A natural prior probability distribution derived from the propositional calculus
Suppose that you regard propositional variables as random quantities with equal chances of being true or false -- i.e., of having the value 0 or 1. A number of approaches to uncertainty begin from just such assumptions. Now consider, not small sentences, but large classes of sentences of the form \(A^ m_ n\), where this denotes the set of formulae using only the propositional variables \(p_ 1 \dots p_ m\) and using at most \(n\) occurrences of \(\vee\) or \(\wedge\). The authors explore the question of whether \[ S(I)= \lim _{\substack{ m\to\infty\\ n\to \infty}} {{|\{ \alpha\in A_ n^ m\mid E(\alpha)\in I\}|} \over {| A_ m^ n|}} \] exists for \(I\) a subinterval of \([0,1]\). They are able to show that this limit always exists, and define a countably additive measure on subintervals of \([0, 1]\). This measure is symmetric about \(1/2\). It also has some peculiarities, as the authors show, including that of having infinite density at each rational point. The proof depends on the use of Catalan numbers \[ b_ n= {1\over {n+1}} {{2n} \choose n} \] and makes use of some rather complicated combinatorial calculations. It is carried out first for a measure \(D\) on sentences in which no propositional variable occurs more than once. The generalization from \(D\) to \(S\) is quite easy. The main theorem is 3.11: For any interval \(I\subset [0,1]\) the measure \[ S(I)= \lim _{\substack{ m\to \infty\\ n\to\infty }} {{|\{ \alpha\in A_ m^ n\mid h(\alpha) \in I\}|} \over m} \] exists and, furthermore, equals the corresponding measure with the restriction of formulas to those in which no sentential variable has more than one occurrence. The last two sections of the paper are concerned with the character of the distribution determined by \(S\). The long first section of the paper is devoted to philosophical motivation, and is not as clear as the rest of the paper. The mathematical results are interesting in their own right; it remains unclear how they bear on uncertain inference or on the question of prior logical probabilities.
countably additive measure on subintervals of \([0, 1]\)
scientific article; zbMATH DE number 722501