentage wise. You can apply this to many different arguments. My suggestion would be to just try it. Analyze slippery slope arguments about other controversial subjects and attach what you would think would be plausible probability and work it out. String the events together and find the probable outcome of the conclusion. We can’t become susceptible to the slippery slope simply because the elements of the argument appear to have likelier probabilities. As we can see, majority probabilities compounded together can lead to an unlikely probability.