How to argue statistics using logic?

Question

Typically in logic, we have the axioms as facts which are 100% true but in statistics we have things which are true sometimes. For example, "the coin is heads" is true 50% of the time when we flip a coin. So, how can one implement such statistical idea in a proof system?

Answer 1

If you have a full specification of a situation, and there are probabilities with known values, then you can use the probability calculus to compute the probabilities you want from those that are given.

For example, if you have a fair coin and an unbiased way of tossing it, then you can calculate that the probability of getting three heads from three tosses is 1/8. What you are doing in such cases is treating the probability calculus as part of a deductive apparatus. Statements about the coin and the experiment of tossing it are like premises in an argument, and the conclusion follows with certainty from the premises. We might call such reasoning probabilistic-deductive. We are deducing one probability from others.

But in the real world, we typically don't have a full specification or known values for the probabilities. We don't know the coin is fair. We don't know how likely it is to rain tomorrow. We don't know how likely it is that such-and-such party will win the election next year.

We would like to be able to calculate such things from known data, but this an intractable problem in general. Reasoning of this kind is often called inductive, though there are other ways of characterising it.

In the simple case of tossing a coin, we can approach the probability of it being fair by tossing it a lot of times and using the observed frequency.

Even then, we are making several assumptions. We are assuming that the tosses are entirely independent, that there is no bias in the tossing mechanism, and that the coin is unchanged each time, i.e. it is not becoming worn. These are reasonable assumptions, though not guaranteed. Dice wear out over time, which is why casinos regularly destroy them and replace them with new ones.

When it comes to predicting the weather, the best we can do is create some computational model based on a rather inexact understanding of atmospheric physics. It works reasonably well much of the time, but when the forecast tells us to expect a 60% chance of rain tomorrow we know it is only approximate.

When it comes to harder things like predicting the outcome of elections, the problem is so difficult, we don't even know what variables are important, what factors to treat as constant, how accurate our data is, or even to what extent past experience is a reliable guide to future events.

This does not prevent us from attaching probabilities to such events. You can always go to a betting shop and ask for the odds.

There is ongoing research to combine logic and probability, called 'progic'. There is a regular conference on this topic, with published papers. There are also probability logics in which validity is probability-preserving rather than truth-preserving, for example Ernest Adams' A Primer of Probability Logic.

Answer 2

First, in formal logic, axioms need not be true. They are merely taken as ground rules which for the purposes of an argument developed in this formal logic are not to be questioned, and so, internally are true. But this does not mean that they are neccessarily true externally, that is in the broader world - that is the proper philosophical sense.

Now the logic in a formal system is usually ordinary classical logic but need not be and you can choose logics that reflect the subject domain under consideration. For example, in statistics you could choose probabilistic logic. Generally this is not done as the logic of probability is built over a classical logic foundation.