Tutorial Nine
Probability Introduction: Axioms and Bayes' Theorem

Probability plays a role in inductive logic that is analogous to the role played by possibility in deductive logic.

For example, a valid deductive argument has premises which, granted as true, make it impossible for the conclusion to be false. Similarly, a strong inductive argument has premises which, granted as true, make it improbable that the conclusion is false.

We won't spell out exactly what probability is however. Probabilities remain a bit mysterious even in the context of a sentence logic. (The notion of possibility is still a philosophical mystery as well. But at least one can spell out the notion of logical possibility for particular formal languages like the ones we develop.)

For present purposes, it's sufficient to know that there are a number of ways that philosophers, statisticians, and mathematicians have interpreted "probability". And this may seem natural to you. After all, when you say that "There's a 50% probability that you'll get an 'A' in logic" this might mean:

  1. The class you're in gives or would give 'A's to about half the people in your position.
  2. You're uncertain, but feel as confident in getting an 'A' as not.
  3. The evidence supports getting an 'A' to degree 1/2.

Here's a very incomplete listing to give the idea of interpreting probability just a little substance.

The Empirical/Objective Interpretation of Probability: Probability is the measure of a physical property but one that admits of degrees of potentiality. These might be approximated from statistical testing.

The Epistemic/Subjective Interpretation of Probability: Probability is a measure of degree of (reasonable) belief.

The Logical Interpretation of Probability: Probability is a measure of the degree to which premises support a conclusion.

Probability doesn't have to be just one thing. Arguably the word is vague and/or ambiguous and might well be interpreted in quite different ways for different contexts.

So, which of the following claims is most plausibly interpreted in the Epistemic/Subjective way:

  1. I'm unsure about the getting an 'A', but my belief is a strong one: I'd say there's a 95% probability that I'll get an 'A'.
  2. I get 'A's about 1 class in 3, so my probability of getting an 'A' here is less than one half.