Basic rule 2 states that probability 1 is assigned to a logical truth, because either is true or its opposite (``tertium non datur''). Indeed represent a logical, tautological certainty (a tautology, usually indicated with ), while is a contradiction, that is something impossible, indicated by .
The first three basic rules are also known the `axioms' of probability,40while the inverses of the fourth one, e.g. , are called in most literature ``definition of conditional probability''. In the approach followed here such a statement has no sense, because probability is always conditional probability (note the ubiquitous `' in all our formulae - for further comments see section 10.3 of Ref. ). Note that when the condition does not change the probability of , i.e. , then and are said to be independent in probability. In this case the joint probability is given by the so-called product rule, i.e. .
These rules are automatically satisfied if probabilities
are evaluated from favorable over possible, equally probably cases.
Also relative frequencies of occurrences in the past respect these
rules, with the little difference that the probabilistic
interpretation of past relative frequencies is not really
straightforward, as briefly discussed in the following appendix.
That beliefs satisfy, in general, the same basic rules can be
proved in several ways. If we calibrate our degrees of beliefs
against `standards', as illustrated in section
3, this is quite easy to understand.
Otherwise it can be proved by the normative principle of the
coherent bet .