The A+ Problem

As promised, here is the first example of Foundational Evidentialism in practice:

The Problem

Your child comes home and claims they got an A+ (100%) on a recent math test. Do you take their word for it or demand more evidence (e.g. the graded exam paper)? Here's the relevant evidence:

• Your child has been historically honest about their exam grades 72/75 times.
• Your child's math grades are typically C (70%) average with a standard deviation of 10%.

If this is starting to sound like a math problem, you'd be right. Typically in real life, our minds will estimate these values (and the answer) for us unconsciously and quickly. We might even include other factors we are unaware of. These subjective estimations are easily thrown off by subtle cues, childhood indoctrination, personal disposition or any degree of altered state of mind. We are using hard numbers in this example to keep the answer as objective as possible.

Cause and Effect

Establishing the accuracy of truth claims like this one requires understanding that testimony is part of a cause and effect relationship. Regardless of weather or not your child actually got the A+, you know that they did claim it. The fact that they claimed it is an effect. Our job, in this logic problem, is to establish the most likely cause of that effect. Here are the most likely causes:

• The child is lying about their grade
• The child actually got an A+

There are, of course, other possible causes... perhaps the child is being mind-controlled by invisible aliens or perhaps the child is a clone impersonator from the 4th dimension. We don't consider these possibilities. Why not? Two reasons:

1. No Knowledge. We don't exactly know anything about "invisible aliens" or "clone impersonators." We don't know anything about their likelihood as candidates for an explanatory cause. By contrast, we know a lot about the honesty of children. In this case, we even have data regarding the specific child's honesty and academic records.
2. No Positive Evidence. I happen to know a little about exobiology (alien life), "extra" dimensions and cloning. My knowledge about these related concepts tell me that these hypotheses are inconsistent with repeated observations. So, given the only evidence about these hypotheses is negative, they shouldn't be considered. It only makes sense to consider explanations with at least some positive evidence.

It's important to note that these excluded explanations are not "ruled out" per se. They are still possibilities... There's just no logical justification for including them in the reasoning process.

Abductive Reasoning

This kind of reasoning (comparing the relative likelihood of competing explanations) is called abductive reasoning. In this form of reasoning, we don't judge the likelihood of each explanation independently. Instead, we compare their relative likelihoods and proceed with the most likely explanation. Let's get to it...

Doing the Math

First, let's consider the odds that the child is lying. Calculating this value is easy, since it is practically given:

Second, let's consider the odds that the child got an A+. This calculation requires a strong understand of probability distribution functions and calculus:

At this point, it should be clear that you should ask to see the graded exam before taking your child at their word. Perhaps if the child was more honest and/or got better grades, the result would have been different. Either way, there is a clear, objective method for figuring out weather or not you should believe someone when they make a claim. That is point of this exercise.