Saturday, July 13, 2002

When I was talking below about T and P values, I said that every statement has a P and a T associated with it . This is not strictly true. First, some definitions. T is the truth-value of a statement. Usually it is binary (i.e. true or false). 'All cats are mammals' has a binary truth-value (it is true). 'Some cats living at 123 New Street are more than five years old' also has a binary truth-value, even if we do not know which one it is (yet). Sometimes a truth-value is continuous. This is really what fuzzy logic is all about.

The P of a statement is its probability-value i.e. its likelihood of being true. Simple statements of fact (or their negations) do not have a P in any interesting sense. P is either 0 or 1 depending on whether the statement is true or false. Statements like 'some cats living at 123 New Street are more than five years old' do have a P, but only until the statement is checked. Then the hitherto-unknown truth-value collapses from its a priori, unknown value to its a posteriori, known value, and the probability-value ceases to be interesting.

Where this all gets complicated is counterfactuals. Counterfactuals are statements about hypothetical situations. There are two types of counterfactual. One, which I will call type A, are statements of the form 'if X occurs, then Y will occur'. For example, 'if you keep driving like that, we are going to crash'. The other type, type B, is a statement like 'if X had happened, then Y would have happened'. For example, 'if you had kept on driving like that, we would have crashed.'*

Type A counterfactuals have both a T and a P. If we carry on doing X, and Y does indeed happen, then T is 1. Before Y happens, we can assign some probability to the likelihood of its occurence. This is P. Type B counterfactuals, on the other hand, do not have a T. It is impossible to assign a truth-value to the statement, 'if Britain had lost the Battle of Britain, Hitler would have invaded England.' We can still assign a P to this statement, but there is no way of actually running the experiment and collapsing it down to give a definite T.

Does this mean that Type B events are uninteresting? On the contrary; there is a large corpus of literature and even academic history devoted to Type B counterfactuals (what if the US had lost the War of Independence, what if aliens invaded during World War II, what if Henry VIII had not disestablished the monasteries, etc.)

We now return you to your normal service. Thank you for your consideration.

* In Spanish this phrase uses a combination of the pluperfect subjunctive and conditional perfect (si hubiera seguido manejando así, habríamos estrellado). English does not draw this distinction.


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