Essay:Dissolving logical fallacies

As I was walking around today, I witnessed the following snippet of conversation, which was presumably part of an argument between a female theist and a male atheist:
 * Theist: All I'm saying is, lots of people believe in God so that's evidence that He exists.
 * Atheist (triumphantly): Argumentum ad populum, bitch!

After reflecting on the incident, I've come to the conclusion that the atheist was not only rude but also wrong. Moreover, this conversation is an example of a common but troublesome practice in religious and political arguments: twisting a logical fallacy to support an illogical position and oppose legitimate arguments. As a case study, I'm going to discuss and dissect this particular conversation.

Dissolving the theist's argument
The theist is essentially arguing that widespread belief in God is evidence for God's existence. Regular readers of this site are probably already thinking "that's a fallacy," and the theist's argument does sound suspicious at first glance. However, there is actually a great deal of valid reasoning behind it. To understand why, let's delve into quantitative analysis:

Let h be some statement about the world that is either true or false. It can be "the sky is green" or "Barack Obama is the current President of the United States" or "the Earth is flat" or any other assertion about the way the world is. Next, let p be the fact that a particular belief is popular - it doesn't have to be universally accepted, but it should be somewhat mainstream. "God exists" or "The US fought in WWII" or "Sweden is liberal" would all be examples of p. Now, imagine that it's possible to take every possible statement about reality (including statements that are true and ones that are false) and put them all into an enormous jar. Next, we will pull out one statement from the jar at random and say that this statement is h. Now, how likely is it that h is a true statement? The answer is "very unlikely," because for each statement about the world that is true, there are an enormous number of ones that are false. For example, if "the sky is blue" is true, then it's obvious that "the sky is green," "the sky is brown," and "the sky is fuchsia" are all false. As a result, the vast majority of statements in the jar are false ones. Thus, the odds that our statement h is true are pretty low.

Now, let's look back into our jar and make a duplicate of all of the statements that are p. This will be a pretty large number of duplicates, but it will still be minuscule compared to the number of statements in the jar. Let's take all of these duplicates of p statements and put them in a small bowl beside our jar. Also, let's take the h we picked and put it back into the jar. Now, let's say that we want to randomly pick a new h, except that this time we want to have a better chance of h being true. Should we pick from the jar or the bowl? This is trickier question, and we need to think about the difference between the bowl and the jar in order to answer it. Remember, the bowl contains only statements that are p, or popular, and the jar contains all statements. So, what we are really asking is, "is there a larger proportion of true statements in the bowl compared to the jar?" The answer is yes. Even though a great deal of statements in the bowl are false, such as homeopathy works or ghosts exist, there is still a larger proportion of true statements than in the jar, which is flooded with countless falsities. As a result, we would be better off picking our new h from the bowl of popular statements.

People familiar with probability theory can probably see where this is going: since we would be better off picking from the bowl than the jar, this means that any random statement in the bowl has better odds of being true than a random statement in the jar. In statistical notation this would be expressed as P(h|p) > P(h), or in plain English, a statement is more likely to be true if it is popular. Thus, the theist's argument has a grain of truth to it, so arguing that belief in God is widespread is indeed a valid reason to take that hypothesis more seriously.

Where the theist's argument fails
Of course, the analysis doesn't stop there, since we can be a lot more specific than just saying that a statement is p or not-p. After all, human beings believe a lot of things, and if we break them down by category we might be able to be more accurate. Let's define e as a statement that is p and that anyone can easily observe whether the statement is true or false. For example, "the sky is blue," "the moon is the largest celestial object in the night sky", and "pavement gets hot in the summer" are examples of e statements. Obviously the number of e statements will be smaller than the number of p statements, since there are a lot of popular beliefs that can't be confirmed easily, such as "ghosts exist." Now, imagine that we've combed our bowl for e statements, duplicated them, and put the duplicates into a cup. Again, let's throw h back to where it came from and pick a new one. If we want to maximize the chances of h being true, where should we pick from this time? The answer, of course, is the cup, since the proportion of e statements that are true is higher than the proportion of p statements that are true and definitely higher than the proportion of statements in the jar that are true. Thus, if we know that a statement is popular and easily verifiable (i.e. from the cup), it is more likely to be true than if we only know that it is popular (i.e. from the bowl).

The moral of the story is that we can do much better than just "popular" and "not popular", because certain types of beliefs are more likely to be true than others. It's obvious that e statements, which anyone can test easily, are much more likely to be true than beliefs that are only p. If we liked, we could also denote a category s for beliefs that are in the scientific consensus and compare this category to p as well. Since the theist's claim, "God exists," is a belief that is p but not e or s, the theist's argument is not as strong as she would like. Also, let's not forget that even though picking from the p bowl is better than picking from the jar, this does not mean that the majority of statements in the bowl are true, it just means that proportionately more of them are true than those in the jar. In all probability, the majority of statements in p aren't true, so even though a statement's popularity is indeed evidence that it is true, it's extremely weak evidence. To put this in terms of our model: drawing from the bowl may be better than drawing from the jar, but it's still pretty unreliable.

In conclusion: though the theist's argument is valid, it is ultimately a weak argument when broken down into quantitative terms. Though the theist's reasoning isn't inherently fallacious, it isn't very powerful either.

Extrapolating to the general case
Although it took a little bit of time, we've gotten to the bottom of the theist's argument and have accurately assessed its strength. Instead of simply reciting the name of the logical fallacy and declaring victory, we took the time to analyze the theist's position and figured out what was really going on in the argument. That said, we still need to talk about what was wrong with the atheist's response in the dialogue above. After all, isn't the theist's argument still a fallacy? The answer is no: we've broken down the argument and seen that its basic framework is indeed valid, and calling it fallacious won't change that. Careful readers may have noticed that the atheist is actually misusing the argumentum ad populum, since it is only supposed to apply to arguments of the form "X is popular and its popularity makes it true." But this distinction isn't always clear until we dissolve the argument and examine its internal structure.

In general, fallacies have a tendency to be misused because our brains like to jump to a certain conclusion instead of being thoughtful. However, as I've (hopefully) shown above, a thoughtful and detailed analysis is often the only way of truly assessing an argument. As W.K. Clifford put it: "But," says one, "I am a busy man; I have no time for the long course of study which would be necessary to make me in any degree a competent judge of certain questions, or even able to understand the nature of the arguments."

Then he should have no time to believe.

So next time you're about to answer someone with a Latin phrase or a quick dismissal, stop for a moment and think. What is really going on in this argument? Is it really a fallacy, or is there a grain of truth hidden somewhere? It may take you more time, but the more you work at a problem, the closer you will be to the truth.