Transitive Evidence

A snippet from a conversation, paraphrased:

A: I’m worried about my posture. People will think I’m not attractive because I slouch.

B: Don’t worry, you can improve your posture because you’re intelligent.

A: What? How does that follow?

B: I notice that rich people tend to be able to improve their posture. Meanwhile, it is usually easy for intelligent people to become rich. Therefore, intelligent people can usually improve their posture.

Regardless of the somewhat-questionable factualness of these assertions, is the statement logically sound? If A is evidence for B and B is evidence for C, is A evidence for C? Mathematically, it is quite easy to see this is not the case. Check out this probability distribution, for example:   A few moments of staring will show you that it’s a counterexample (A is evidence for B, B is evidence for C, but A is not evidence for C). Good thing, too! Imagine if it were true:

  • Being a Native Hawaiian is evidence for being in Hawaii. Being in Hawaii is evidence for being a tourist. Therefore, being a Native Hawaiian is evidence for being a tourist.
  • If an object is an insect, that’s evidence that it can fly. If an object can fly, that’s evidence that it’s an airplane. Therefore, being an insect is evidence that an object is an airplane.
  • Having sex is evidence that you are breathing hard. Breathing hard is evidence that you’re jogging. Therefore, having sex is evidence that you’re jogging.
  • If it’s raining, that’s evidence that there are umbrellas around. If there are umbrellas around, that’s evidence that you’re in an umbrella factory. Therefore, rain is evidence that you’re in an umbrella factory.

2 Responses to “Transitive Evidence”

  1. MG Says:

    What is the definition of evidence? In your example Venn diagram, how is A an evidence for B if there are elements of A which do not imply existence of B?

  2. Mark Eichenlaub Says:

    The statement “A is evidence for B” means P(B|A) > P(B)

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